path: root/ovr_sdk_win_23.0.0/LibOVR/Include/Extras/OVR_Math.h

/********************************************************************************/ /**
 \file      OVR_Math.h
 \brief     Implementation of 3D primitives such as vectors, matrices.
 \copyright Copyright (c) Facebook Technologies, LLC and its affiliates. All rights reserved.
 *************************************************************************************/

#ifndef OVR_Math_h
#define OVR_Math_h

// This file is intended to be independent of the rest of LibOVR and LibOVRKernel and thus
// has no #include dependencies on either.

#include <float.h>
#include <math.h>
#include <stdint.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>

#ifndef OVR_EXCLUDE_CAPI_FROM_MATH
#include "../OVR_CAPI.h" // Required due to a dependence on the ovrFovPort_ declaration.
#endif

#if defined(_MSC_VER)
#pragma warning(push)
#pragma warning(disable : 4127) // conditional expression is constant

#if _MSC_VER < 1800 // isfinite was introduced in VS2013
#define isfinite(x) _finite((x))
#endif
#endif

#if defined(_MSC_VER)
#define OVRMath_sprintf sprintf_s
#else
#define OVRMath_sprintf snprintf
#endif

//-------------------------------------------------------------------------------------
// ***** OVR_MATH_ASSERT
//
// Independent debug break implementation for OVR_Math.h.

#if !defined(OVR_MATH_DEBUG_BREAK)
#if defined(_DEBUG)
#if defined(_MSC_VER)
#define OVR_MATH_DEBUG_BREAK __debugbreak()
#else
#define OVR_MATH_DEBUG_BREAK __builtin_trap()
#endif
#else
#define OVR_MATH_DEBUG_BREAK ((void)0)
#endif
#endif

//-------------------------------------------------------------------------------------
// ***** OVR_MATH_ASSERT
//
// Independent OVR_MATH_ASSERT implementation for OVR_Math.h.

#if !defined(OVR_MATH_ASSERT)
#if defined(_DEBUG)
#define OVR_MATH_ASSERT(p) \
  if (!(p)) {              \
    OVR_MATH_DEBUG_BREAK;  \
  }
#else
#define OVR_MATH_ASSERT(p) ((void)0)
#endif
#endif

//-------------------------------------------------------------------------------------
// ***** OVR_MATH_STATIC_ASSERT
//
// Independent OVR_MATH_ASSERT implementation for OVR_Math.h.

#if !defined(OVR_MATH_STATIC_ASSERT)
#if defined(__cplusplus) &&                                                                       \
    ((defined(_MSC_VER) && (defined(_MSC_VER) >= 1600)) || defined(__GXX_EXPERIMENTAL_CXX0X__) || \
     (__cplusplus >= 201103L))
#define OVR_MATH_STATIC_ASSERT static_assert
#else
#if !defined(OVR_SA_UNUSED)
#if defined(__GNUC__) || defined(__clang__)
#define OVR_SA_UNUSED __attribute__((unused))
#else
#define OVR_SA_UNUSED
#endif
#define OVR_SA_PASTE(a, b) a##b
#define OVR_SA_HELP(a, b) OVR_SA_PASTE(a, b)
#endif

#define OVR_MATH_STATIC_ASSERT(expression, msg) \
  typedef char OVR_SA_HELP(compileTimeAssert, __LINE__)[((expression) != 0) ? 1 : -1] OVR_SA_UNUSED
#endif
#endif

namespace OVR {

template <class T>
const T OVRMath_Min(const T a, const T b) {
  return (a < b) ? a : b;
}

template <class T>
const T OVRMath_Max(const T a, const T b) {
  return (b < a) ? a : b;
}

template <class T>
void OVRMath_Swap(T& a, T& b) {
  T temp(a);
  a = b;
  b = temp;
}

//-------------------------------------------------------------------------------------
// ***** Constants for 3D world/axis definitions.

// Definitions of axes for coordinate and rotation conversions.
enum Axis { Axis_X = 0, Axis_Y = 1, Axis_Z = 2 };

// RotateDirection describes the rotation direction around an axis, interpreted as follows:
//  CW  - Clockwise while looking "down" from positive axis towards the origin.
//  CCW - Counter-clockwise while looking from the positive axis towards the origin,
//        which is in the negative axis direction.
//  CCW is the default for the RHS coordinate system. Oculus standard RHS coordinate
//  system defines Y up, X right, and Z back (pointing out from the screen). In this
//  system Rotate_CCW around Z will specifies counter-clockwise rotation in XY plane.
enum RotateDirection { Rotate_CCW = 1, Rotate_CW = -1 };

// Constants for right handed and left handed coordinate systems
enum HandedSystem { Handed_R = 1, Handed_L = -1 };

// AxisDirection describes which way the coordinate axis points. Used by WorldAxes.
enum AxisDirection {
  Axis_Up = 2,
  Axis_Down = -2,
  Axis_Right = 1,
  Axis_Left = -1,
  Axis_In = 3,
  Axis_Out = -3
};

struct WorldAxes {
  AxisDirection XAxis, YAxis, ZAxis;

  WorldAxes(AxisDirection x, AxisDirection y, AxisDirection z) : XAxis(x), YAxis(y), ZAxis(z) {
    OVR_MATH_ASSERT(abs(x) != abs(y) && abs(y) != abs(z) && abs(z) != abs(x));
  }
};

} // namespace OVR

//------------------------------------------------------------------------------------//
// ***** C Compatibility Types

// These declarations are used to support conversion between C types used in
// LibOVR C interfaces and their C++ versions. As an example, they allow passing
// Vector3f into a function that expects ovrVector3f.

typedef struct ovrQuatf_ ovrQuatf;
typedef struct ovrQuatd_ ovrQuatd;
typedef struct ovrSizei_ ovrSizei;
typedef struct ovrSizef_ ovrSizef;
typedef struct ovrSized_ ovrSized;
typedef struct ovrRecti_ ovrRecti;
typedef struct ovrVector2i_ ovrVector2i;
typedef struct ovrVector2f_ ovrVector2f;
typedef struct ovrVector2d_ ovrVector2d;
typedef struct ovrVector3f_ ovrVector3f;
typedef struct ovrVector3d_ ovrVector3d;
typedef struct ovrVector4f_ ovrVector4f;
typedef struct ovrVector4d_ ovrVector4d;
typedef struct ovrMatrix2f_ ovrMatrix2f;
typedef struct ovrMatrix2d_ ovrMatrix2d;
typedef struct ovrMatrix3f_ ovrMatrix3f;
typedef struct ovrMatrix3d_ ovrMatrix3d;
typedef struct ovrMatrix4f_ ovrMatrix4f;
typedef struct ovrMatrix4d_ ovrMatrix4d;
typedef struct ovrPosef_ ovrPosef;
typedef struct ovrPosed_ ovrPosed;
typedef struct ovrPoseStatef_ ovrPoseStatef;
typedef struct ovrPoseStated_ ovrPoseStated;
typedef struct ovrFovPort_ ovrFovPort;

namespace OVR {

// Forward-declare our templates.
template <class T>
class Quat;
template <class T>
class Size;
template <class T>
class Rect;
template <class T>
class Vector2;
template <class T>
class Vector3;
template <class T>
class Vector4;
template <class T>
class Matrix2;
template <class T>
class Matrix3;
template <class T>
class Matrix4;
template <class T>
class Pose;
template <class T>
class PoseState;
struct FovPort;

// CompatibleTypes::Type is used to lookup a compatible C-version of a C++ class.
template <class C>
struct CompatibleTypes {
  // Declaration here seems necessary for MSVC; specializations are
  // used instead.
  typedef struct {
  } Type;
};

// Specializations providing CompatibleTypes::Type value.
template <>
struct CompatibleTypes<Quat<float>> {
  typedef ovrQuatf Type;
};
template <>
struct CompatibleTypes<Quat<double>> {
  typedef ovrQuatd Type;
};
template <>
struct CompatibleTypes<Matrix2<float>> {
  typedef ovrMatrix2f Type;
};
template <>
struct CompatibleTypes<Matrix2<double>> {
  typedef ovrMatrix2d Type;
};
template <>
struct CompatibleTypes<Matrix3<float>> {
  typedef ovrMatrix3f Type;
};
template <>
struct CompatibleTypes<Matrix3<double>> {
  typedef ovrMatrix3d Type;
};
template <>
struct CompatibleTypes<Matrix4<float>> {
  typedef ovrMatrix4f Type;
};
template <>
struct CompatibleTypes<Matrix4<double>> {
  typedef ovrMatrix4d Type;
};
template <>
struct CompatibleTypes<Size<int>> {
  typedef ovrSizei Type;
};
template <>
struct CompatibleTypes<Size<float>> {
  typedef ovrSizef Type;
};
template <>
struct CompatibleTypes<Size<double>> {
  typedef ovrSized Type;
};
template <>
struct CompatibleTypes<Rect<int>> {
  typedef ovrRecti Type;
};
template <>
struct CompatibleTypes<Vector2<int>> {
  typedef ovrVector2i Type;
};
template <>
struct CompatibleTypes<Vector2<float>> {
  typedef ovrVector2f Type;
};
template <>
struct CompatibleTypes<Vector2<double>> {
  typedef ovrVector2d Type;
};
template <>
struct CompatibleTypes<Vector3<float>> {
  typedef ovrVector3f Type;
};
template <>
struct CompatibleTypes<Vector3<double>> {
  typedef ovrVector3d Type;
};
template <>
struct CompatibleTypes<Vector4<float>> {
  typedef ovrVector4f Type;
};
template <>
struct CompatibleTypes<Vector4<double>> {
  typedef ovrVector4d Type;
};
template <>
struct CompatibleTypes<Pose<float>> {
  typedef ovrPosef Type;
};
template <>
struct CompatibleTypes<Pose<double>> {
  typedef ovrPosed Type;
};
template <>
struct CompatibleTypes<FovPort> {
  typedef ovrFovPort Type;
};

//------------------------------------------------------------------------------------//
// ***** Math
//
// Math class contains constants and functions. This class is a template specialized
// per type, with Math<float> and Math<double> being distinct.
template <class T>
class Math {
 public:
  // By default, support explicit conversion to float. This allows Vector2<int> to
  // compile, for example.
  typedef float OtherFloatType;

  static int Tolerance() {
    return 0;
  } // Default value so integer types compile
};

//------------------------------------------------------------------------------------//
// ***** double constants
#define MATH_DOUBLE_PI 3.14159265358979323846
#define MATH_DOUBLE_TWOPI (2 * MATH_DOUBLE_PI)
#define MATH_DOUBLE_PIOVER2 (0.5 * MATH_DOUBLE_PI)
#define MATH_DOUBLE_PIOVER4 (0.25 * MATH_DOUBLE_PI)
#define MATH_FLOAT_MAXVALUE (FLT_MAX)

#define MATH_DOUBLE_RADTODEGREEFACTOR (360.0 / MATH_DOUBLE_TWOPI)
#define MATH_DOUBLE_DEGREETORADFACTOR (MATH_DOUBLE_TWOPI / 360.0)

#define MATH_DOUBLE_E 2.71828182845904523536
#define MATH_DOUBLE_LOG2E 1.44269504088896340736
#define MATH_DOUBLE_LOG10E 0.434294481903251827651
#define MATH_DOUBLE_LN2 0.693147180559945309417
#define MATH_DOUBLE_LN10 2.30258509299404568402

#define MATH_DOUBLE_SQRT2 1.41421356237309504880
#define MATH_DOUBLE_SQRT1_2 0.707106781186547524401

#define MATH_DOUBLE_TOLERANCE \
  1e-12 // a default number for value equality tolerance: about 4500*Epsilon;
#define MATH_DOUBLE_SINGULARITYRADIUS \
  1e-12 // about 1-cos(.0001 degree), for gimbal lock numerical problems

#define MATH_DOUBLE_HUGENUMBER 1.3407807929942596e+154
#define MATH_DOUBLE_SMALLESTNONDENORMAL 2.2250738585072014e-308

//------------------------------------------------------------------------------------//
// ***** float constants
#define MATH_FLOAT_PI float(MATH_DOUBLE_PI)
#define MATH_FLOAT_TWOPI float(MATH_DOUBLE_TWOPI)
#define MATH_FLOAT_PIOVER2 float(MATH_DOUBLE_PIOVER2)
#define MATH_FLOAT_PIOVER4 float(MATH_DOUBLE_PIOVER4)

#define MATH_FLOAT_RADTODEGREEFACTOR float(MATH_DOUBLE_RADTODEGREEFACTOR)
#define MATH_FLOAT_DEGREETORADFACTOR float(MATH_DOUBLE_DEGREETORADFACTOR)

#define MATH_FLOAT_E float(MATH_DOUBLE_E)
#define MATH_FLOAT_LOG2E float(MATH_DOUBLE_LOG2E)
#define MATH_FLOAT_LOG10E float(MATH_DOUBLE_LOG10E)
#define MATH_FLOAT_LN2 float(MATH_DOUBLE_LN2)
#define MATH_FLOAT_LN10 float(MATH_DOUBLE_LN10)

#define MATH_FLOAT_SQRT2 float(MATH_DOUBLE_SQRT2)
#define MATH_FLOAT_SQRT1_2 float(MATH_DOUBLE_SQRT1_2)

#define MATH_FLOAT_TOLERANCE \
  1e-5f // a default number for value equality tolerance: 1e-5, about 84*EPSILON;
#define MATH_FLOAT_SINGULARITYRADIUS \
  1e-7f // about 1-cos(.025 degree), for gimbal lock numerical problems

#define MATH_FLOAT_HUGENUMBER 1.8446742974197924e+019f
#define MATH_FLOAT_SMALLESTNONDENORMAL 1.1754943508222875e-038f

// Single-precision Math constants class.
template <>
class Math<float> {
 public:
  typedef double OtherFloatType;

  static inline float MaxValue() {
    return FLT_MAX;
  };
  static inline float Tolerance() {
    return MATH_FLOAT_TOLERANCE;
  }; // a default number for value equality tolerance
  static inline float SingularityRadius() {
    return MATH_FLOAT_SINGULARITYRADIUS;
  }; // for gimbal lock numerical problems
  static inline float HugeNumber() {
    return MATH_FLOAT_HUGENUMBER;
  }
  static inline float SmallestNonDenormal() {
    return MATH_FLOAT_SMALLESTNONDENORMAL;
  }
};

// Double-precision Math constants class
template <>
class Math<double> {
 public:
  typedef float OtherFloatType;

  static inline double Tolerance() {
    return MATH_DOUBLE_TOLERANCE;
  }; // a default number for value equality tolerance
  static inline double SingularityRadius() {
    return MATH_DOUBLE_SINGULARITYRADIUS;
  }; // for gimbal lock numerical problems
  static inline double HugeNumber() {
    return MATH_DOUBLE_HUGENUMBER;
  }
  static inline double SmallestNonDenormal() {
    return MATH_DOUBLE_SMALLESTNONDENORMAL;
  }
};

typedef Math<float> Mathf;
typedef Math<double> Mathd;

// Conversion functions between degrees and radians
// (non-templated to ensure passing int arguments causes warning)
inline float RadToDegree(float rad) {
  return rad * MATH_FLOAT_RADTODEGREEFACTOR;
}
inline double RadToDegree(double rad) {
  return rad * MATH_DOUBLE_RADTODEGREEFACTOR;
}

inline float DegreeToRad(float deg) {
  return deg * MATH_FLOAT_DEGREETORADFACTOR;
}
inline double DegreeToRad(double deg) {
  return deg * MATH_DOUBLE_DEGREETORADFACTOR;
}

// Square function
template <class T>
inline T Sqr(T x) {
  return x * x;
}

// MERGE_MOBILE_SDK
// Safe reciprocal square root.
template <class T>
T RcpSqrt(const T f) {
  return (f >= Math<T>::SmallestNonDenormal()) ? static_cast<T>(1.0 / sqrt(f))
                                               : Math<T>::HugeNumber();
}
// MERGE_MOBILE_SDK

// Sign: returns 0 if x == 0, -1 if x < 0, and 1 if x > 0
template <class T>
inline T Sign(T x) {
  return (x != T(0)) ? (x < T(0) ? T(-1) : T(1)) : T(0);
}

// Numerically stable acos function
inline float Acos(float x) {
  return (x > 1.0f) ? 0.0f : (x < -1.0f) ? MATH_FLOAT_PI : acosf(x);
}
inline double Acos(double x) {
  return (x > 1.0) ? 0.0 : (x < -1.0) ? MATH_DOUBLE_PI : acos(x);
}

// Numerically stable asin function
inline float Asin(float x) {
  return (x > 1.0f) ? MATH_FLOAT_PIOVER2 : (x < -1.0f) ? -MATH_FLOAT_PIOVER2 : asinf(x);
}
inline double Asin(double x) {
  return (x > 1.0) ? MATH_DOUBLE_PIOVER2 : (x < -1.0) ? -MATH_DOUBLE_PIOVER2 : asin(x);
}

template <class T>
class Quat;

//-------------------------------------------------------------------------------------
// ***** Vector2<>

// Vector2f (Vector2d) represents a 2-dimensional vector or point in space,
// consisting of coordinates x and y

template <class T>
class Vector2 {
 public:
  typedef T ElementType;
  static const size_t ElementCount = 2;

  T x, y;

  Vector2() : x(0), y(0) {}
  Vector2(T x_, T y_) : x(x_), y(y_) {}
  explicit Vector2(T s) : x(s), y(s) {}
  explicit Vector2(const Vector2<typename Math<T>::OtherFloatType>& src)
      : x((T)src.x), y((T)src.y) {}

  static Vector2 Zero() {
    return Vector2(0, 0);
  }

  // C-interop support.
  typedef typename CompatibleTypes<Vector2<T>>::Type CompatibleType;

  Vector2(const CompatibleType& s) : x(s.x), y(s.y) {}

  operator const CompatibleType&() const {
    OVR_MATH_STATIC_ASSERT(
        sizeof(Vector2<T>) == sizeof(CompatibleType), "sizeof(Vector2<T>) failure");
    return reinterpret_cast<const CompatibleType&>(*this);
  }

  bool operator==(const Vector2& b) const {
    return x == b.x && y == b.y;
  }
  bool operator!=(const Vector2& b) const {
    return x != b.x || y != b.y;
  }

  Vector2 operator+(const Vector2& b) const {
    return Vector2(x + b.x, y + b.y);
  }
  Vector2& operator+=(const Vector2& b) {
    x += b.x;
    y += b.y;
    return *this;
  }
  Vector2 operator-(const Vector2& b) const {
    return Vector2(x - b.x, y - b.y);
  }
  Vector2& operator-=(const Vector2& b) {
    x -= b.x;
    y -= b.y;
    return *this;
  }
  Vector2 operator-() const {
    return Vector2(-x, -y);
  }

  // Scalar multiplication/division scales vector.
  Vector2 operator*(T s) const {
    return Vector2(x * s, y * s);
  }
  Vector2& operator*=(T s) {
    x *= s;
    y *= s;
    return *this;
  }

  Vector2 operator/(T s) const {
    T rcp = T(1) / s;
    return Vector2(x * rcp, y * rcp);
  }
  Vector2& operator/=(T s) {
    T rcp = T(1) / s;
    x *= rcp;
    y *= rcp;
    return *this;
  }

  static Vector2 Min(const Vector2& a, const Vector2& b) {
    return Vector2((a.x < b.x) ? a.x : b.x, (a.y < b.y) ? a.y : b.y);
  }
  static Vector2 Max(const Vector2& a, const Vector2& b) {
    return Vector2((a.x > b.x) ? a.x : b.x, (a.y > b.y) ? a.y : b.y);
  }

  Vector2 Clamped(T maxMag) const {
    T magSquared = LengthSq();
    if (magSquared <= Sqr(maxMag))
      return *this;
    else
      return *this * (maxMag / sqrt(magSquared));
  }

  // Compare two vectors for equality with tolerance. Returns true if vectors match within
  // tolerance.
  bool IsEqual(const Vector2& b, T tolerance = Math<T>::Tolerance()) const {
    return (fabs(b.x - x) <= tolerance) && (fabs(b.y - y) <= tolerance);
  }
  bool Compare(const Vector2& b, T tolerance = Math<T>::Tolerance()) const {
    return IsEqual(b, tolerance);
  }

  // Access element by index
  T& operator[](int idx) {
    OVR_MATH_ASSERT(0 <= idx && idx < 2);
    return *(&x + idx);
  }
  const T& operator[](int idx) const {
    OVR_MATH_ASSERT(0 <= idx && idx < 2);
    return *(&x + idx);
  }

  // Entry-wise product of two vectors
  Vector2 EntrywiseMultiply(const Vector2& b) const {
    return Vector2(x * b.x, y * b.y);
  }

  // Multiply and divide operators do entry-wise math. Used Dot() for dot product.
  Vector2 operator*(const Vector2& b) const {
    return Vector2(x * b.x, y * b.y);
  }
  Vector2 operator/(const Vector2& b) const {
    return Vector2(x / b.x, y / b.y);
  }

  // Dot product
  // Used to calculate angle q between two vectors among other things,
  // as (A dot B) = |a||b|cos(q).
  T Dot(const Vector2& b) const {
    return x * b.x + y * b.y;
  }

  // Returns the angle from this vector to b, in radians.
  T Angle(const Vector2& b) const {
    T div = LengthSq() * b.LengthSq();
    OVR_MATH_ASSERT(div != T(0));
    T result = Acos((this->Dot(b)) / sqrt(div));
    return result;
  }

  // Return Length of the vector squared.
  T LengthSq() const {
    return (x * x + y * y);
  }

  // Return vector length.
  T Length() const {
    return sqrt(LengthSq());
  }

  // Returns squared distance between two points represented by vectors.
  T DistanceSq(const Vector2& b) const {
    return (*this - b).LengthSq();
  }

  // Returns distance between two points represented by vectors.
  T Distance(const Vector2& b) const {
    return (*this - b).Length();
  }

  // Determine if this a unit vector.
  bool IsNormalized() const {
    return fabs(LengthSq() - T(1)) < Math<T>::Tolerance();
  }

  // Normalize, convention vector length to 1.
  void Normalize() {
    T s = Length();
    if (s != T(0))
      s = T(1) / s;
    *this *= s;
  }

  // Returns normalized (unit) version of the vector without modifying itself.
  Vector2 Normalized() const {
    T s = Length();
    if (s != T(0))
      s = T(1) / s;
    return *this * s;
  }

  // Linearly interpolates from this vector to another.
  // Factor should be between 0.0 and 1.0, with 0 giving full value to this.
  Vector2 Lerp(const Vector2& b, T f) const {
    return *this * (T(1) - f) + b * f;
  }

  // Projects this vector onto the argument; in other words,
  // A.Project(B) returns projection of vector A onto B.
  Vector2 ProjectTo(const Vector2& b) const {
    T l2 = b.LengthSq();
    OVR_MATH_ASSERT(l2 != T(0));
    return b * (Dot(b) / l2);
  }

  // returns true if vector b is clockwise from this vector
  bool IsClockwise(const Vector2& b) const {
    return (x * b.y - y * b.x) < 0;
  }
};

typedef Vector2<float> Vector2f;
typedef Vector2<double> Vector2d;
typedef Vector2<int> Vector2i;

typedef Vector2<float> Point2f;
typedef Vector2<double> Point2d;
typedef Vector2<int> Point2i;

//-------------------------------------------------------------------------------------
// ***** Vector3<> - 3D vector of {x, y, z}

//
// Vector3f (Vector3d) represents a 3-dimensional vector or point in space,
// consisting of coordinates x, y and z.

template <class T>
class Vector3 {
 public:
  typedef T ElementType;
  static const size_t ElementCount = 3;

  T x, y, z;

  // FIXME: default initialization of a vector class can be very expensive in a full-blown
  // application.  A few hundred thousand vector constructions is not unlikely and can add
  // up to milliseconds of time on processors like the PS3 PPU.
  Vector3() : x(0), y(0), z(0) {}
  Vector3(T x_, T y_, T z_ = 0) : x(x_), y(y_), z(z_) {}
  explicit Vector3(T s) : x(s), y(s), z(s) {}
  explicit Vector3(const Vector3<typename Math<T>::OtherFloatType>& src)
      : x((T)src.x), y((T)src.y), z((T)src.z) {}

  static Vector3 Zero() {
    return Vector3(0, 0, 0);
  }

  // C-interop support.
  typedef typename CompatibleTypes<Vector3<T>>::Type CompatibleType;

  Vector3(const CompatibleType& s) : x(s.x), y(s.y), z(s.z) {}

  operator const CompatibleType&() const {
    OVR_MATH_STATIC_ASSERT(
        sizeof(Vector3<T>) == sizeof(CompatibleType), "sizeof(Vector3<T>) failure");
    return reinterpret_cast<const CompatibleType&>(*this);
  }

  bool operator==(const Vector3& b) const {
    return x == b.x && y == b.y && z == b.z;
  }
  bool operator!=(const Vector3& b) const {
    return x != b.x || y != b.y || z != b.z;
  }

  Vector3 operator+(const Vector3& b) const {
    return Vector3(x + b.x, y + b.y, z + b.z);
  }
  Vector3& operator+=(const Vector3& b) {
    x += b.x;
    y += b.y;
    z += b.z;
    return *this;
  }
  Vector3 operator-(const Vector3& b) const {
    return Vector3(x - b.x, y - b.y, z - b.z);
  }
  Vector3& operator-=(const Vector3& b) {
    x -= b.x;
    y -= b.y;
    z -= b.z;
    return *this;
  }
  Vector3 operator-() const {
    return Vector3(-x, -y, -z);
  }

  // Scalar multiplication/division scales vector.
  Vector3 operator*(T s) const {
    return Vector3(x * s, y * s, z * s);
  }
  Vector3& operator*=(T s) {
    x *= s;
    y *= s;
    z *= s;
    return *this;
  }

  Vector3 operator/(T s) const {
    T rcp = T(1) / s;
    return Vector3(x * rcp, y * rcp, z * rcp);
  }
  Vector3& operator/=(T s) {
    T rcp = T(1) / s;
    x *= rcp;
    y *= rcp;
    z *= rcp;
    return *this;
  }

  static Vector3 Min(const Vector3& a, const Vector3& b) {
    return Vector3((a.x < b.x) ? a.x : b.x, (a.y < b.y) ? a.y : b.y, (a.z < b.z) ? a.z : b.z);
  }
  static Vector3 Max(const Vector3& a, const Vector3& b) {
    return Vector3((a.x > b.x) ? a.x : b.x, (a.y > b.y) ? a.y : b.y, (a.z > b.z) ? a.z : b.z);
  }

  Vector3 Clamped(T maxMag) const {
    T magSquared = LengthSq();
    if (magSquared <= Sqr(maxMag))
      return *this;
    else
      return *this * (maxMag / sqrt(magSquared));
  }

  // Compare two vectors for equality with tolerance. Returns true if vectors match within
  // tolerance.
  bool IsEqual(const Vector3& b, T tolerance = Math<T>::Tolerance()) const {
    return (fabs(b.x - x) <= tolerance) && (fabs(b.y - y) <= tolerance) &&
        (fabs(b.z - z) <= tolerance);
  }
  bool Compare(const Vector3& b, T tolerance = Math<T>::Tolerance()) const {
    return IsEqual(b, tolerance);
  }

  T& operator[](int idx) {
    OVR_MATH_ASSERT(0 <= idx && idx < 3);
    return *(&x + idx);
  }

  const T& operator[](int idx) const {
    OVR_MATH_ASSERT(0 <= idx && idx < 3);
    return *(&x + idx);
  }

  // Entrywise product of two vectors
  Vector3 EntrywiseMultiply(const Vector3& b) const {
    return Vector3(x * b.x, y * b.y, z * b.z);
  }

  // Multiply and divide operators do entry-wise math
  Vector3 operator*(const Vector3& b) const {
    return Vector3(x * b.x, y * b.y, z * b.z);
  }

  Vector3 operator/(const Vector3& b) const {
    return Vector3(x / b.x, y / b.y, z / b.z);
  }

  // Dot product
  // Used to calculate angle q between two vectors among other things,
  // as (A dot B) = |a||b|cos(q).
  T Dot(const Vector3& b) const {
    return x * b.x + y * b.y + z * b.z;
  }

  // Compute cross product, which generates a normal vector.
  // Direction vector can be determined by right-hand rule: Pointing index finder in
  // direction a and middle finger in direction b, thumb will point in a.Cross(b).
  Vector3 Cross(const Vector3& b) const {
    return Vector3(y * b.z - z * b.y, z * b.x - x * b.z, x * b.y - y * b.x);
  }

  // Returns the angle from this vector to b, in radians.
  T Angle(const Vector3& b) const {
    T div = LengthSq() * b.LengthSq();
    OVR_MATH_ASSERT(div != T(0));
    T result = Acos((this->Dot(b)) / sqrt(div));
    return result;
  }

  // Return Length of the vector squared.
  T LengthSq() const {
    return (x * x + y * y + z * z);
  }

  // Return vector length.
  T Length() const {
    return (T)sqrt(LengthSq());
  }

  // Returns squared distance between two points represented by vectors.
  T DistanceSq(Vector3 const& b) const {
    return (*this - b).LengthSq();
  }

  // Returns distance between two points represented by vectors.
  T Distance(Vector3 const& b) const {
    return (*this - b).Length();
  }

  bool IsNormalized() const {
    return fabs(LengthSq() - T(1)) < Math<T>::Tolerance();
  }

  // Normalize, convention vector length to 1.
  void Normalize() {
    T s = Length();
    if (s != T(0))
      s = T(1) / s;
    *this *= s;
  }

  // Returns normalized (unit) version of the vector without modifying itself.
  Vector3 Normalized() const {
    T s = Length();
    if (s != T(0))
      s = T(1) / s;
    return *this * s;
  }

  // Linearly interpolates from this vector to another.
  // Factor should be between 0.0 and 1.0, with 0 giving full value to this.
  Vector3 Lerp(const Vector3& b, T f) const {
    return *this * (T(1) - f) + b * f;
  }

  // Projects this vector onto the argument; in other words,
  // A.Project(B) returns projection of vector A onto B.
  Vector3 ProjectTo(const Vector3& b) const {
    T l2 = b.LengthSq();
    OVR_MATH_ASSERT(l2 != T(0));
    return b * (Dot(b) / l2);
  }

  // Projects this vector onto a plane defined by a normal vector
  Vector3 ProjectToPlane(const Vector3& normal) const {
    return *this - this->ProjectTo(normal);
  }

  bool IsNan() const {
    return !isfinite(x + y + z);
  }
  bool IsFinite() const {
    return isfinite(x + y + z);
  }
};

typedef Vector3<float> Vector3f;
typedef Vector3<double> Vector3d;
typedef Vector3<int32_t> Vector3i;

OVR_MATH_STATIC_ASSERT((sizeof(Vector3f) == 3 * sizeof(float)), "sizeof(Vector3f) failure");
OVR_MATH_STATIC_ASSERT((sizeof(Vector3d) == 3 * sizeof(double)), "sizeof(Vector3d) failure");
OVR_MATH_STATIC_ASSERT((sizeof(Vector3i) == 3 * sizeof(int32_t)), "sizeof(Vector3i) failure");

typedef Vector3<float> Point3f;
typedef Vector3<double> Point3d;
typedef Vector3<int32_t> Point3i;

//-------------------------------------------------------------------------------------
// ***** Vector4<> - 4D vector of {x, y, z, w}

//
// Vector4f (Vector4d) represents a 3-dimensional vector or point in space,
// consisting of coordinates x, y, z and w.

template <class T>
class Vector4 {
 public:
  typedef T ElementType;
  static const size_t ElementCount = 4;

  T x, y, z, w;

  // FIXME: default initialization of a vector class can be very expensive in a full-blown
  // application.  A few hundred thousand vector constructions is not unlikely and can add
  // up to milliseconds of time on processors like the PS3 PPU.
  Vector4() : x(0), y(0), z(0), w(0) {}
  Vector4(T x_, T y_, T z_, T w_) : x(x_), y(y_), z(z_), w(w_) {}
  explicit Vector4(T s) : x(s), y(s), z(s), w(s) {}
  explicit Vector4(const Vector3<T>& v, const T w_ = T(1)) : x(v.x), y(v.y), z(v.z), w(w_) {}
  explicit Vector4(const Vector4<typename Math<T>::OtherFloatType>& src)
      : x((T)src.x), y((T)src.y), z((T)src.z), w((T)src.w) {}

  static Vector4 Zero() {
    return Vector4(0, 0, 0, 0);
  }

  // C-interop support.
  typedef typename CompatibleTypes<Vector4<T>>::Type CompatibleType;

  Vector4(const CompatibleType& s) : x(s.x), y(s.y), z(s.z), w(s.w) {}

  operator const CompatibleType&() const {
    OVR_MATH_STATIC_ASSERT(
        sizeof(Vector4<T>) == sizeof(CompatibleType), "sizeof(Vector4<T>) failure");
    return reinterpret_cast<const CompatibleType&>(*this);
  }

  Vector4& operator=(const Vector3<T>& other) {
    x = other.x;
    y = other.y;
    z = other.z;
    w = 1;
    return *this;
  }
  bool operator==(const Vector4& b) const {
    return x == b.x && y == b.y && z == b.z && w == b.w;
  }
  bool operator!=(const Vector4& b) const {
    return x != b.x || y != b.y || z != b.z || w != b.w;
  }

  Vector4 operator+(const Vector4& b) const {
    return Vector4(x + b.x, y + b.y, z + b.z, w + b.w);
  }
  Vector4& operator+=(const Vector4& b) {
    x += b.x;
    y += b.y;
    z += b.z;
    w += b.w;
    return *this;
  }
  Vector4 operator-(const Vector4& b) const {
    return Vector4(x - b.x, y - b.y, z - b.z, w - b.w);
  }
  Vector4& operator-=(const Vector4& b) {
    x -= b.x;
    y -= b.y;
    z -= b.z;
    w -= b.w;
    return *this;
  }
  Vector4 operator-() const {
    return Vector4(-x, -y, -z, -w);
  }

  // Scalar multiplication/division scales vector.
  Vector4 operator*(T s) const {
    return Vector4(x * s, y * s, z * s, w * s);
  }
  Vector4& operator*=(T s) {
    x *= s;
    y *= s;
    z *= s;
    w *= s;
    return *this;
  }

  Vector4 operator/(T s) const {
    T rcp = T(1) / s;
    return Vector4(x * rcp, y * rcp, z * rcp, w * rcp);
  }
  Vector4& operator/=(T s) {
    T rcp = T(1) / s;
    x *= rcp;
    y *= rcp;
    z *= rcp;
    w *= rcp;
    return *this;
  }

  static Vector4 Min(const Vector4& a, const Vector4& b) {
    return Vector4(
        (a.x < b.x) ? a.x : b.x,
        (a.y < b.y) ? a.y : b.y,
        (a.z < b.z) ? a.z : b.z,
        (a.w < b.w) ? a.w : b.w);
  }
  static Vector4 Max(const Vector4& a, const Vector4& b) {
    return Vector4(
        (a.x > b.x) ? a.x : b.x,
        (a.y > b.y) ? a.y : b.y,
        (a.z > b.z) ? a.z : b.z,
        (a.w > b.w) ? a.w : b.w);
  }

  Vector4 Clamped(T maxMag) const {
    T magSquared = LengthSq();
    if (magSquared <= Sqr(maxMag))
      return *this;
    else
      return *this * (maxMag / sqrt(magSquared));
  }

  // Compare two vectors for equality with tolerance. Returns true if vectors match within
  // tolerance.
  bool IsEqual(const Vector4& b, T tolerance = Math<T>::Tolerance()) const {
    return (fabs(b.x - x) <= tolerance) && (fabs(b.y - y) <= tolerance) &&
        (fabs(b.z - z) <= tolerance) && (fabs(b.w - w) <= tolerance);
  }
  bool Compare(const Vector4& b, T tolerance = Math<T>::Tolerance()) const {
    return IsEqual(b, tolerance);
  }

  T& operator[](int idx) {
    OVR_MATH_ASSERT(0 <= idx && idx < 4);
    return *(&x + idx);
  }

  const T& operator[](int idx) const {
    OVR_MATH_ASSERT(0 <= idx && idx < 4);
    return *(&x + idx);
  }

  // Entry wise product of two vectors
  Vector4 EntrywiseMultiply(const Vector4& b) const {
    return Vector4(x * b.x, y * b.y, z * b.z, w * b.w);
  }

  // Multiply and divide operators do entry-wise math
  Vector4 operator*(const Vector4& b) const {
    return Vector4(x * b.x, y * b.y, z * b.z, w * b.w);
  }

  Vector4 operator/(const Vector4& b) const {
    return Vector4(x / b.x, y / b.y, z / b.z, w / b.w);
  }

  // Dot product
  T Dot(const Vector4& b) const {
    return x * b.x + y * b.y + z * b.z + w * b.w;
  }

  // Return Length of the vector squared.
  T LengthSq() const {
    return (x * x + y * y + z * z + w * w);
  }

  // Return vector length.
  T Length() const {
    return sqrt(LengthSq());
  }

  bool IsNormalized() const {
    return fabs(LengthSq() - T(1)) < Math<T>::Tolerance();
  }

  // Normalize, convention vector length to 1.
  void Normalize() {
    T s = Length();
    if (s != T(0))
      s = T(1) / s;
    *this *= s;
  }

  // Returns normalized (unit) version of the vector without modifying itself.
  Vector4 Normalized() const {
    T s = Length();
    if (s != T(0))
      s = T(1) / s;
    return *this * s;
  }

  // Linearly interpolates from this vector to another.
  // Factor should be between 0.0 and 1.0, with 0 giving full value to this.
  Vector4 Lerp(const Vector4& b, T f) const {
    return *this * (T(1) - f) + b * f;
  }
};

typedef Vector4<float> Vector4f;
typedef Vector4<double> Vector4d;
typedef Vector4<int> Vector4i;

//-------------------------------------------------------------------------------------
// ***** Bounds3

// Bounds class used to describe a 3D axis aligned bounding box.

template <class T>
class Bounds3 {
 public:
  Vector3<T> b[2];

  Bounds3() {
    Clear();
  }

  Bounds3(const Vector3<T>& mins, const Vector3<T>& maxs) {
    b[0] = mins;
    b[1] = maxs;
  }

  void Clear() {
    b[0].x = b[0].y = b[0].z = Math<T>::MaxValue();
    b[1].x = b[1].y = b[1].z = -Math<T>::MaxValue();
  }

  void AddPoint(const Vector3<T>& v) {
    b[0].x = (b[0].x < v.x ? b[0].x : v.x);
    b[0].y = (b[0].y < v.y ? b[0].y : v.y);
    b[0].z = (b[0].z < v.z ? b[0].z : v.z);
    b[1].x = (v.x < b[1].x ? b[1].x : v.x);
    b[1].y = (v.y < b[1].y ? b[1].y : v.y);
    b[1].z = (v.z < b[1].z ? b[1].z : v.z);
  }

  bool Excludes(const Vector3<T>& v) const {
    bool testing = false;
    for (int32_t t = 0; t < 3; ++t) {
      testing |= v[t] > b[1][t];
      testing |= v[t] < b[0][t];
    }
    return testing;
  }

  // exludes, ignoring vertical
  bool ExcludesXZ(const Vector3<T>& v) const {
    bool testing = false;
    testing |= v[0] > b[1][0];
    testing |= v[0] < b[0][0];
    testing |= v[2] > b[1][2];
    testing |= v[2] < b[0][2];
    return testing;
  }

  bool Excludes(const Bounds3<T>& bounds) const {
    bool testing = false;
    for (int32_t t = 0; t < 3; ++t) {
      testing |= bounds.b[0][t] > b[1][t];
      testing |= bounds.b[1][t] < b[0][t];
    }
    return testing;
  }

  const Vector3<T>& GetMins() const {
    return b[0];
  }
  const Vector3<T>& GetMaxs() const {
    return b[1];
  }

  Vector3<T>& GetMins() {
    return b[0];
  }
  Vector3<T>& GetMaxs() {
    return b[1];
  }
};

typedef Bounds3<float> Bounds3f;
typedef Bounds3<double> Bounds3d;

//-------------------------------------------------------------------------------------
// ***** Size

// Size class represents 2D size with Width, Height components.
// Used to describe distentions of render targets, etc.

template <class T>
class Size {
 public:
  T w, h;

  Size() : w(0), h(0) {}
  Size(T w_, T h_) : w(w_), h(h_) {}
  explicit Size(T s) : w(s), h(s) {}
  explicit Size(const Size<typename Math<T>::OtherFloatType>& src) : w((T)src.w), h((T)src.h) {}

  // C-interop support.
  typedef typename CompatibleTypes<Size<T>>::Type CompatibleType;

  Size(const CompatibleType& s) : w(s.w), h(s.h) {}

  operator const CompatibleType&() const {
    OVR_MATH_STATIC_ASSERT(sizeof(Size<T>) == sizeof(CompatibleType), "sizeof(Size<T>) failure");
    return reinterpret_cast<const CompatibleType&>(*this);
  }

  bool operator==(const Size& b) const {
    return w == b.w && h == b.h;
  }
  bool operator!=(const Size& b) const {
    return w != b.w || h != b.h;
  }

  Size operator+(const Size& b) const {
    return Size(w + b.w, h + b.h);
  }
  Size& operator+=(const Size& b) {
    w += b.w;
    h += b.h;
    return *this;
  }
  Size operator-(const Size& b) const {
    return Size(w - b.w, h - b.h);
  }
  Size& operator-=(const Size& b) {
    w -= b.w;
    h -= b.h;
    return *this;
  }
  Size operator-() const {
    return Size(-w, -h);
  }
  Size operator*(const Size& b) const {
    return Size(w * b.w, h * b.h);
  }
  Size& operator*=(const Size& b) {
    w *= b.w;
    h *= b.h;
    return *this;
  }
  Size operator/(const Size& b) const {
    return Size(w / b.w, h / b.h);
  }
  Size& operator/=(const Size& b) {
    w /= b.w;
    h /= b.h;
    return *this;
  }

  // Scalar multiplication/division scales both components.
  Size operator*(T s) const {
    return Size(w * s, h * s);
  }
  Size& operator*=(T s) {
    w *= s;
    h *= s;
    return *this;
  }
  Size operator/(T s) const {
    return Size(w / s, h / s);
  }
  Size& operator/=(T s) {
    w /= s;
    h /= s;
    return *this;
  }

  static Size Min(const Size& a, const Size& b) {
    return Size((a.w < b.w) ? a.w : b.w, (a.h < b.h) ? a.h : b.h);
  }
  static Size Max(const Size& a, const Size& b) {
    return Size((a.w > b.w) ? a.w : b.w, (a.h > b.h) ? a.h : b.h);
  }

  T Area() const {
    return w * h;
  }

  inline Vector2<T> ToVector() const {
    return Vector2<T>(w, h);
  }
};

typedef Size<int> Sizei;
typedef Size<unsigned> Sizeu;
typedef Size<float> Sizef;
typedef Size<double> Sized;

//-------------------------------------------------------------------------------------
// ***** Size3

// Size3 class represents 3D size with Width, Height, Depth components.
// Used to describe distentions of render targets, etc.

template <class T>
class Size3 {
 public:
  T w, h, d;

  Size3() : w(0), h(0), d(0) {}
  Size3(T w_, T h_, T d_) : w(w_), h(h_), d(d_) {}
  explicit Size3(T s) : w(s), h(s), d(s) {}
  explicit Size3(const Size<typename Math<T>::OtherFloatType>& src)
      : w((T)src.w), h((T)src.h), d((T)1) {}
  explicit Size3(const Size3<typename Math<T>::OtherFloatType>& src)
      : w((T)src.w), h((T)src.h), d((T)src.d) {}

  // C-interop support.
  typedef typename CompatibleTypes<Size3<T>>::Type CompatibleType;

  Size3(const CompatibleType& s) : w(s.w), h(s.h), d(s.d) {}

  operator const CompatibleType&() const {
    OVR_MATH_STATIC_ASSERT(sizeof(Size3<T>) == sizeof(CompatibleType), "sizeof(Size3<T>) failure");
    return reinterpret_cast<const CompatibleType&>(*this);
  }

  bool operator==(const Size3& b) const {
    return w == b.w && h == b.h && d == b.d;
  }
  bool operator!=(const Size3& b) const {
    return w != b.w || h != b.h || d != b.d;
  }

  Size3 operator+(const Size3& b) const {
    return Size3(w + b.w, h + b.h, d + b.d);
  }
  Size3& operator+=(const Size3& b) {
    w += b.w;
    h += b.h;
    d += b.d;
    return *this;
  }
  Size3 operator-(const Size3& b) const {
    return Size3(w - b.w, h - b.h, d - b.d);
  }
  Size3& operator-=(const Size3& b) {
    w -= b.w;
    h -= b.h;
    d -= b.d;
    return *this;
  }
  Size3 operator-() const {
    return Size3(-w, -h, -d);
  }
  Size3 operator*(const Size3& b) const {
    return Size3(w * b.w, h * b.h, d * b.d);
  }
  Size3& operator*=(const Size3& b) {
    w *= b.w;
    h *= b.h;
    d *= b.d;
    return *this;
  }
  Size3 operator/(const Size3& b) const {
    return Size3(w / b.w, h / b.h, d / b.d);
  }
  Size3& operator/=(const Size3& b) {
    w /= b.w;
    h /= b.h;
    d /= b.d;
    return *this;
  }

  // Scalar multiplication/division scales both components.
  Size3 operator*(T s) const {
    return Size3(w * s, h * s, d * s);
  }
  Size3& operator*=(T s) {
    w *= s;
    h *= s;
    d *= s;
    return *this;
  }
  Size3 operator/(T s) const {
    return Size3(w / s, h / s, d / s);
  }
  Size3& operator/=(T s) {
    w /= s;
    h /= s;
    d /= s;
    return *this;
  }

  static Size3 Min(const Size3& a, const Size3& b) {
    return Size3((a.w < b.w) ? a.w : b.w, (a.h < b.h) ? a.h : b.h, (a.d < b.d) ? a.d : b.d);
  }
  static Size3 Max(const Size3& a, const Size3& b) {
    return Size3((a.w > b.w) ? a.w : b.w, (a.h > b.h) ? a.h : b.h, (a.d > b.d) ? a.d : b.d);
  }

  T Volume() const {
    return w * h * d;
  }

  inline Vector3<T> ToVector() const {
    return Vector3<T>(w, h, d);
  }
};

typedef Size3<unsigned> Size3u;

//-----------------------------------------------------------------------------------
// ***** Rect

// Rect describes a rectangular area for rendering, that includes position and size.
template <class T>
class Rect {
 public:
  T x, y;
  T w, h;

  Rect() {}
  Rect(T x1, T y1, T w1, T h1) : x(x1), y(y1), w(w1), h(h1) {}
  Rect(const Vector2<T>& pos, const Size<T>& sz) : x(pos.x), y(pos.y), w(sz.w), h(sz.h) {}
  Rect(const Size<T>& sz) : x(0), y(0), w(sz.w), h(sz.h) {}

  // C-interop support.
  typedef typename CompatibleTypes<Rect<T>>::Type CompatibleType;

  Rect(const CompatibleType& s) : x(s.Pos.x), y(s.Pos.y), w(s.Size.w), h(s.Size.h) {}

  operator const CompatibleType&() const {
    OVR_MATH_STATIC_ASSERT(sizeof(Rect<T>) == sizeof(CompatibleType), "sizeof(Rect<T>) failure");
    return reinterpret_cast<const CompatibleType&>(*this);
  }

  Vector2<T> GetPos() const {
    return Vector2<T>(x, y);
  }
  Size<T> GetSize() const {
    return Size<T>(w, h);
  }
  void SetPos(const Vector2<T>& pos) {
    x = pos.x;
    y = pos.y;
  }
  void SetSize(const Size<T>& sz) {
    w = sz.w;
    h = sz.h;
  }

  bool operator==(const Rect& vp) const {
    return (x == vp.x) && (y == vp.y) && (w == vp.w) && (h == vp.h);
  }
  bool operator!=(const Rect& vp) const {
    return !operator==(vp);
  }
};

typedef Rect<int> Recti;

//-------------------------------------------------------------------------------------//
// ***** Quat
//
// Quatf represents a quaternion class used for rotations.
//
// Quaternion multiplications are done in right-to-left order, to match the
// behavior of matrices.

template <class T>
class Quat {
 public:
  typedef T ElementType;
  static const size_t ElementCount = 4;

  // x,y,z = axis*sin(angle), w = cos(angle)
  T x, y, z, w;

  Quat() : x(0), y(0), z(0), w(1) {}
  Quat(T x_, T y_, T z_, T w_) : x(x_), y(y_), z(z_), w(w_) {}
  explicit Quat(const Quat<typename Math<T>::OtherFloatType>& src)
      : x((T)src.x), y((T)src.y), z((T)src.z), w((T)src.w) {
    // NOTE: Converting a normalized Quat<float> to Quat<double>
    // will generally result in an un-normalized quaternion.
    // But we don't normalize here in case the quaternion
    // being converted is not a normalized rotation quaternion.
  }

  typedef typename CompatibleTypes<Quat<T>>::Type CompatibleType;

  // C-interop support.
  Quat(const CompatibleType& s) : x(s.x), y(s.y), z(s.z), w(s.w) {}

  operator CompatibleType() const {
    CompatibleType result;
    result.x = x;
    result.y = y;
    result.z = z;
    result.w = w;
    return result;
  }

  // Constructs quaternion for rotation around the axis by an angle.
  Quat(const Vector3<T>& axis, T angle) {
    // Make sure we don't divide by zero.
    if (axis.LengthSq() == T(0)) {
      // Assert if the axis is zero, but the angle isn't
      OVR_MATH_ASSERT(angle == T(0));
      x = y = z = T(0);
      w = T(1);
      return;
    }

    Vector3<T> unitAxis = axis.Normalized();
    T sinHalfAngle = sin(angle * T(0.5));

    w = cos(angle * T(0.5));
    x = unitAxis.x * sinHalfAngle;
    y = unitAxis.y * sinHalfAngle;
    z = unitAxis.z * sinHalfAngle;
  }

  // Constructs quaternion for rotation around one of the coordinate axis by an angle.
  Quat(Axis A, T angle, RotateDirection d = Rotate_CCW, HandedSystem s = Handed_R) {
    T sinHalfAngle = s * d * sin(angle * T(0.5));
    T v[3];
    v[0] = v[1] = v[2] = T(0);
    v[A] = sinHalfAngle;

    w = cos(angle * T(0.5));
    x = v[0];
    y = v[1];
    z = v[2];
  }

  Quat operator-() {
    return Quat(-x, -y, -z, -w);
  } // unary minus

  static Quat Identity() {
    return Quat(0, 0, 0, 1);
  }

  // Compute axis and angle from quaternion
  void GetAxisAngle(Vector3<T>* axis, T* angle) const {
    if (x * x + y * y + z * z > Math<T>::Tolerance() * Math<T>::Tolerance()) {
      *axis = Vector3<T>(x, y, z).Normalized();
      *angle = 2 * Acos(w);
      if (*angle > ((T)MATH_DOUBLE_PI)) // Reduce the magnitude of the angle, if necessary
      {
        *angle = ((T)MATH_DOUBLE_TWOPI) - *angle;
        *axis = *axis * (-1);
      }
    } else {
      *axis = Vector3<T>(1, 0, 0);
      *angle = T(0);
    }
  }

  // Convert a quaternion to a rotation vector, also known as
  // Rodrigues vector, AxisAngle vector, SORA vector, exponential map.
  // A rotation vector describes a rotation about an axis:
  // the axis of rotation is the vector normalized,
  // the angle of rotation is the magnitude of the vector.
  Vector3<T> ToRotationVector() const {
    // OVR_MATH_ASSERT(IsNormalized()); // If this fires, caller has a quat math bug
    T s = T(0);
    T sinHalfAngle = sqrt(x * x + y * y + z * z);
    if (sinHalfAngle > T(0)) {
      T cosHalfAngle = w;
      T halfAngle = atan2(sinHalfAngle, cosHalfAngle);

      // Ensure minimum rotation magnitude
      if (cosHalfAngle < 0)
        halfAngle -= T(MATH_DOUBLE_PI);

      s = T(2) * halfAngle / sinHalfAngle;
    }
    return Vector3<T>(x * s, y * s, z * s);
  }

  // Faster version of the above, optimized for use with small rotations, where rotation angle ~=
  // sin(angle)
  inline OVR::Vector3<T> FastToRotationVector() const {
    OVR_MATH_ASSERT(IsNormalized()); // If this fires, caller has a quat math bug
    T s;
    T sinHalfSquared = x * x + y * y + z * z;
    if (sinHalfSquared < T(.0037)) // =~ sin(7/2 degrees)^2
    {
      // Max rotation magnitude error is about .062% at 7 degrees rotation, or about .0043 degrees
      s = T(2) * Sign(w);
    } else {
      T sinHalfAngle = sqrt(sinHalfSquared);
      T cosHalfAngle = w;
      T halfAngle = atan2(sinHalfAngle, cosHalfAngle);

      // Ensure minimum rotation magnitude
      if (cosHalfAngle < 0)
        halfAngle -= T(MATH_DOUBLE_PI);

      s = T(2) * halfAngle / sinHalfAngle;
    }
    return Vector3<T>(x * s, y * s, z * s);
  }

  // Given a rotation vector of form unitRotationAxis * angle,
  // returns the equivalent quaternion (unitRotationAxis * sin(angle), cos(Angle)).
  static Quat FromRotationVector(const Vector3<T>& v) {
    T angleSquared = v.LengthSq();
    T s = T(0);
    T c = T(1);
    if (angleSquared > T(0)) {
      T angle = sqrt(angleSquared);
      s = sin(angle * T(0.5)) / angle; // normalize
      c = cos(angle * T(0.5));
    }
    return Quat(s * v.x, s * v.y, s * v.z, c);
  }

  // Faster version of above, optimized for use with small rotation magnitudes, where rotation angle
  // =~ sin(angle).
  // If normalize is false, small-angle quaternions are returned un-normalized.
  inline static Quat FastFromRotationVector(const OVR::Vector3<T>& v, bool normalize = true) {
    T s, c;
    T angleSquared = v.LengthSq();
    if (angleSquared < T(0.0076)) // =~ (5 degrees*pi/180)^2
    {
      s = T(0.5);
      c = T(1.0);
      // Max rotation magnitude error (after normalization) is about .064% at 5 degrees rotation, or
      // .0032 degrees
      if (normalize && angleSquared > 0) {
        // sin(angle/2)^2 ~= (angle/2)^2 and cos(angle/2)^2 ~= 1
        T invLen = T(1) / sqrt(angleSquared * T(0.25) + T(1)); // normalize
        s = s * invLen;
        c = c * invLen;
      }
    } else {
      T angle = sqrt(angleSquared);
      s = sin(angle * T(0.5)) / angle;
      c = cos(angle * T(0.5));
    }
    return Quat(s * v.x, s * v.y, s * v.z, c);
  }

  // Constructs the quaternion from a rotation matrix
  explicit Quat(const Matrix4<T>& m) {
    T trace = m.M[0][0] + m.M[1][1] + m.M[2][2];

    // In almost all cases, the first part is executed.
    // However, if the trace is not positive, the other
    // cases arise.
    if (trace > T(0)) {
      T s = sqrt(trace + T(1)) * T(2); // s=4*qw
      w = T(0.25) * s;
      x = (m.M[2][1] - m.M[1][2]) / s;
      y = (m.M[0][2] - m.M[2][0]) / s;
      z = (m.M[1][0] - m.M[0][1]) / s;
    } else if ((m.M[0][0] > m.M[1][1]) && (m.M[0][0] > m.M[2][2])) {
      T s = sqrt(T(1) + m.M[0][0] - m.M[1][1] - m.M[2][2]) * T(2);
      w = (m.M[2][1] - m.M[1][2]) / s;
      x = T(0.25) * s;
      y = (m.M[0][1] + m.M[1][0]) / s;
      z = (m.M[2][0] + m.M[0][2]) / s;
    } else if (m.M[1][1] > m.M[2][2]) {
      T s = sqrt(T(1) + m.M[1][1] - m.M[0][0] - m.M[2][2]) * T(2); // S=4*qy
      w = (m.M[0][2] - m.M[2][0]) / s;
      x = (m.M[0][1] + m.M[1][0]) / s;
      y = T(0.25) * s;
      z = (m.M[1][2] + m.M[2][1]) / s;
    } else {
      T s = sqrt(T(1) + m.M[2][2] - m.M[0][0] - m.M[1][1]) * T(2); // S=4*qz
      w = (m.M[1][0] - m.M[0][1]) / s;
      x = (m.M[0][2] + m.M[2][0]) / s;
      y = (m.M[1][2] + m.M[2][1]) / s;
      z = T(0.25) * s;
    }
    OVR_MATH_ASSERT(IsNormalized()); // Ensure input matrix is orthogonal
  }

  // Constructs the quaternion from a rotation matrix
  explicit Quat(const Matrix3<T>& m) {
    T trace = m.M[0][0] + m.M[1][1] + m.M[2][2];

    // In almost all cases, the first part is executed.
    // However, if the trace is not positive, the other
    // cases arise.
    if (trace > T(0)) {
      T s = sqrt(trace + T(1)) * T(2); // s=4*qw
      w = T(0.25) * s;
      x = (m.M[2][1] - m.M[1][2]) / s;
      y = (m.M[0][2] - m.M[2][0]) / s;
      z = (m.M[1][0] - m.M[0][1]) / s;
    } else if ((m.M[0][0] > m.M[1][1]) && (m.M[0][0] > m.M[2][2])) {
      T s = sqrt(T(1) + m.M[0][0] - m.M[1][1] - m.M[2][2]) * T(2);
      w = (m.M[2][1] - m.M[1][2]) / s;
      x = T(0.25) * s;
      y = (m.M[0][1] + m.M[1][0]) / s;
      z = (m.M[2][0] + m.M[0][2]) / s;
    } else if (m.M[1][1] > m.M[2][2]) {
      T s = sqrt(T(1) + m.M[1][1] - m.M[0][0] - m.M[2][2]) * T(2); // S=4*qy
      w = (m.M[0][2] - m.M[2][0]) / s;
      x = (m.M[0][1] + m.M[1][0]) / s;
      y = T(0.25) * s;
      z = (m.M[1][2] + m.M[2][1]) / s;
    } else {
      T s = sqrt(T(1) + m.M[2][2] - m.M[0][0] - m.M[1][1]) * T(2); // S=4*qz
      w = (m.M[1][0] - m.M[0][1]) / s;
      x = (m.M[0][2] + m.M[2][0]) / s;
      y = (m.M[1][2] + m.M[2][1]) / s;
      z = T(0.25) * s;
    }
    OVR_MATH_ASSERT(IsNormalized()); // Ensure input matrix is orthogonal
  }

  // MERGE_MOBILE_SDK
  // Constructs a quaternion that rotates 'from' to line up with 'to'.
  explicit Quat(const Vector3<T>& from, const Vector3<T>& to) {
    const T cx = from.y * to.z - from.z * to.y;
    const T cy = from.z * to.x - from.x * to.z;
    const T cz = from.x * to.y - from.y * to.x;
    const T dot = from.x * to.x + from.y * to.y + from.z * to.z;
    const T crossLengthSq = cx * cx + cy * cy + cz * cz;
    const T magnitude = static_cast<T>(sqrt(crossLengthSq + dot * dot));
    const T cw = dot + magnitude;
    if (cw < Math<T>::SmallestNonDenormal()) {
      const T sx = to.y * to.y + to.z * to.z;
      const T sz = to.x * to.x + to.y * to.y;
      if (sx > sz) {
        const T rcpLength = RcpSqrt(sx);
        x = T(0);
        y = to.z * rcpLength;
        z = -to.y * rcpLength;
        w = T(0);
      } else {
        const T rcpLength = RcpSqrt(sz);
        x = to.y * rcpLength;
        y = -to.x * rcpLength;
        z = T(0);
        w = T(0);
      }
      return;
    }
    const T rcpLength = RcpSqrt(crossLengthSq + cw * cw);
    x = cx * rcpLength;
    y = cy * rcpLength;
    z = cz * rcpLength;
    w = cw * rcpLength;
  }
  // MERGE_MOBILE_SDK

  bool operator==(const Quat& b) const {
    return x == b.x && y == b.y && z == b.z && w == b.w;
  }
  bool operator!=(const Quat& b) const {
    return x != b.x || y != b.y || z != b.z || w != b.w;
  }

  Quat operator+(const Quat& b) const {
    return Quat(x + b.x, y + b.y, z + b.z, w + b.w);
  }
  Quat& operator+=(const Quat& b) {
    w += b.w;
    x += b.x;
    y += b.y;
    z += b.z;
    return *this;
  }
  Quat operator-(const Quat& b) const {
    return Quat(x - b.x, y - b.y, z - b.z, w - b.w);
  }
  Quat& operator-=(const Quat& b) {
    w -= b.w;
    x -= b.x;
    y -= b.y;
    z -= b.z;
    return *this;
  }

  Quat operator*(T s) const {
    return Quat(x * s, y * s, z * s, w * s);
  }
  Quat& operator*=(T s) {
    w *= s;
    x *= s;
    y *= s;
    z *= s;
    return *this;
  }
  Quat operator/(T s) const {
    T rcp = T(1) / s;
    return Quat(x * rcp, y * rcp, z * rcp, w * rcp);
  }
  Quat& operator/=(T s) {
    T rcp = T(1) / s;
    w *= rcp;
    x *= rcp;
    y *= rcp;
    z *= rcp;
    return *this;
  }

  // MERGE_MOBILE_SDK
  Vector3<T> operator*(const Vector3<T>& v) const {
    return Rotate(v);
  }
  // MERGE_MOBILE_SDK

  // Compare two quats for equality within tolerance. Returns true if quats match within tolerance.
  bool IsEqual(const Quat& b, T tolerance = Math<T>::Tolerance()) const {
    return Abs(Dot(b)) >= T(1) - tolerance;
  }

  // Compare two quats for equality within tolerance while checking matching hemispheres. Returns
  // true if quats match within tolerance.
  bool IsEqualMatchHemisphere(Quat b, T tolerance = Math<T>::Tolerance()) const {
    b.EnsureSameHemisphere(*this);
    return Abs(Dot(b)) >= T(1) - tolerance;
  }

  static T Abs(const T v) {
    return (v >= 0) ? v : -v;
  }

  // Get Imaginary part vector
  Vector3<T> Imag() const {
    return Vector3<T>(x, y, z);
  }

  // Get quaternion length.
  T Length() const {
    return sqrt(LengthSq());
  }

  // Get quaternion length squared.
  T LengthSq() const {
    return (x * x + y * y + z * z + w * w);
  }

  // Simple Euclidean distance in R^4 (not SLERP distance, but at least respects Haar measure)
  T Distance(const Quat& q) const {
    T d1 = (*this - q).Length();
    T d2 = (*this + q).Length(); // Antipodal point check
    return (d1 < d2) ? d1 : d2;
  }

  T DistanceSq(const Quat& q) const {
    T d1 = (*this - q).LengthSq();
    T d2 = (*this + q).LengthSq(); // Antipodal point check
    return (d1 < d2) ? d1 : d2;
  }

  T Dot(const Quat& q) const {
    return x * q.x + y * q.y + z * q.z + w * q.w;
  }

  // Angle between two quaternions in radians
  T Angle(const Quat& q) const {
    return T(2) * Acos(Abs(Dot(q)));
  }

  // Angle of quaternion
  T Angle() const {
    return T(2) * Acos(Abs(w));
  }

  // Normalize
  bool IsNormalized() const {
    return fabs(LengthSq() - T(1)) < Math<T>::Tolerance();
  }

  void Normalize() {
    T s = Length();
    if (s != T(0))
      s = T(1) / s;
    *this *= s;
  }

  Quat Normalized() const {
    T s = Length();
    if (s != T(0))
      s = T(1) / s;
    return *this * s;
  }

  inline void EnsureSameHemisphere(const Quat& o) {
    if (Dot(o) < T(0)) {
      x = -x;
      y = -y;
      z = -z;
      w = -w;
    }
  }

  // Returns conjugate of the quaternion. Produces inverse rotation if quaternion is normalized.
  Quat Conj() const {
    return Quat(-x, -y, -z, w);
  }

  // Quaternion multiplication. Combines quaternion rotations, performing the one on the
  // right hand side first.
  Quat operator*(const Quat& b) const {
    return Quat(
        w * b.x + x * b.w + y * b.z - z * b.y,
        w * b.y - x * b.z + y * b.w + z * b.x,
        w * b.z + x * b.y - y * b.x + z * b.w,
        w * b.w - x * b.x - y * b.y - z * b.z);
  }
  const Quat& operator*=(const Quat& b) {
    *this = *this * b;
    return *this;
  }

  //
  // this^p normalized; same as rotating by this p times.
  Quat PowNormalized(T p) const {
    Vector3<T> v;
    T a;
    GetAxisAngle(&v, &a);
    return Quat(v, a * p);
  }

  // Compute quaternion that rotates v into alignTo: alignTo = Quat::Align(alignTo, v).Rotate(v).
  // NOTE: alignTo and v must be normalized.
  static Quat Align(const Vector3<T>& alignTo, const Vector3<T>& v) {
    OVR_MATH_ASSERT(alignTo.IsNormalized() && v.IsNormalized());
    Vector3<T> bisector = (v + alignTo);
    bisector.Normalize();
    T cosHalfAngle = v.Dot(bisector); // 0..1
    if (cosHalfAngle > T(0)) {
      Vector3<T> imag = v.Cross(bisector);
      return Quat(imag.x, imag.y, imag.z, cosHalfAngle);
    } else {
      // cosHalfAngle == 0: a 180 degree rotation.
      // sinHalfAngle == 1, rotation axis is any axis perpendicular
      // to alignTo.  Choose axis to include largest magnitude components
      if (fabs(v.x) > fabs(v.y)) {
        // x or z is max magnitude component
        // = Cross(v, (0,1,0)).Normalized();
        T invLen = sqrt(v.x * v.x + v.z * v.z);
        if (invLen > T(0))
          invLen = T(1) / invLen;
        return Quat(-v.z * invLen, 0, v.x * invLen, 0);
      } else {
        // y or z is max magnitude component
        // = Cross(v, (1,0,0)).Normalized();
        T invLen = sqrt(v.y * v.y + v.z * v.z);
        if (invLen > T(0))
          invLen = T(1) / invLen;
        return Quat(0, v.z * invLen, -v.y * invLen, 0);
      }
    }
  }

  // Decompose a quat into quat = swing * twist, where twist is a rotation about axis,
  // and swing is a rotation perpendicular to axis.
  Quat GetSwingTwist(const Vector3<T>& axis, Quat* twist) const {
    OVR_MATH_ASSERT(twist);
    OVR_MATH_ASSERT(axis.IsNormalized());

    // Create a normalized quaternion from projection of (x,y,z) onto axis
    T d = axis.Dot(Vector3<T>(x, y, z));
    *twist = Quat(axis.x * d, axis.y * d, axis.z * d, w);
    T len = twist->Length();
    if (len == 0)
      twist->w = T(1); // identity
    else
      *twist /= len; // normalize

    return *this * twist->Inverted();
  }

  // Normalized linear interpolation of quaternions
  // NOTE: This function is a bad approximation of Slerp()
  // when the angle between the *this and b is large.
  // Use FastSlerp() or Slerp() instead.
  Quat Lerp(const Quat& b, T s) const {
    return (*this * (T(1) - s) + b * (Dot(b) < 0 ? -s : s)).Normalized();
  }

  // Spherical linear interpolation between rotations
  Quat Slerp(const Quat& b, T s) const {
    Vector3<T> delta = (b * this->Inverted()).ToRotationVector();
    return (FromRotationVector(delta * s) * *this)
        .Normalized(); // normalize so errors don't accumulate
  }

  // Spherical linear interpolation: much faster for small rotations, accurate for large rotations.
  // See FastTo/FromRotationVector
  Quat FastSlerp(const Quat& b, T s) const {
    Vector3<T> delta = (b * this->Inverted()).FastToRotationVector();
    return (FastFromRotationVector(delta * s, false) * *this).Normalized();
  }

  // MERGE_MOBILE_SDK
  // FIXME: This is opposite of Lerp for some reason.  It goes from 1 to 0 instead of 0 to 1.
  // Leaving it as a gift for future generations to deal with.
  Quat Nlerp(const Quat& other, T a) const {
    T sign = (Dot(other) >= 0.0f) ? 1.0f : -1.0f;
    return (*this * sign * a + other * (1 - a)).Normalized();
  }
  // MERGE_MOBILE_SDK

  // Rotate transforms vector in a manner that matches Matrix rotations (counter-clockwise,
  // assuming negative direction of the axis). Standard formula: q(t) * V * q(t)^-1.
  Vector3<T> Rotate(const Vector3<T>& v) const {
    OVR_MATH_ASSERT(IsNormalized()); // If this fires, caller has a quat math bug

    // rv = q * (v,0) * q'
    // Same as rv = v + real * cross(imag,v)*2 + cross(imag, cross(imag,v)*2);

    // uv = 2 * Imag().Cross(v);
    T uvx = T(2) * (y * v.z - z * v.y);
    T uvy = T(2) * (z * v.x - x * v.z);
    T uvz = T(2) * (x * v.y - y * v.x);

    // return v + Real()*uv + Imag().Cross(uv);
    return Vector3<T>(
        v.x + w * uvx + y * uvz - z * uvy,
        v.y + w * uvy + z * uvx - x * uvz,
        v.z + w * uvz + x * uvy - y * uvx);
  }

  // Rotation by inverse of *this
  Vector3<T> InverseRotate(const Vector3<T>& v) const {
    OVR_MATH_ASSERT(IsNormalized()); // If this fires, caller has a quat math bug

    // rv = q' * (v,0) * q
    // Same as rv = v + real * cross(-imag,v)*2 + cross(-imag, cross(-imag,v)*2);
    //      or rv = v - real * cross(imag,v)*2 + cross(imag, cross(imag,v)*2);

    // uv = 2 * Imag().Cross(v);
    T uvx = T(2) * (y * v.z - z * v.y);
    T uvy = T(2) * (z * v.x - x * v.z);
    T uvz = T(2) * (x * v.y - y * v.x);

    // return v - Real()*uv + Imag().Cross(uv);
    return Vector3<T>(
        v.x - w * uvx + y * uvz - z * uvy,
        v.y - w * uvy + z * uvx - x * uvz,
        v.z - w * uvz + x * uvy - y * uvx);
  }

  // Inversed quaternion rotates in the opposite direction.
  Quat Inverted() const {
    return Quat(-x, -y, -z, w);
  }

  Quat Inverse() const {
    return Quat(-x, -y, -z, w);
  }

  // Sets this quaternion to the one rotates in the opposite direction.
  void Invert() {
    *this = Quat(-x, -y, -z, w);
  }

  // Time integration of constant angular velocity over dt
  Quat TimeIntegrate(const Vector3<T>& angularVelocity, T dt) const {
    // solution is: this * exp( omega*dt/2 ); FromRotationVector(v) gives exp(v*.5).
    return (*this * FastFromRotationVector(angularVelocity * dt, false)).Normalized();
  }

  // Time integration of constant angular acceleration and velocity over dt
  // These are the first two terms of the "Magnus expansion" of the solution
  //
  //   o = o * exp( W=(W1 + W2 + W3+...) * 0.5 );
  //
  //  omega1 = (omega + omegaDot*dt)
  //  W1 = (omega + omega1)*dt/2
  //  W2 = cross(omega, omega1)/12*dt^2 % (= -cross(omega_dot, omega)/12*dt^3)
  // Terms 3 and beyond are vanishingly small:
  //  W3 = cross(omega_dot, cross(omega_dot, omega))/240*dt^5
  //
  Quat TimeIntegrate(const Vector3<T>& angularVelocity, const Vector3<T>& angularAcceleration, T dt)
      const {
    const Vector3<T>& omega = angularVelocity;
    const Vector3<T>& omegaDot = angularAcceleration;

    Vector3<T> omega1 = (omega + omegaDot * dt);
    Vector3<T> W = ((omega + omega1) + omega.Cross(omega1) * (dt / T(6))) * (dt / T(2));

    // FromRotationVector(v) is exp(v*.5)
    return (*this * FastFromRotationVector(W, false)).Normalized();
  }

  // Decompose rotation into three rotations:
  // roll radians about Z axis, then pitch radians about X axis, then yaw radians about Y axis.
  // Call with nullptr if a return value is not needed.
  void GetYawPitchRoll(T* yaw, T* pitch, T* roll) const {
    return GetEulerAngles<Axis_Y, Axis_X, Axis_Z, Rotate_CCW, Handed_R>(yaw, pitch, roll);
  }

  // GetEulerAngles extracts Euler angles from the quaternion, in the specified order of
  // axis rotations and the specified coordinate system. Right-handed coordinate system
  // is the default, with CCW rotations while looking in the negative axis direction.
  // Here a,b,c, are the Yaw/Pitch/Roll angles to be returned.
  // Rotation order is c, b, a:
  // rotation c around axis A3
  // is followed by rotation b around axis A2
  // is followed by rotation a around axis A1
  // rotations are CCW or CW (D) in LH or RH coordinate system (S)
  //
  template <Axis A1, Axis A2, Axis A3, RotateDirection D, HandedSystem S>
  void GetEulerAngles(T* a, T* b, T* c) const {
    OVR_MATH_ASSERT(IsNormalized()); // If this fires, caller has a quat math bug
    OVR_MATH_STATIC_ASSERT(
        (A1 != A2) && (A2 != A3) && (A1 != A3), "(A1 != A2) && (A2 != A3) && (A1 != A3)");

    T Q[3] = {x, y, z}; // Quaternion components x,y,z

    T ww = w * w;
    T Q11 = Q[A1] * Q[A1];
    T Q22 = Q[A2] * Q[A2];
    T Q33 = Q[A3] * Q[A3];

    T psign = T(-1);
    // Determine whether even permutation
    if (((A1 + 1) % 3 == A2) && ((A2 + 1) % 3 == A3))
      psign = T(1);

    T s2 = psign * T(2) * (psign * w * Q[A2] + Q[A1] * Q[A3]);

    T singularityRadius = Math<T>::SingularityRadius();
    if (s2 < T(-1) + singularityRadius) { // South pole singularity
      if (a)
        *a = T(0);
      if (b)
        *b = -S * D * ((T)MATH_DOUBLE_PIOVER2);
      if (c)
        *c = S * D * atan2(T(2) * (psign * Q[A1] * Q[A2] + w * Q[A3]), ww + Q22 - Q11 - Q33);
    } else if (s2 > T(1) - singularityRadius) { // North pole singularity
      if (a)
        *a = T(0);
      if (b)
        *b = S * D * ((T)MATH_DOUBLE_PIOVER2);
      if (c)
        *c = S * D * atan2(T(2) * (psign * Q[A1] * Q[A2] + w * Q[A3]), ww + Q22 - Q11 - Q33);
    } else {
      if (a)
        *a = -S * D * atan2(T(-2) * (w * Q[A1] - psign * Q[A2] * Q[A3]), ww + Q33 - Q11 - Q22);
      if (b)
        *b = S * D * asin(s2);
      if (c)
        *c = S * D * atan2(T(2) * (w * Q[A3] - psign * Q[A1] * Q[A2]), ww + Q11 - Q22 - Q33);
    }
  }

  template <Axis A1, Axis A2, Axis A3, RotateDirection D>
  void GetEulerAngles(T* a, T* b, T* c) const {
    GetEulerAngles<A1, A2, A3, D, Handed_R>(a, b, c);
  }

  template <Axis A1, Axis A2, Axis A3>
  void GetEulerAngles(T* a, T* b, T* c) const {
    GetEulerAngles<A1, A2, A3, Rotate_CCW, Handed_R>(a, b, c);
  }

  // GetEulerAnglesABA extracts Euler angles from the quaternion, in the specified order of
  // axis rotations and the specified coordinate system. Right-handed coordinate system
  // is the default, with CCW rotations while looking in the negative axis direction.
  // Here a,b,c, are the Yaw/Pitch/Roll angles to be returned.
  // rotation a around axis A1
  // is followed by rotation b around axis A2
  // is followed by rotation c around axis A1
  // Rotations are CCW or CW (D) in LH or RH coordinate system (S)
  template <Axis A1, Axis A2, RotateDirection D, HandedSystem S>
  void GetEulerAnglesABA(T* a, T* b, T* c) const {
    OVR_MATH_ASSERT(IsNormalized()); // If this fires, caller has a quat math bug
    OVR_MATH_STATIC_ASSERT(A1 != A2, "A1 != A2");

    T Q[3] = {x, y, z}; // Quaternion components

    // Determine the missing axis that was not supplied
    int m = 3 - A1 - A2;

    T ww = w * w;
    T Q11 = Q[A1] * Q[A1];
    T Q22 = Q[A2] * Q[A2];
    T Qmm = Q[m] * Q[m];

    T psign = T(-1);
    if ((A1 + 1) % 3 == A2) // Determine whether even permutation
    {
      psign = T(1);
    }

    T c2 = ww + Q11 - Q22 - Qmm;
    T singularityRadius = Math<T>::SingularityRadius();
    if (c2 < T(-1) + singularityRadius) { // South pole singularity
      if (a)
        *a = T(0);
      if (b)
        *b = S * D * ((T)MATH_DOUBLE_PI);
      if (c)
        *c = S * D * atan2(T(2) * (w * Q[A1] - psign * Q[A2] * Q[m]), ww + Q22 - Q11 - Qmm);
    } else if (c2 > T(1) - singularityRadius) { // North pole singularity
      if (a)
        *a = T(0);
      if (b)
        *b = T(0);
      if (c)
        *c = S * D * atan2(T(2) * (w * Q[A1] - psign * Q[A2] * Q[m]), ww + Q22 - Q11 - Qmm);
    } else {
      if (a)
        *a = S * D * atan2(psign * w * Q[m] + Q[A1] * Q[A2], w * Q[A2] - psign * Q[A1] * Q[m]);
      if (b)
        *b = S * D * acos(c2);
      if (c)
        *c = S * D * atan2(-psign * w * Q[m] + Q[A1] * Q[A2], w * Q[A2] + psign * Q[A1] * Q[m]);
    }
  }

  bool IsNan() const {
    return !isfinite(x + y + z + w);
  }
  bool IsFinite() const {
    return isfinite(x + y + z + w);
  }
};

typedef Quat<float> Quatf;
typedef Quat<double> Quatd;

OVR_MATH_STATIC_ASSERT((sizeof(Quatf) == 4 * sizeof(float)), "sizeof(Quatf) failure");
OVR_MATH_STATIC_ASSERT((sizeof(Quatd) == 4 * sizeof(double)), "sizeof(Quatd) failure");

//-------------------------------------------------------------------------------------
// ***** Pose
//
// Position and orientation combined.
//
// This structure needs to be the same size and layout on 32-bit and 64-bit arch.
// Update OVR_PadCheck.cpp when updating this object.
template <class T>
class Pose {
 public:
  typedef typename CompatibleTypes<Pose<T>>::Type CompatibleType;

  Pose() {}
  Pose(const Quat<T>& orientation, const Vector3<T>& pos)
      : Rotation(orientation), Translation(pos) {}
  Pose(const Pose& s) : Rotation(s.Rotation), Translation(s.Translation) {}
  Pose(const Matrix3<T>& R, const Vector3<T>& t) : Rotation((Quat<T>)R), Translation(t) {}
  Pose(const CompatibleType& s) : Rotation(s.Orientation), Translation(s.Position) {}

  explicit Pose(const Pose<typename Math<T>::OtherFloatType>& s)
      : Rotation(s.Rotation), Translation(s.Translation) {
    // Ensure normalized rotation if converting from float to double
    if (sizeof(T) > sizeof(typename Math<T>::OtherFloatType))
      Rotation.Normalize();
  }

  static Pose Identity() {
    return Pose(Quat<T>(0, 0, 0, 1), Vector3<T>(0, 0, 0));
  }

  void SetIdentity() {
    Rotation = Quat<T>(0, 0, 0, 1);
    Translation = Vector3<T>(0, 0, 0);
  }

  // used to make things obviously broken if someone tries to use the value
  void SetInvalid() {
    Rotation = Quat<T>(NAN, NAN, NAN, NAN);
    Translation = Vector3<T>(NAN, NAN, NAN);
  }

  bool IsEqual(const Pose& b, T tolerance = Math<T>::Tolerance()) const {
    return Translation.IsEqual(b.Translation, tolerance) && Rotation.IsEqual(b.Rotation, tolerance);
  }

  bool IsEqualMatchHemisphere(const Pose& b, T tolerance = Math<T>::Tolerance()) const {
    return Translation.IsEqual(b.Translation, tolerance) &&
        Rotation.IsEqualMatchHemisphere(b.Rotation, tolerance);
  }

  operator typename CompatibleTypes<Pose<T>>::Type() const {
    typename CompatibleTypes<Pose<T>>::Type result;
    result.Orientation = Rotation;
    result.Position = Translation;
    return result;
  }

  Quat<T> Rotation;
  Vector3<T> Translation;

  OVR_MATH_STATIC_ASSERT(
      (sizeof(T) == sizeof(double) || sizeof(T) == sizeof(float)),
      "(sizeof(T) == sizeof(double) || sizeof(T) == sizeof(float))");

  void ToArray(T* arr) const {
    T temp[7] = {Rotation.x,
                 Rotation.y,
                 Rotation.z,
                 Rotation.w,
                 Translation.x,
                 Translation.y,
                 Translation.z};
    for (int i = 0; i < 7; i++)
      arr[i] = temp[i];
  }

  static Pose<T> FromArray(const T* v) {
    Quat<T> rotation(v[0], v[1], v[2], v[3]);
    Vector3<T> translation(v[4], v[5], v[6]);
    // Ensure rotation is normalized, in case it was originally a float, stored in a .json file,
    // etc.
    return Pose<T>(rotation.Normalized(), translation);
  }

  Vector3<T> Rotate(const Vector3<T>& v) const {
    return Rotation.Rotate(v);
  }

  Vector3<T> InverseRotate(const Vector3<T>& v) const {
    return Rotation.InverseRotate(v);
  }

  Vector3<T> Translate(const Vector3<T>& v) const {
    return v + Translation;
  }

  Vector3<T> Transform(const Vector3<T>& v) const {
    return Rotate(v) + Translation;
  }

  Vector3<T> InverseTransform(const Vector3<T>& v) const {
    return InverseRotate(v - Translation);
  }

  Vector3<T> TransformNormal(const Vector3<T>& v) const {
    return Rotate(v);
  }

  Vector3<T> InverseTransformNormal(const Vector3<T>& v) const {
    return InverseRotate(v);
  }

  Vector3<T> Apply(const Vector3<T>& v) const {
    return Transform(v);
  }

  Pose operator*(const Pose& other) const {
    return Pose(Rotation * other.Rotation, Apply(other.Translation));
  }

  Pose Inverted() const {
    Quat<T> inv = Rotation.Inverted();
    return Pose(inv, inv.Rotate(-Translation));
  }

  // Interpolation between two poses: translation is interpolated with Lerp(),
  // and rotations are interpolated with Slerp().
  Pose Lerp(const Pose& b, T s) const {
    return Pose(Rotation.Slerp(b.Rotation, s), Translation.Lerp(b.Translation, s));
  }

  // Similar to Lerp above, except faster in case of small rotation differences.  See
  // Quat<T>::FastSlerp.
  Pose FastLerp(const Pose& b, T s) const {
    return Pose(Rotation.FastSlerp(b.Rotation, s), Translation.Lerp(b.Translation, s));
  }

  Pose TimeIntegrate(const Vector3<T>& linearVelocity, const Vector3<T>& angularVelocity, T dt)
      const {
    return Pose(
        (Rotation * Quat<T>::FastFromRotationVector(angularVelocity * dt, false)).Normalized(),
        Translation + linearVelocity * dt);
  }

  Pose TimeIntegrate(
      const Vector3<T>& linearVelocity,
      const Vector3<T>& linearAcceleration,
      const Vector3<T>& angularVelocity,
      const Vector3<T>& angularAcceleration,
      T dt) const {
    return Pose(
        Rotation.TimeIntegrate(angularVelocity, angularAcceleration, dt),
        Translation + linearVelocity * dt + linearAcceleration * dt * dt * T(0.5));
  }

  Pose Normalized() const {
    return Pose(Rotation.Normalized(), Translation);
  }
  void Normalize() {
    Rotation.Normalize();
  }

  bool IsNan() const {
    return Translation.IsNan() || Rotation.IsNan();
  }
  bool IsFinite() const {
    return Translation.IsFinite() && Rotation.IsFinite();
  }
};

typedef Pose<float> Posef;
typedef Pose<double> Posed;

OVR_MATH_STATIC_ASSERT(
    (sizeof(Posed) == sizeof(Quatd) + sizeof(Vector3d)),
    "sizeof(Posed) failure");
OVR_MATH_STATIC_ASSERT(
    (sizeof(Posef) == sizeof(Quatf) + sizeof(Vector3f)),
    "sizeof(Posef) failure");

//-------------------------------------------------------------------------------------
// ***** Matrix4
//
// Matrix4 is a 4x4 matrix used for 3d transformations and projections.
// Translation stored in the last column.
// The matrix is stored in row-major order in memory, meaning that values
// of the first row are stored before the next one.
//
// The arrangement of the matrix is chosen to be in Right-Handed
// coordinate system and counterclockwise rotations when looking down
// the axis
//
// Transformation Order:
//   - Transformations are applied from right to left, so the expression
//     M1 * M2 * M3 * V means that the vector V is transformed by M3 first,
//     followed by M2 and M1.
//
// Coordinate system: Right Handed
//
// Rotations: Counterclockwise when looking down the axis. All angles are in radians.
//
//  | sx   01   02   tx |    // First column  (sx, 10, 20): Axis X basis vector.
//  | 10   sy   12   ty |    // Second column (01, sy, 21): Axis Y basis vector.
//  | 20   21   sz   tz |    // Third columnt (02, 12, sz): Axis Z basis vector.
//  | 30   31   32   33 |
//
//  The basis vectors are first three columns.

template <class T>
class Matrix4 {
 public:
  typedef T ElementType;
  static const size_t Dimension = 4;

  T M[4][4];

  enum NoInitType { NoInit };

  // Construct with no memory initialization.
  Matrix4(NoInitType) {}

  // By default, we construct identity matrix.
  Matrix4() {
    M[0][0] = M[1][1] = M[2][2] = M[3][3] = T(1);
    M[0][1] = M[1][0] = M[2][3] = M[3][1] = T(0);
    M[0][2] = M[1][2] = M[2][0] = M[3][2] = T(0);
    M[0][3] = M[1][3] = M[2][1] = M[3][0] = T(0);
  }

  Matrix4(
      T m11,
      T m12,
      T m13,
      T m14,
      T m21,
      T m22,
      T m23,
      T m24,
      T m31,
      T m32,
      T m33,
      T m34,
      T m41,
      T m42,
      T m43,
      T m44) {
    M[0][0] = m11;
    M[0][1] = m12;
    M[0][2] = m13;
    M[0][3] = m14;
    M[1][0] = m21;
    M[1][1] = m22;
    M[1][2] = m23;
    M[1][3] = m24;
    M[2][0] = m31;
    M[2][1] = m32;
    M[2][2] = m33;
    M[2][3] = m34;
    M[3][0] = m41;
    M[3][1] = m42;
    M[3][2] = m43;
    M[3][3] = m44;
  }

  Matrix4(T m11, T m12, T m13, T m21, T m22, T m23, T m31, T m32, T m33) {
    M[0][0] = m11;
    M[0][1] = m12;
    M[0][2] = m13;
    M[0][3] = T(0);
    M[1][0] = m21;
    M[1][1] = m22;
    M[1][2] = m23;
    M[1][3] = T(0);
    M[2][0] = m31;
    M[2][1] = m32;
    M[2][2] = m33;
    M[2][3] = T(0);
    M[3][0] = T(0);
    M[3][1] = T(0);
    M[3][2] = T(0);
    M[3][3] = T(1);
  }

  explicit Matrix4(const Matrix3<T>& m) {
    M[0][0] = m.M[0][0];
    M[0][1] = m.M[0][1];
    M[0][2] = m.M[0][2];
    M[0][3] = T(0);
    M[1][0] = m.M[1][0];
    M[1][1] = m.M[1][1];
    M[1][2] = m.M[1][2];
    M[1][3] = T(0);
    M[2][0] = m.M[2][0];
    M[2][1] = m.M[2][1];
    M[2][2] = m.M[2][2];
    M[2][3] = T(0);
    M[3][0] = T(0);
    M[3][1] = T(0);
    M[3][2] = T(0);
    M[3][3] = T(1);
  }

  explicit Matrix4(const Quat<T>& q) {
    OVR_MATH_ASSERT(q.IsNormalized()); // If this fires, caller has a quat math bug
    T ww = q.w * q.w;
    T xx = q.x * q.x;
    T yy = q.y * q.y;
    T zz = q.z * q.z;

    M[0][0] = ww + xx - yy - zz;
    M[0][1] = 2 * (q.x * q.y - q.w * q.z);
    M[0][2] = 2 * (q.x * q.z + q.w * q.y);
    M[0][3] = T(0);
    M[1][0] = 2 * (q.x * q.y + q.w * q.z);
    M[1][1] = ww - xx + yy - zz;
    M[1][2] = 2 * (q.y * q.z - q.w * q.x);
    M[1][3] = T(0);
    M[2][0] = 2 * (q.x * q.z - q.w * q.y);
    M[2][1] = 2 * (q.y * q.z + q.w * q.x);
    M[2][2] = ww - xx - yy + zz;
    M[2][3] = T(0);
    M[3][0] = T(0);
    M[3][1] = T(0);
    M[3][2] = T(0);
    M[3][3] = T(1);
  }

  explicit Matrix4(const Pose<T>& p) {
    Matrix4 result(p.Rotation);
    result.SetTranslation(p.Translation);
    *this = result;
  }

  // C-interop support
  explicit Matrix4(const Matrix4<typename Math<T>::OtherFloatType>& src) {
    for (int i = 0; i < 4; i++)
      for (int j = 0; j < 4; j++)
        M[i][j] = (T)src.M[i][j];
  }

  // C-interop support.
  Matrix4(const typename CompatibleTypes<Matrix4<T>>::Type& s) {
    OVR_MATH_STATIC_ASSERT(sizeof(s) == sizeof(Matrix4), "sizeof(s) == sizeof(Matrix4)");
    memcpy(M, s.M, sizeof(M));
  }

  operator typename CompatibleTypes<Matrix4<T>>::Type() const {
    typename CompatibleTypes<Matrix4<T>>::Type result;
    OVR_MATH_STATIC_ASSERT(sizeof(result) == sizeof(Matrix4), "sizeof(result) == sizeof(Matrix4)");
    memcpy(result.M, M, sizeof(M));
    return result;
  }

  void ToString(char* dest, size_t destsize) const {
    size_t pos = 0;
    for (int r = 0; r < 4; r++) {
      for (int c = 0; c < 4; c++) {
        pos += OVRMath_sprintf(dest + pos, destsize - pos, "%g ", M[r][c]);
      }
    }
  }

  static Matrix4 FromString(const char* src) {
    Matrix4 result;
    if (src) {
      for (int r = 0; r < 4; r++) {
        for (int c = 0; c < 4; c++) {
          result.M[r][c] = (T)atof(src);
          while (*src && *src != ' ') {
            src++;
          }
          while (*src && *src == ' ') {
            src++;
          }
        }
      }
    }
    return result;
  }

  static Matrix4 Identity() {
    return Matrix4();
  }

  void SetIdentity() {
    M[0][0] = M[1][1] = M[2][2] = M[3][3] = T(1);
    M[0][1] = M[1][0] = M[2][3] = M[3][1] = T(0);
    M[0][2] = M[1][2] = M[2][0] = M[3][2] = T(0);
    M[0][3] = M[1][3] = M[2][1] = M[3][0] = T(0);
  }

  void SetXBasis(const Vector3<T>& v) {
    M[0][0] = v.x;
    M[1][0] = v.y;
    M[2][0] = v.z;
  }
  Vector3<T> GetXBasis() const {
    return Vector3<T>(M[0][0], M[1][0], M[2][0]);
  }

  void SetYBasis(const Vector3<T>& v) {
    M[0][1] = v.x;
    M[1][1] = v.y;
    M[2][1] = v.z;
  }
  Vector3<T> GetYBasis() const {
    return Vector3<T>(M[0][1], M[1][1], M[2][1]);
  }

  void SetZBasis(const Vector3<T>& v) {
    M[0][2] = v.x;
    M[1][2] = v.y;
    M[2][2] = v.z;
  }
  Vector3<T> GetZBasis() const {
    return Vector3<T>(M[0][2], M[1][2], M[2][2]);
  }

  bool operator==(const Matrix4& b) const {
    bool isEqual = true;
    for (int i = 0; i < 4; i++)
      for (int j = 0; j < 4; j++)
        isEqual &= (M[i][j] == b.M[i][j]);

    return isEqual;
  }

  Matrix4 operator+(const Matrix4& b) const {
    Matrix4 result(*this);
    result += b;
    return result;
  }

  Matrix4& operator+=(const Matrix4& b) {
    for (int i = 0; i < 4; i++)
      for (int j = 0; j < 4; j++)
        M[i][j] += b.M[i][j];
    return *this;
  }

  Matrix4 operator-(const Matrix4& b) const {
    Matrix4 result(*this);
    result -= b;
    return result;
  }

  Matrix4& operator-=(const Matrix4& b) {
    for (int i = 0; i < 4; i++)
      for (int j = 0; j < 4; j++)
        M[i][j] -= b.M[i][j];
    return *this;
  }

  // Multiplies two matrices into destination with minimum copying.
  static Matrix4& Multiply(Matrix4* d, const Matrix4& a, const Matrix4& b) {
    OVR_MATH_ASSERT((d != &a) && (d != &b));
    int i = 0;
    do {
      d->M[i][0] = a.M[i][0] * b.M[0][0] + a.M[i][1] * b.M[1][0] + a.M[i][2] * b.M[2][0] +
          a.M[i][3] * b.M[3][0];
      d->M[i][1] = a.M[i][0] * b.M[0][1] + a.M[i][1] * b.M[1][1] + a.M[i][2] * b.M[2][1] +
          a.M[i][3] * b.M[3][1];
      d->M[i][2] = a.M[i][0] * b.M[0][2] + a.M[i][1] * b.M[1][2] + a.M[i][2] * b.M[2][2] +
          a.M[i][3] * b.M[3][2];
      d->M[i][3] = a.M[i][0] * b.M[0][3] + a.M[i][1] * b.M[1][3] + a.M[i][2] * b.M[2][3] +
          a.M[i][3] * b.M[3][3];
    } while ((++i) < 4);

    return *d;
  }

  Matrix4 operator*(const Matrix4& b) const {
    Matrix4 result(Matrix4::NoInit);
    Multiply(&result, *this, b);
    return result;
  }

  Matrix4& operator*=(const Matrix4& b) {
    return Multiply(this, Matrix4(*this), b);
  }

  Matrix4 operator*(T s) const {
    Matrix4 result(*this);
    result *= s;
    return result;
  }

  Matrix4& operator*=(T s) {
    for (int i = 0; i < 4; i++)
      for (int j = 0; j < 4; j++)
        M[i][j] *= s;
    return *this;
  }

  Matrix4 operator/(T s) const {
    Matrix4 result(*this);
    result /= s;
    return result;
  }

  Matrix4& operator/=(T s) {
    for (int i = 0; i < 4; i++)
      for (int j = 0; j < 4; j++)
        M[i][j] /= s;
    return *this;
  }

  T operator()(int i, int j) const {
    return M[i][j];
  }
  T& operator()(int i, int j) {
    return M[i][j];
  }

  Vector4<T> operator*(const Vector4<T>& b) const {
    return Transform(b);
  }

  Vector3<T> Transform(const Vector3<T>& v) const {
    const T rcpW = T(1) / (M[3][0] * v.x + M[3][1] * v.y + M[3][2] * v.z + M[3][3]);
    return Vector3<T>(
        (M[0][0] * v.x + M[0][1] * v.y + M[0][2] * v.z + M[0][3]) * rcpW,
        (M[1][0] * v.x + M[1][1] * v.y + M[1][2] * v.z + M[1][3]) * rcpW,
        (M[2][0] * v.x + M[2][1] * v.y + M[2][2] * v.z + M[2][3]) * rcpW);
  }

  Vector4<T> Transform(const Vector4<T>& v) const {
    return Vector4<T>(
        M[0][0] * v.x + M[0][1] * v.y + M[0][2] * v.z + M[0][3] * v.w,
        M[1][0] * v.x + M[1][1] * v.y + M[1][2] * v.z + M[1][3] * v.w,
        M[2][0] * v.x + M[2][1] * v.y + M[2][2] * v.z + M[2][3] * v.w,
        M[3][0] * v.x + M[3][1] * v.y + M[3][2] * v.z + M[3][3] * v.w);
  }

  Matrix4 Transposed() const {
    return Matrix4(
        M[0][0],
        M[1][0],
        M[2][0],
        M[3][0],
        M[0][1],
        M[1][1],
        M[2][1],
        M[3][1],
        M[0][2],
        M[1][2],
        M[2][2],
        M[3][2],
        M[0][3],
        M[1][3],
        M[2][3],
        M[3][3]);
  }

  void Transpose() {
    *this = Transposed();
  }

  T SubDet(const size_t* rows, const size_t* cols) const {
    return M[rows[0]][cols[0]] *
        (M[rows[1]][cols[1]] * M[rows[2]][cols[2]] - M[rows[1]][cols[2]] * M[rows[2]][cols[1]]) -
        M[rows[0]][cols[1]] *
        (M[rows[1]][cols[0]] * M[rows[2]][cols[2]] - M[rows[1]][cols[2]] * M[rows[2]][cols[0]]) +
        M[rows[0]][cols[2]] *
        (M[rows[1]][cols[0]] * M[rows[2]][cols[1]] - M[rows[1]][cols[1]] * M[rows[2]][cols[0]]);
  }

  T Cofactor(size_t I, size_t J) const {
    const size_t indices[4][3] = {{1, 2, 3}, {0, 2, 3}, {0, 1, 3}, {0, 1, 2}};
    return ((I + J) & 1) ? -SubDet(indices[I], indices[J]) : SubDet(indices[I], indices[J]);
  }

  T Determinant() const {
    return M[0][0] * Cofactor(0, 0) + M[0][1] * Cofactor(0, 1) + M[0][2] * Cofactor(0, 2) +
        M[0][3] * Cofactor(0, 3);
  }

  Matrix4 Adjugated() const {
    return Matrix4(
        Cofactor(0, 0),
        Cofactor(1, 0),
        Cofactor(2, 0),
        Cofactor(3, 0),
        Cofactor(0, 1),
        Cofactor(1, 1),
        Cofactor(2, 1),
        Cofactor(3, 1),
        Cofactor(0, 2),
        Cofactor(1, 2),
        Cofactor(2, 2),
        Cofactor(3, 2),
        Cofactor(0, 3),
        Cofactor(1, 3),
        Cofactor(2, 3),
        Cofactor(3, 3));
  }

  Matrix4 Inverted() const {
    T det = Determinant();
    OVR_MATH_ASSERT(det != 0);
    return Adjugated() * (T(1) / det);
  }

  void Invert() {
    *this = Inverted();
  }

  // This is more efficient than general inverse, but ONLY works
  // correctly if it is a homogeneous transform matrix (rot + trans)
  Matrix4 InvertedHomogeneousTransform() const {
    // Make the inverse rotation matrix
    Matrix4 rinv = this->Transposed();
    rinv.M[3][0] = rinv.M[3][1] = rinv.M[3][2] = T(0);
    // Make the inverse translation matrix
    Vector3<T> tvinv(-M[0][3], -M[1][3], -M[2][3]);
    Matrix4 tinv = Matrix4::Translation(tvinv);
    return rinv * tinv; // "untranslate", then "unrotate"
  }

  // This is more efficient than general inverse, but ONLY works
  // correctly if it is a homogeneous transform matrix (rot + trans)
  void InvertHomogeneousTransform() {
    *this = InvertedHomogeneousTransform();
  }

  // Matrix to Euler Angles conversion
  // a,b,c, are the YawPitchRoll angles to be returned
  // rotation a around axis A1
  // is followed by rotation b around axis A2
  // is followed by rotation c around axis A3
  // rotations are CCW or CW (D) in LH or RH coordinate system (S)
  template <Axis A1, Axis A2, Axis A3, RotateDirection D, HandedSystem S>
  void ToEulerAngles(T* a, T* b, T* c) const {
    OVR_MATH_STATIC_ASSERT(
        (A1 != A2) && (A2 != A3) && (A1 != A3), "(A1 != A2) && (A2 != A3) && (A1 != A3)");

    T psign = T(-1);
    if (((A1 + 1) % 3 == A2) && ((A2 + 1) % 3 == A3)) // Determine whether even permutation
      psign = T(1);

    T pm = psign * M[A1][A3];
    T singularityRadius = Math<T>::SingularityRadius();
    if (pm < T(-1) + singularityRadius) { // South pole singularity
      *a = T(0);
      *b = -S * D * ((T)MATH_DOUBLE_PIOVER2);
      *c = S * D * atan2(psign * M[A2][A1], M[A2][A2]);
    } else if (pm > T(1) - singularityRadius) { // North pole singularity
      *a = T(0);
      *b = S * D * ((T)MATH_DOUBLE_PIOVER2);
      *c = S * D * atan2(psign * M[A2][A1], M[A2][A2]);
    } else { // Normal case (nonsingular)
      *a = S * D * atan2(-psign * M[A2][A3], M[A3][A3]);
      *b = S * D * asin(pm);
      *c = S * D * atan2(-psign * M[A1][A2], M[A1][A1]);
    }
  }

  // Matrix to Euler Angles conversion
  // a,b,c, are the YawPitchRoll angles to be returned
  // rotation a around axis A1
  // is followed by rotation b around axis A2
  // is followed by rotation c around axis A1
  // rotations are CCW or CW (D) in LH or RH coordinate system (S)
  template <Axis A1, Axis A2, RotateDirection D, HandedSystem S>
  void ToEulerAnglesABA(T* a, T* b, T* c) const {
    OVR_MATH_STATIC_ASSERT(A1 != A2, "A1 != A2");

    // Determine the axis that was not supplied
    int m = 3 - A1 - A2;

    T psign = T(-1);
    if ((A1 + 1) % 3 == A2) // Determine whether even permutation
      psign = T(1);

    T c2 = M[A1][A1];
    T singularityRadius = Math<T>::SingularityRadius();
    if (c2 < T(-1) + singularityRadius) { // South pole singularity
      *a = T(0);
      *b = S * D * ((T)MATH_DOUBLE_PI);
      *c = S * D * atan2(-psign * M[A2][m], M[A2][A2]);
    } else if (c2 > T(1) - singularityRadius) { // North pole singularity
      *a = T(0);
      *b = T(0);
      *c = S * D * atan2(-psign * M[A2][m], M[A2][A2]);
    } else { // Normal case (nonsingular)
      *a = S * D * atan2(M[A2][A1], -psign * M[m][A1]);
      *b = S * D * acos(c2);
      *c = S * D * atan2(M[A1][A2], psign * M[A1][m]);
    }
  }

  // Creates a matrix that converts the vertices from one coordinate system
  // to another.
  static Matrix4 AxisConversion(const WorldAxes& to, const WorldAxes& from) {
    // Holds axis values from the 'to' structure
    int toArray[3] = {to.XAxis, to.YAxis, to.ZAxis};

    // The inverse of the toArray
    int inv[4];
    inv[0] = inv[abs(to.XAxis)] = 0;
    inv[abs(to.YAxis)] = 1;
    inv[abs(to.ZAxis)] = 2;

    Matrix4 m(0, 0, 0, 0, 0, 0, 0, 0, 0);

    // Only three values in the matrix need to be changed to 1 or -1.
    m.M[inv[abs(from.XAxis)]][0] = T(from.XAxis / toArray[inv[abs(from.XAxis)]]);
    m.M[inv[abs(from.YAxis)]][1] = T(from.YAxis / toArray[inv[abs(from.YAxis)]]);
    m.M[inv[abs(from.ZAxis)]][2] = T(from.ZAxis / toArray[inv[abs(from.ZAxis)]]);
    return m;
  }

  // Creates a matrix for translation by vector
  static Matrix4 Translation(const Vector3<T>& v) {
    Matrix4 t;
    t.M[0][3] = v.x;
    t.M[1][3] = v.y;
    t.M[2][3] = v.z;
    return t;
  }

  // Creates a matrix for translation by vector
  static Matrix4 Translation(T x, T y, T z = T(0)) {
    Matrix4 t;
    t.M[0][3] = x;
    t.M[1][3] = y;
    t.M[2][3] = z;
    return t;
  }

  // Sets the translation part
  void SetTranslation(const Vector3<T>& v) {
    M[0][3] = v.x;
    M[1][3] = v.y;
    M[2][3] = v.z;
  }

  Vector3<T> GetTranslation() const {
    return Vector3<T>(M[0][3], M[1][3], M[2][3]);
  }

  // Creates a matrix for scaling by vector
  static Matrix4 Scaling(const Vector3<T>& v) {
    Matrix4 t;
    t.M[0][0] = v.x;
    t.M[1][1] = v.y;
    t.M[2][2] = v.z;
    return t;
  }

  // Creates a matrix for scaling by vector
  static Matrix4 Scaling(T x, T y, T z) {
    Matrix4 t;
    t.M[0][0] = x;
    t.M[1][1] = y;
    t.M[2][2] = z;
    return t;
  }

  // Creates a matrix for scaling by constant
  static Matrix4 Scaling(T s) {
    Matrix4 t;
    t.M[0][0] = s;
    t.M[1][1] = s;
    t.M[2][2] = s;
    return t;
  }

  // Simple L1 distance in R^12
  T Distance(const Matrix4& m2) const {
    T d = fabs(M[0][0] - m2.M[0][0]) + fabs(M[0][1] - m2.M[0][1]);
    d += fabs(M[0][2] - m2.M[0][2]) + fabs(M[0][3] - m2.M[0][3]);
    d += fabs(M[1][0] - m2.M[1][0]) + fabs(M[1][1] - m2.M[1][1]);
    d += fabs(M[1][2] - m2.M[1][2]) + fabs(M[1][3] - m2.M[1][3]);
    d += fabs(M[2][0] - m2.M[2][0]) + fabs(M[2][1] - m2.M[2][1]);
    d += fabs(M[2][2] - m2.M[2][2]) + fabs(M[2][3] - m2.M[2][3]);
    d += fabs(M[3][0] - m2.M[3][0]) + fabs(M[3][1] - m2.M[3][1]);
    d += fabs(M[3][2] - m2.M[3][2]) + fabs(M[3][3] - m2.M[3][3]);
    return d;
  }

  // Creates a rotation matrix rotating around the X axis by 'angle' radians.
  // Just for quick testing.  Not for final API.  Need to remove case.
  static Matrix4 RotationAxis(Axis A, T angle, RotateDirection d, HandedSystem s) {
    T sina = s * d * sin(angle);
    T cosa = cos(angle);

    switch (A) {
      case Axis_X:
        return Matrix4(1, 0, 0, 0, cosa, -sina, 0, sina, cosa);
      case Axis_Y:
        return Matrix4(cosa, 0, sina, 0, 1, 0, -sina, 0, cosa);
      case Axis_Z:
        return Matrix4(cosa, -sina, 0, sina, cosa, 0, 0, 0, 1);
      default:
        return Matrix4();
    }
  }

  // Creates a rotation matrix rotating around the X axis by 'angle' radians.
  // Rotation direction is depends on the coordinate system:
  // RHS (Oculus default): Positive angle values rotate Counter-clockwise (CCW),
  //                        while looking in the negative axis direction. This is the
  //                        same as looking down from positive axis values towards origin.
  // LHS: Positive angle values rotate clock-wise (CW), while looking in the
  //       negative axis direction.
  static Matrix4 RotationX(T angle) {
    T sina = sin(angle);
    T cosa = cos(angle);
    return Matrix4(1, 0, 0, 0, cosa, -sina, 0, sina, cosa);
  }

  // Creates a rotation matrix rotating around the Y axis by 'angle' radians.
  // Rotation direction is depends on the coordinate system:
  //  RHS (Oculus default): Positive angle values rotate Counter-clockwise (CCW),
  //                        while looking in the negative axis direction. This is the
  //                        same as looking down from positive axis values towards origin.
  //  LHS: Positive angle values rotate clock-wise (CW), while looking in the
  //       negative axis direction.
  static Matrix4 RotationY(T angle) {
    T sina = (T)sin(angle);
    T cosa = (T)cos(angle);
    return Matrix4(cosa, 0, sina, 0, 1, 0, -sina, 0, cosa);
  }

  // Creates a rotation matrix rotating around the Z axis by 'angle' radians.
  // Rotation direction is depends on the coordinate system:
  //  RHS (Oculus default): Positive angle values rotate Counter-clockwise (CCW),
  //                        while looking in the negative axis direction. This is the
  //                        same as looking down from positive axis values towards origin.
  //  LHS: Positive angle values rotate clock-wise (CW), while looking in the
  //       negative axis direction.
  static Matrix4 RotationZ(T angle) {
    T sina = sin(angle);
    T cosa = cos(angle);
    return Matrix4(cosa, -sina, 0, sina, cosa, 0, 0, 0, 1);
  }

  // LookAtRH creates a View transformation matrix for right-handed coordinate system.
  // The resulting matrix points camera from 'eye' towards 'at' direction, with 'up'
  // specifying the up vector. The resulting matrix should be used with PerspectiveRH
  // projection.
  static Matrix4 LookAtRH(const Vector3<T>& eye, const Vector3<T>& at, const Vector3<T>& up) {
    Vector3<T> z = (eye - at).Normalized(); // Forward
    Vector3<T> x = up.Cross(z).Normalized(); // Right
    Vector3<T> y = z.Cross(x);

    Matrix4 m(
        x.x,
        x.y,
        x.z,
        -(x.Dot(eye)),
        y.x,
        y.y,
        y.z,
        -(y.Dot(eye)),
        z.x,
        z.y,
        z.z,
        -(z.Dot(eye)),
        0,
        0,
        0,
        1);
    return m;
  }

  // LookAtLH creates a View transformation matrix for left-handed coordinate system.
  // The resulting matrix points camera from 'eye' towards 'at' direction, with 'up'
  // specifying the up vector.
  static Matrix4 LookAtLH(const Vector3<T>& eye, const Vector3<T>& at, const Vector3<T>& up) {
    Vector3<T> z = (at - eye).Normalized(); // Forward
    Vector3<T> x = up.Cross(z).Normalized(); // Right
    Vector3<T> y = z.Cross(x);

    Matrix4 m(
        x.x,
        x.y,
        x.z,
        -(x.Dot(eye)),
        y.x,
        y.y,
        y.z,
        -(y.Dot(eye)),
        z.x,
        z.y,
        z.z,
        -(z.Dot(eye)),
        0,
        0,
        0,
        1);
    return m;
  }

  // PerspectiveRH creates a right-handed perspective projection matrix that can be
  // used with the Oculus sample renderer.
  //  yfov   - Specifies vertical field of view in radians.
  //  aspect - Screen aspect ration, which is usually width/height for square pixels.
  //           Note that xfov = yfov * aspect.
  //  znear  - Absolute value of near Z clipping clipping range.
  //  zfar   - Absolute value of far  Z clipping clipping range (larger then near).
  // Even though RHS usually looks in the direction of negative Z, positive values
  // are expected for znear and zfar.
  static Matrix4 PerspectiveRH(T yfov, T aspect, T znear, T zfar) {
    Matrix4 m;
    T tanHalfFov = (T)tan(yfov * T(0.5));

    m.M[0][0] = T(1) / (aspect * tanHalfFov);
    m.M[1][1] = T(1) / tanHalfFov;
    m.M[2][2] = zfar / (znear - zfar);
    m.M[3][2] = T(-1);
    m.M[2][3] = (zfar * znear) / (znear - zfar);
    m.M[3][3] = T(0);

    // Note: Post-projection matrix result assumes Left-Handed coordinate system,
    //       with Y up, X right and Z forward. This supports positive z-buffer values.
    // This is the case even for RHS coordinate input.
    return m;
  }

  // PerspectiveLH creates a left-handed perspective projection matrix that can be
  // used with the Oculus sample renderer.
  //  yfov   - Specifies vertical field of view in radians.
  //  aspect - Screen aspect ration, which is usually width/height for square pixels.
  //           Note that xfov = yfov * aspect.
  //  znear  - Absolute value of near Z clipping clipping range.
  //  zfar   - Absolute value of far  Z clipping clipping range (larger then near).
  static Matrix4 PerspectiveLH(T yfov, T aspect, T znear, T zfar) {
    Matrix4 m;
    T tanHalfFov = (T)tan(yfov * T(0.5));

    m.M[0][0] = T(1) / (aspect * tanHalfFov);
    m.M[1][1] = T(1) / tanHalfFov;
    // m.M[2][2] = zfar / (znear - zfar);
    m.M[2][2] = zfar / (zfar - znear);
    m.M[3][2] = T(-1);
    m.M[2][3] = (zfar * znear) / (znear - zfar);
    m.M[3][3] = T(0);

    // Note: Post-projection matrix result assumes Left-Handed coordinate system,
    //       with Y up, X right and Z forward. This supports positive z-buffer values.
    // This is the case even for RHS coordinate input.
    return m;
  }

  static Matrix4 Ortho2D(T w, T h) {
    Matrix4 m;
    m.M[0][0] = T(2.0) / w;
    m.M[1][1] = T(-2.0) / h;
    m.M[0][3] = T(-1.0);
    m.M[1][3] = T(1.0);
    m.M[2][2] = T(0);
    return m;
  }
};

typedef Matrix4<float> Matrix4f;
typedef Matrix4<double> Matrix4d;

//-------------------------------------------------------------------------------------
// ***** Matrix3
//
// Matrix3 is a 3x3 matrix used for representing a rotation matrix.
// The matrix is stored in row-major order in memory, meaning that values
// of the first row are stored before the next one.
//
// The arrangement of the matrix is chosen to be in Right-Handed
// coordinate system and counterclockwise rotations when looking down
// the axis
//
// Transformation Order:
//   - Transformations are applied from right to left, so the expression
//     M1 * M2 * M3 * V means that the vector V is transformed by M3 first,
//     followed by M2 and M1.
//
// Coordinate system: Right Handed
//
// Rotations: Counterclockwise when looking down the axis. All angles are in radians.

template <class T>
class Matrix3 {
 public:
  typedef T ElementType;
  static const size_t Dimension = 3;

  T M[3][3];

  enum NoInitType { NoInit };

  // Construct with no memory initialization.
  Matrix3(NoInitType) {}

  // By default, we construct identity matrix.
  Matrix3() {
    M[0][0] = M[1][1] = M[2][2] = T(1);
    M[0][1] = M[1][0] = M[2][0] = T(0);
    M[0][2] = M[1][2] = M[2][1] = T(0);
  }

  Matrix3(T m11, T m12, T m13, T m21, T m22, T m23, T m31, T m32, T m33) {
    M[0][0] = m11;
    M[0][1] = m12;
    M[0][2] = m13;
    M[1][0] = m21;
    M[1][1] = m22;
    M[1][2] = m23;
    M[2][0] = m31;
    M[2][1] = m32;
    M[2][2] = m33;
  }

  // Construction from X, Y, Z basis vectors
  Matrix3(const Vector3<T>& xBasis, const Vector3<T>& yBasis, const Vector3<T>& zBasis) {
    M[0][0] = xBasis.x;
    M[0][1] = yBasis.x;
    M[0][2] = zBasis.x;
    M[1][0] = xBasis.y;
    M[1][1] = yBasis.y;
    M[1][2] = zBasis.y;
    M[2][0] = xBasis.z;
    M[2][1] = yBasis.z;
    M[2][2] = zBasis.z;
  }

  explicit Matrix3(const Quat<T>& q) {
    OVR_MATH_ASSERT(q.IsNormalized()); // If this fires, caller has a quat math bug
    const T tx = q.x + q.x, ty = q.y + q.y, tz = q.z + q.z;
    const T twx = q.w * tx, twy = q.w * ty, twz = q.w * tz;
    const T txx = q.x * tx, txy = q.x * ty, txz = q.x * tz;
    const T tyy = q.y * ty, tyz = q.y * tz, tzz = q.z * tz;
    M[0][0] = T(1) - (tyy + tzz);
    M[0][1] = txy - twz;
    M[0][2] = txz + twy;
    M[1][0] = txy + twz;
    M[1][1] = T(1) - (txx + tzz);
    M[1][2] = tyz - twx;
    M[2][0] = txz - twy;
    M[2][1] = tyz + twx;
    M[2][2] = T(1) - (txx + tyy);
  }

  inline explicit Matrix3(T s) {
    M[0][0] = M[1][1] = M[2][2] = s;
    M[0][1] = M[0][2] = M[1][0] = M[1][2] = M[2][0] = M[2][1] = T(0);
  }

  Matrix3(T m11, T m22, T m33) {
    M[0][0] = m11;
    M[0][1] = T(0);
    M[0][2] = T(0);
    M[1][0] = T(0);
    M[1][1] = m22;
    M[1][2] = T(0);
    M[2][0] = T(0);
    M[2][1] = T(0);
    M[2][2] = m33;
  }

  explicit Matrix3(const Matrix3<typename Math<T>::OtherFloatType>& src) {
    for (int i = 0; i < 3; i++)
      for (int j = 0; j < 3; j++)
        M[i][j] = (T)src.M[i][j];
  }

  // C-interop support.
  Matrix3(const typename CompatibleTypes<Matrix3<T>>::Type& s) {
    OVR_MATH_STATIC_ASSERT(sizeof(s) == sizeof(Matrix3), "sizeof(s) == sizeof(Matrix3)");
    memcpy(M, s.M, sizeof(M));
  }

  operator const typename CompatibleTypes<Matrix3<T>>::Type() const {
    typename CompatibleTypes<Matrix3<T>>::Type result;
    OVR_MATH_STATIC_ASSERT(sizeof(result) == sizeof(Matrix3), "sizeof(result) == sizeof(Matrix3)");
    memcpy(result.M, M, sizeof(M));
    return result;
  }

  T operator()(int i, int j) const {
    return M[i][j];
  }
  T& operator()(int i, int j) {
    return M[i][j];
  }

  void ToString(char* dest, size_t destsize) const {
    size_t pos = 0;
    for (int r = 0; r < 3; r++) {
      for (int c = 0; c < 3; c++)
        pos += OVRMath_sprintf(dest + pos, destsize - pos, "%g ", M[r][c]);
    }
  }

  static Matrix3 FromString(const char* src) {
    Matrix3 result;
    if (src) {
      for (int r = 0; r < 3; r++) {
        for (int c = 0; c < 3; c++) {
          result.M[r][c] = (T)atof(src);
          while (*src && *src != ' ')
            src++;
          while (*src && *src == ' ')
            src++;
        }
      }
    }
    return result;
  }

  static Matrix3 Identity() {
    return Matrix3();
  }

  void SetIdentity() {
    M[0][0] = M[1][1] = M[2][2] = T(1);
    M[0][1] = M[1][0] = M[2][0] = T(0);
    M[0][2] = M[1][2] = M[2][1] = T(0);
  }

  static Matrix3 Diagonal(T m00, T m11, T m22) {
    return Matrix3(m00, 0, 0, 0, m11, 0, 0, 0, m22);
  }
  static Matrix3 Diagonal(const Vector3<T>& v) {
    return Diagonal(v.x, v.y, v.z);
  }

  T Trace() const {
    return M[0][0] + M[1][1] + M[2][2];
  }

  bool operator==(const Matrix3& b) const {
    bool isEqual = true;
    for (int i = 0; i < 3; i++) {
      for (int j = 0; j < 3; j++)
        isEqual &= (M[i][j] == b.M[i][j]);
    }

    return isEqual;
  }

  Matrix3 operator+(const Matrix3& b) const {
    Matrix3<T> result(*this);
    result += b;
    return result;
  }

  Matrix3& operator+=(const Matrix3& b) {
    for (int i = 0; i < 3; i++)
      for (int j = 0; j < 3; j++)
        M[i][j] += b.M[i][j];
    return *this;
  }

  void operator=(const Matrix3& b) {
    for (int i = 0; i < 3; i++)
      for (int j = 0; j < 3; j++)
        M[i][j] = b.M[i][j];
  }

  Matrix3 operator-(const Matrix3& b) const {
    Matrix3 result(*this);
    result -= b;
    return result;
  }

  Matrix3& operator-=(const Matrix3& b) {
    for (int i = 0; i < 3; i++) {
      for (int j = 0; j < 3; j++)
        M[i][j] -= b.M[i][j];
    }

    return *this;
  }

  // Multiplies two matrices into destination with minimum copying.
  static Matrix3& Multiply(Matrix3* d, const Matrix3& a, const Matrix3& b) {
    OVR_MATH_ASSERT((d != &a) && (d != &b));
    int i = 0;
    do {
      d->M[i][0] = a.M[i][0] * b.M[0][0] + a.M[i][1] * b.M[1][0] + a.M[i][2] * b.M[2][0];
      d->M[i][1] = a.M[i][0] * b.M[0][1] + a.M[i][1] * b.M[1][1] + a.M[i][2] * b.M[2][1];
      d->M[i][2] = a.M[i][0] * b.M[0][2] + a.M[i][1] * b.M[1][2] + a.M[i][2] * b.M[2][2];
    } while ((++i) < 3);

    return *d;
  }

  Matrix3 operator*(const Matrix3& b) const {
    Matrix3 result(Matrix3::NoInit);
    Multiply(&result, *this, b);
    return result;
  }

  Matrix3& operator*=(const Matrix3& b) {
    return Multiply(this, Matrix3(*this), b);
  }

  Matrix3 operator*(T s) const {
    Matrix3 result(*this);
    result *= s;
    return result;
  }

  Matrix3& operator*=(T s) {
    for (int i = 0; i < 3; i++) {
      for (int j = 0; j < 3; j++)
        M[i][j] *= s;
    }

    return *this;
  }

  Vector3<T> operator*(const Vector3<T>& b) const {
    Vector3<T> result;
    result.x = M[0][0] * b.x + M[0][1] * b.y + M[0][2] * b.z;
    result.y = M[1][0] * b.x + M[1][1] * b.y + M[1][2] * b.z;
    result.z = M[2][0] * b.x + M[2][1] * b.y + M[2][2] * b.z;

    return result;
  }

  Matrix3 operator/(T s) const {
    Matrix3 result(*this);
    result /= s;
    return result;
  }

  Matrix3& operator/=(T s) {
    for (int i = 0; i < 3; i++) {
      for (int j = 0; j < 3; j++)
        M[i][j] /= s;
    }

    return *this;
  }

  Vector2<T> Transform(const Vector2<T>& v) const {
    const T rcpZ = T(1) / (M[2][0] * v.x + M[2][1] * v.y + M[2][2]);
    return Vector2<T>(
        (M[0][0] * v.x + M[0][1] * v.y + M[0][2]) * rcpZ,
        (M[1][0] * v.x + M[1][1] * v.y + M[1][2]) * rcpZ);
  }

  Vector3<T> Transform(const Vector3<T>& v) const {
    return Vector3<T>(
        M[0][0] * v.x + M[0][1] * v.y + M[0][2] * v.z,
        M[1][0] * v.x + M[1][1] * v.y + M[1][2] * v.z,
        M[2][0] * v.x + M[2][1] * v.y + M[2][2] * v.z);
  }

  Matrix3 Transposed() const {
    return Matrix3(M[0][0], M[1][0], M[2][0], M[0][1], M[1][1], M[2][1], M[0][2], M[1][2], M[2][2]);
  }

  void Transpose() {
    *this = Transposed();
  }

  T SubDet(const size_t* rows, const size_t* cols) const {
    return M[rows[0]][cols[0]] *
        (M[rows[1]][cols[1]] * M[rows[2]][cols[2]] - M[rows[1]][cols[2]] * M[rows[2]][cols[1]]) -
        M[rows[0]][cols[1]] *
        (M[rows[1]][cols[0]] * M[rows[2]][cols[2]] - M[rows[1]][cols[2]] * M[rows[2]][cols[0]]) +
        M[rows[0]][cols[2]] *
        (M[rows[1]][cols[0]] * M[rows[2]][cols[1]] - M[rows[1]][cols[1]] * M[rows[2]][cols[0]]);
  }

  // M += a*b.t()
  inline void Rank1Add(const Vector3<T>& a, const Vector3<T>& b) {
    M[0][0] += a.x * b.x;
    M[0][1] += a.x * b.y;
    M[0][2] += a.x * b.z;
    M[1][0] += a.y * b.x;
    M[1][1] += a.y * b.y;
    M[1][2] += a.y * b.z;
    M[2][0] += a.z * b.x;
    M[2][1] += a.z * b.y;
    M[2][2] += a.z * b.z;
  }

  // M -= a*b.t()
  inline void Rank1Sub(const Vector3<T>& a, const Vector3<T>& b) {
    M[0][0] -= a.x * b.x;
    M[0][1] -= a.x * b.y;
    M[0][2] -= a.x * b.z;
    M[1][0] -= a.y * b.x;
    M[1][1] -= a.y * b.y;
    M[1][2] -= a.y * b.z;
    M[2][0] -= a.z * b.x;
    M[2][1] -= a.z * b.y;
    M[2][2] -= a.z * b.z;
  }

  inline Vector3<T> Col(int c) const {
    return Vector3<T>(M[0][c], M[1][c], M[2][c]);
  }

  inline Vector3<T> Row(int r) const {
    return Vector3<T>(M[r][0], M[r][1], M[r][2]);
  }

  inline Vector3<T> GetColumn(int c) const {
    return Vector3<T>(M[0][c], M[1][c], M[2][c]);
  }

  inline Vector3<T> GetRow(int r) const {
    return Vector3<T>(M[r][0], M[r][1], M[r][2]);
  }

  inline void SetColumn(int c, const Vector3<T>& v) {
    M[0][c] = v.x;
    M[1][c] = v.y;
    M[2][c] = v.z;
  }

  inline void SetRow(int r, const Vector3<T>& v) {
    M[r][0] = v.x;
    M[r][1] = v.y;
    M[r][2] = v.z;
  }

  inline T Determinant() const {
    const Matrix3<T>& m = *this;
    T d;

    d = m.M[0][0] * (m.M[1][1] * m.M[2][2] - m.M[1][2] * m.M[2][1]);
    d -= m.M[0][1] * (m.M[1][0] * m.M[2][2] - m.M[1][2] * m.M[2][0]);
    d += m.M[0][2] * (m.M[1][0] * m.M[2][1] - m.M[1][1] * m.M[2][0]);

    return d;
  }

  inline Matrix3<T> Inverse() const {
    Matrix3<T> a;
    const Matrix3<T>& m = *this;
    T d = Determinant();

    OVR_MATH_ASSERT(d != 0);
    T s = T(1) / d;

    a.M[0][0] = s * (m.M[1][1] * m.M[2][2] - m.M[1][2] * m.M[2][1]);
    a.M[1][0] = s * (m.M[1][2] * m.M[2][0] - m.M[1][0] * m.M[2][2]);
    a.M[2][0] = s * (m.M[1][0] * m.M[2][1] - m.M[1][1] * m.M[2][0]);

    a.M[0][1] = s * (m.M[0][2] * m.M[2][1] - m.M[0][1] * m.M[2][2]);
    a.M[1][1] = s * (m.M[0][0] * m.M[2][2] - m.M[0][2] * m.M[2][0]);
    a.M[2][1] = s * (m.M[0][1] * m.M[2][0] - m.M[0][0] * m.M[2][1]);

    a.M[0][2] = s * (m.M[0][1] * m.M[1][2] - m.M[0][2] * m.M[1][1]);
    a.M[1][2] = s * (m.M[0][2] * m.M[1][0] - m.M[0][0] * m.M[1][2]);
    a.M[2][2] = s * (m.M[0][0] * m.M[1][1] - m.M[0][1] * m.M[1][0]);

    return a;
  }

  // Outer Product of two column vectors: a * b.Transpose()
  static Matrix3 OuterProduct(const Vector3<T>& a, const Vector3<T>& b) {
    return Matrix3(
        a.x * b.x,
        a.x * b.y,
        a.x * b.z,
        a.y * b.x,
        a.y * b.y,
        a.y * b.z,
        a.z * b.x,
        a.z * b.y,
        a.z * b.z);
  }

  // Vector cross product as a premultiply matrix:
  // L.Cross(R) = LeftCrossAsMatrix(L) * R
  static Matrix3 LeftCrossAsMatrix(const Vector3<T>& L) {
    return Matrix3(T(0), -L.z, +L.y, +L.z, T(0), -L.x, -L.y, +L.x, T(0));
  }

  // Vector cross product as a premultiply matrix:
  // L.Cross(R) = RightCrossAsMatrix(R) * L
  static Matrix3 RightCrossAsMatrix(const Vector3<T>& R) {
    return Matrix3(T(0), +R.z, -R.y, -R.z, T(0), +R.x, +R.y, -R.x, T(0));
  }

  // Angle in radians of a rotation matrix
  // Uses identity trace(a) = 2*cos(theta) + 1
  T Angle() const {
    return Acos((Trace() - T(1)) * T(0.5));
  }

  // Angle in radians between two rotation matrices
  T Angle(const Matrix3& b) const {
    // Compute trace of (this->Transposed() * b)
    // This works out to sum of products of elements.
    T trace = T(0);
    for (int i = 0; i < 3; i++) {
      for (int j = 0; j < 3; j++) {
        trace += M[i][j] * b.M[i][j];
      }
    }
    return Acos((trace - T(1)) * T(0.5));
  }
};

typedef Matrix3<float> Matrix3f;
typedef Matrix3<double> Matrix3d;

//-------------------------------------------------------------------------------------
// ***** Matrix2

template <class T>
class Matrix2 {
 public:
  typedef T ElementType;
  static const size_t Dimension = 2;

  T M[2][2];

  enum NoInitType { NoInit };

  // Construct with no memory initialization.
  Matrix2(NoInitType) {}

  // By default, we construct identity matrix.
  Matrix2() {
    M[0][0] = M[1][1] = T(1);
    M[0][1] = M[1][0] = T(0);
  }

  Matrix2(T m11, T m12, T m21, T m22) {
    M[0][0] = m11;
    M[0][1] = m12;
    M[1][0] = m21;
    M[1][1] = m22;
  }

  // Construction from X, Y basis vectors
  Matrix2(const Vector2<T>& xBasis, const Vector2<T>& yBasis) {
    M[0][0] = xBasis.x;
    M[0][1] = yBasis.x;
    M[1][0] = xBasis.y;
    M[1][1] = yBasis.y;
  }

  explicit Matrix2(T s) {
    M[0][0] = M[1][1] = s;
    M[0][1] = M[1][0] = T(0);
  }

  Matrix2(T m11, T m22) {
    M[0][0] = m11;
    M[0][1] = T(0);
    M[1][0] = T(0);
    M[1][1] = m22;
  }

  explicit Matrix2(const Matrix2<typename Math<T>::OtherFloatType>& src) {
    M[0][0] = T(src.M[0][0]);
    M[0][1] = T(src.M[0][1]);
    M[1][0] = T(src.M[1][0]);
    M[1][1] = T(src.M[1][1]);
  }

  // C-interop support
  Matrix2(const typename CompatibleTypes<Matrix2<T>>::Type& s) {
    OVR_MATH_STATIC_ASSERT(sizeof(s) == sizeof(Matrix2), "sizeof(s) == sizeof(Matrix2)");
    memcpy(M, s.M, sizeof(M));
  }

  operator const typename CompatibleTypes<Matrix2<T>>::Type() const {
    typename CompatibleTypes<Matrix2<T>>::Type result;
    OVR_MATH_STATIC_ASSERT(sizeof(result) == sizeof(Matrix2), "sizeof(result) == sizeof(Matrix2)");
    memcpy(result.M, M, sizeof(M));
    return result;
  }

  T operator()(int i, int j) const {
    return M[i][j];
  }
  T& operator()(int i, int j) {
    return M[i][j];
  }
  const T* operator[](int i) const {
    return M[i];
  }
  T* operator[](int i) {
    return M[i];
  }

  static Matrix2 Identity() {
    return Matrix2();
  }

  void SetIdentity() {
    M[0][0] = M[1][1] = T(1);
    M[0][1] = M[1][0] = T(0);
  }

  static Matrix2 Diagonal(T m00, T m11) {
    return Matrix2(m00, m11);
  }
  static Matrix2 Diagonal(const Vector2<T>& v) {
    return Matrix2(v.x, v.y);
  }

  T Trace() const {
    return M[0][0] + M[1][1];
  }

  bool operator==(const Matrix2& b) const {
    return M[0][0] == b.M[0][0] && M[0][1] == b.M[0][1] && M[1][0] == b.M[1][0] &&
        M[1][1] == b.M[1][1];
  }

  Matrix2 operator+(const Matrix2& b) const {
    return Matrix2(
        M[0][0] + b.M[0][0], M[0][1] + b.M[0][1], M[1][0] + b.M[1][0], M[1][1] + b.M[1][1]);
  }

  Matrix2& operator+=(const Matrix2& b) {
    M[0][0] += b.M[0][0];
    M[0][1] += b.M[0][1];
    M[1][0] += b.M[1][0];
    M[1][1] += b.M[1][1];
    return *this;
  }

  void operator=(const Matrix2& b) {
    M[0][0] = b.M[0][0];
    M[0][1] = b.M[0][1];
    M[1][0] = b.M[1][0];
    M[1][1] = b.M[1][1];
  }

  Matrix2 operator-(const Matrix2& b) const {
    return Matrix2(
        M[0][0] - b.M[0][0], M[0][1] - b.M[0][1], M[1][0] - b.M[1][0], M[1][1] - b.M[1][1]);
  }

  Matrix2& operator-=(const Matrix2& b) {
    M[0][0] -= b.M[0][0];
    M[0][1] -= b.M[0][1];
    M[1][0] -= b.M[1][0];
    M[1][1] -= b.M[1][1];
    return *this;
  }

  Matrix2 operator*(const Matrix2& b) const {
    return Matrix2(
        M[0][0] * b.M[0][0] + M[0][1] * b.M[1][0],
        M[0][0] * b.M[0][1] + M[0][1] * b.M[1][1],
        M[1][0] * b.M[0][0] + M[1][1] * b.M[1][0],
        M[1][0] * b.M[0][1] + M[1][1] * b.M[1][1]);
  }

  Matrix2& operator*=(const Matrix2& b) {
    *this = *this * b;
    return *this;
  }

  Matrix2 operator*(T s) const {
    return Matrix2(M[0][0] * s, M[0][1] * s, M[1][0] * s, M[1][1] * s);
  }

  Matrix2& operator*=(T s) {
    M[0][0] *= s;
    M[0][1] *= s;
    M[1][0] *= s;
    M[1][1] *= s;
    return *this;
  }

  Matrix2 operator/(T s) const {
    return *this * (T(1) / s);
  }

  Matrix2& operator/=(T s) {
    return *this *= (T(1) / s);
  }

  Vector2<T> operator*(const Vector2<T>& b) const {
    return Vector2<T>(M[0][0] * b.x + M[0][1] * b.y, M[1][0] * b.x + M[1][1] * b.y);
  }

  Vector2<T> Transform(const Vector2<T>& v) const {
    return Vector2<T>(M[0][0] * v.x + M[0][1] * v.y, M[1][0] * v.x + M[1][1] * v.y);
  }

  Matrix2 Transposed() const {
    return Matrix2(M[0][0], M[1][0], M[0][1], M[1][1]);
  }

  void Transpose() {
    OVRMath_Swap(M[1][0], M[0][1]);
  }

  Vector2<T> GetColumn(int c) const {
    return Vector2<T>(M[0][c], M[1][c]);
  }

  Vector2<T> GetRow(int r) const {
    return Vector2<T>(M[r][0], M[r][1]);
  }

  void SetColumn(int c, const Vector2<T>& v) {
    M[0][c] = v.x;
    M[1][c] = v.y;
  }

  void SetRow(int r, const Vector2<T>& v) {
    M[r][0] = v.x;
    M[r][1] = v.y;
  }

  T Determinant() const {
    return M[0][0] * M[1][1] - M[0][1] * M[1][0];
  }

  Matrix2 Inverse() const {
    T rcpDet = T(1) / Determinant();
    return Matrix2(M[1][1] * rcpDet, -M[0][1] * rcpDet, -M[1][0] * rcpDet, M[0][0] * rcpDet);
  }

  // Outer Product of two column vectors: a * b.Transpose()
  static Matrix2 OuterProduct(const Vector2<T>& a, const Vector2<T>& b) {
    return Matrix2(a.x * b.x, a.x * b.y, a.y * b.x, a.y * b.y);
  }

  // Angle in radians between two rotation matrices
  T Angle(const Matrix2& b) const {
    const Matrix2& a = *this;
    return Acos(a(0, 0) * b(0, 0) + a(1, 0) * b(1, 0));
  }
};

typedef Matrix2<float> Matrix2f;
typedef Matrix2<double> Matrix2d;

//-------------------------------------------------------------------------------------

template <class T>
class SymMat3 {
 private:
  typedef SymMat3<T> this_type;

 public:
  typedef T Value_t;
  // Upper symmetric
  T v[6]; // _00 _01 _02 _11 _12 _22

  inline SymMat3() {}

  inline explicit SymMat3(T s) {
    v[0] = v[3] = v[5] = s;
    v[1] = v[2] = v[4] = T(0);
  }

  inline explicit SymMat3(T a00, T a01, T a02, T a11, T a12, T a22) {
    v[0] = a00;
    v[1] = a01;
    v[2] = a02;
    v[3] = a11;
    v[4] = a12;
    v[5] = a22;
  }

  // Cast to symmetric Matrix3
  operator Matrix3<T>() const {
    return Matrix3<T>(v[0], v[1], v[2], v[1], v[3], v[4], v[2], v[4], v[5]);
  }

  static inline int Index(unsigned int i, unsigned int j) {
    return (i <= j) ? (3 * i - i * (i + 1) / 2 + j) : (3 * j - j * (j + 1) / 2 + i);
  }

  inline T operator()(int i, int j) const {
    return v[Index(i, j)];
  }

  inline T& operator()(int i, int j) {
    return v[Index(i, j)];
  }

  inline this_type& operator+=(const this_type& b) {
    v[0] += b.v[0];
    v[1] += b.v[1];
    v[2] += b.v[2];
    v[3] += b.v[3];
    v[4] += b.v[4];
    v[5] += b.v[5];
    return *this;
  }

  inline this_type& operator-=(const this_type& b) {
    v[0] -= b.v[0];
    v[1] -= b.v[1];
    v[2] -= b.v[2];
    v[3] -= b.v[3];
    v[4] -= b.v[4];
    v[5] -= b.v[5];

    return *this;
  }

  inline this_type& operator*=(T s) {
    v[0] *= s;
    v[1] *= s;
    v[2] *= s;
    v[3] *= s;
    v[4] *= s;
    v[5] *= s;

    return *this;
  }

  inline SymMat3 operator*(T s) const {
    SymMat3 d;
    d.v[0] = v[0] * s;
    d.v[1] = v[1] * s;
    d.v[2] = v[2] * s;
    d.v[3] = v[3] * s;
    d.v[4] = v[4] * s;
    d.v[5] = v[5] * s;

    return d;
  }

  // Multiplies two matrices into destination with minimum copying.
  static SymMat3& Multiply(SymMat3* d, const SymMat3& a, const SymMat3& b) {
    // _00 _01 _02 _11 _12 _22

    d->v[0] = a.v[0] * b.v[0];
    d->v[1] = a.v[0] * b.v[1] + a.v[1] * b.v[3];
    d->v[2] = a.v[0] * b.v[2] + a.v[1] * b.v[4];

    d->v[3] = a.v[3] * b.v[3];
    d->v[4] = a.v[3] * b.v[4] + a.v[4] * b.v[5];

    d->v[5] = a.v[5] * b.v[5];

    return *d;
  }

  inline T Determinant() const {
    const this_type& m = *this;
    T d;

    d = m(0, 0) * (m(1, 1) * m(2, 2) - m(1, 2) * m(2, 1));
    d -= m(0, 1) * (m(1, 0) * m(2, 2) - m(1, 2) * m(2, 0));
    d += m(0, 2) * (m(1, 0) * m(2, 1) - m(1, 1) * m(2, 0));

    return d;
  }

  inline this_type Inverse() const {
    this_type a;
    const this_type& m = *this;
    T d = Determinant();

    OVR_MATH_ASSERT(d != 0);
    T s = T(1) / d;

    a(0, 0) = s * (m(1, 1) * m(2, 2) - m(1, 2) * m(2, 1));

    a(0, 1) = s * (m(0, 2) * m(2, 1) - m(0, 1) * m(2, 2));
    a(1, 1) = s * (m(0, 0) * m(2, 2) - m(0, 2) * m(2, 0));

    a(0, 2) = s * (m(0, 1) * m(1, 2) - m(0, 2) * m(1, 1));
    a(1, 2) = s * (m(0, 2) * m(1, 0) - m(0, 0) * m(1, 2));
    a(2, 2) = s * (m(0, 0) * m(1, 1) - m(0, 1) * m(1, 0));

    return a;
  }

  inline T Trace() const {
    return v[0] + v[3] + v[5];
  }

  // M = a*a.t()
  inline void Rank1(const Vector3<T>& a) {
    v[0] = a.x * a.x;
    v[1] = a.x * a.y;
    v[2] = a.x * a.z;
    v[3] = a.y * a.y;
    v[4] = a.y * a.z;
    v[5] = a.z * a.z;
  }

  // M += a*a.t()
  inline void Rank1Add(const Vector3<T>& a) {
    v[0] += a.x * a.x;
    v[1] += a.x * a.y;
    v[2] += a.x * a.z;
    v[3] += a.y * a.y;
    v[4] += a.y * a.z;
    v[5] += a.z * a.z;
  }

  // M -= a*a.t()
  inline void Rank1Sub(const Vector3<T>& a) {
    v[0] -= a.x * a.x;
    v[1] -= a.x * a.y;
    v[2] -= a.x * a.z;
    v[3] -= a.y * a.y;
    v[4] -= a.y * a.z;
    v[5] -= a.z * a.z;
  }
};

typedef SymMat3<float> SymMat3f;
typedef SymMat3<double> SymMat3d;

template <class T>
inline Matrix3<T> operator*(const SymMat3<T>& a, const SymMat3<T>& b) {
#define AJB_ARBC(r, c) (a(r, 0) * b(0, c) + a(r, 1) * b(1, c) + a(r, 2) * b(2, c))
  return Matrix3<T>(
      AJB_ARBC(0, 0),
      AJB_ARBC(0, 1),
      AJB_ARBC(0, 2),
      AJB_ARBC(1, 0),
      AJB_ARBC(1, 1),
      AJB_ARBC(1, 2),
      AJB_ARBC(2, 0),
      AJB_ARBC(2, 1),
      AJB_ARBC(2, 2));
#undef AJB_ARBC
}

template <class T>
inline Matrix3<T> operator*(const Matrix3<T>& a, const SymMat3<T>& b) {
#define AJB_ARBC(r, c) (a(r, 0) * b(0, c) + a(r, 1) * b(1, c) + a(r, 2) * b(2, c))
  return Matrix3<T>(
      AJB_ARBC(0, 0),
      AJB_ARBC(0, 1),
      AJB_ARBC(0, 2),
      AJB_ARBC(1, 0),
      AJB_ARBC(1, 1),
      AJB_ARBC(1, 2),
      AJB_ARBC(2, 0),
      AJB_ARBC(2, 1),
      AJB_ARBC(2, 2));
#undef AJB_ARBC
}

//-------------------------------------------------------------------------------------
// ***** Angle

// Cleanly representing the algebra of 2D rotations.
// The operations maintain the angle between -Pi and Pi, the same range as atan2.

template <class T>
class Angle {
 public:
  enum AngularUnits { Radians = 0, Degrees = 1 };

  Angle() : a(0) {}

  // Fix the range to be between -Pi and Pi
  Angle(T a_, AngularUnits u = Radians)
      : a((u == Radians) ? a_ : a_ * ((T)MATH_DOUBLE_DEGREETORADFACTOR)) {
    FixRange();
  }

  T Get(AngularUnits u = Radians) const {
    return (u == Radians) ? a : a * ((T)MATH_DOUBLE_RADTODEGREEFACTOR);
  }
  void Set(const T& x, AngularUnits u = Radians) {
    a = (u == Radians) ? x : x * ((T)MATH_DOUBLE_DEGREETORADFACTOR);
    FixRange();
  }
  int Sign() const {
    if (a == 0)
      return 0;
    else
      return (a > 0) ? 1 : -1;
  }
  T Abs() const {
    return (a >= 0) ? a : -a;
  }

  bool operator==(const Angle& b) const {
    return a == b.a;
  }
  bool operator!=(const Angle& b) const {
    return a != b.a;
  }
  //    bool operator<  (const Angle& b) const    { return a < a.b; }
  //    bool operator>  (const Angle& b) const    { return a > a.b; }
  //    bool operator<= (const Angle& b) const    { return a <= a.b; }
  //    bool operator>= (const Angle& b) const    { return a >= a.b; }
  //    bool operator= (const T& x)               { a = x; FixRange(); }

  // These operations assume a is already between -Pi and Pi.
  Angle& operator+=(const Angle& b) {
    a = a + b.a;
    FastFixRange();
    return *this;
  }
  Angle& operator+=(const T& x) {
    a = a + x;
    FixRange();
    return *this;
  }
  Angle operator+(const Angle& b) const {
    Angle res = *this;
    res += b;
    return res;
  }
  Angle operator+(const T& x) const {
    Angle res = *this;
    res += x;
    return res;
  }
  Angle& operator-=(const Angle& b) {
    a = a - b.a;
    FastFixRange();
    return *this;
  }
  Angle& operator-=(const T& x) {
    a = a - x;
    FixRange();
    return *this;
  }
  Angle operator-(const Angle& b) const {
    Angle res = *this;
    res -= b;
    return res;
  }
  Angle operator-(const T& x) const {
    Angle res = *this;
    res -= x;
    return res;
  }
  Angle& operator*=(const T& x) {
    a = a * x;
    FixRange();
    return *this;
  }
  Angle operator*(const T& x) const {
    Angle res = *this;
    res *= x;
    return res;
  }

  T Distance(const Angle& b) {
    T c = fabs(a - b.a);
    return (c <= ((T)MATH_DOUBLE_PI)) ? c : ((T)MATH_DOUBLE_TWOPI) - c;
  }

  Angle Lerp(const Angle& b, T f) const {
    return *this + (b - *this) * f;
  }

 private:
  // The stored angle, which should be maintained between -Pi and Pi
  T a;

  // Fixes the angle range to [-Pi,Pi], but assumes no more than 2Pi away on either side
  inline void FastFixRange() {
    if (a < -((T)MATH_DOUBLE_PI))
      a += ((T)MATH_DOUBLE_TWOPI);
    else if (a > ((T)MATH_DOUBLE_PI))
      a -= ((T)MATH_DOUBLE_TWOPI);
  }

  // Fixes the angle range to [-Pi,Pi] for any given range, but slower then the fast method
  inline void FixRange() {
    // do nothing if the value is already in the correct range, since fmod call is expensive
    if (a >= -((T)MATH_DOUBLE_PI) && a <= ((T)MATH_DOUBLE_PI))
      return;
    a = fmod(a, ((T)MATH_DOUBLE_TWOPI));
    if (a < -((T)MATH_DOUBLE_PI))
      a += ((T)MATH_DOUBLE_TWOPI);
    else if (a > ((T)MATH_DOUBLE_PI))
      a -= ((T)MATH_DOUBLE_TWOPI);
  }
};

typedef Angle<float> Anglef;
typedef Angle<double> Angled;

//-------------------------------------------------------------------------------------
// ***** Plane

// Consists of a normal vector and distance from the origin where the plane is located.

template <class T>
class Plane {
 public:
  Vector3<T> N;
  T D;

  Plane() : D(0) {}

  // Normals must already be normalized
  Plane(const Vector3<T>& n, T d) : N(n), D(d) {}
  Plane(T x, T y, T z, T d) : N(x, y, z), D(d) {}

  // construct from a point on the plane and the normal
  Plane(const Vector3<T>& p, const Vector3<T>& n) : N(n), D(-(p.Dot(n))) {}

  // Find the point to plane distance. The sign indicates what side of the plane the point is on (0
  // = point on plane).
  T TestSide(const Vector3<T>& p) const {
    return (N.Dot(p)) + D;
  }

  Plane<T> Flipped() const {
    return Plane(-N, -D);
  }

  void Flip() {
    N = -N;
    D = -D;
  }

  bool operator==(const Plane<T>& rhs) const {
    return (this->D == rhs.D && this->N == rhs.N);
  }
};

typedef Plane<float> Planef;
typedef Plane<double> Planed;

//-----------------------------------------------------------------------------------
// ***** ScaleAndOffset

template <class T>
struct ScaleAndOffset {
  T Scale;
  T Offset;

  ScaleAndOffset(float sx = 0.0f, float sy = 0.0f, float ox = 0.0f, float oy = 0.0f)
      : Scale(sx, sy), Offset(ox, oy) {}

  T ApplyTo(const T& input) const {
    return input * Scale + Offset;
  }

  ScaleAndOffset Invert() const {
    ScaleAndOffset inverted;
    inverted.Scale = T(1.0f) / this->Scale;
    inverted.Offset = -(this->Offset) * inverted.Scale;
    return inverted;
  }

  // nextSO is the other scale-offset operation that would have normally followed this scale-offset.
  // Result is a single scale-offset operation that can be applied to an input instead of
  // two or more separate scale-offset applications on a given Vector2f input.
  //
  // So this:
  //  ScaleAndOffset2D so1, so2, so3; // initialized to some values
  //  Vector2f input = Vector2f(2.0f, -1.0f);
  //  input = so1.ApplyTo(input);
  //  input = so2.ApplyTo(input);
  //  input = so3.ApplyTo(input);
  //
  // equals this:
  //  ScaleAndOffset2D so1, so2, so3; // initialized to some values
  //  so1 = so1.Combine(so2);
  //  so1 = so1.Combine(so3);
  //  Vector2f input = Vector2f(2.0f, -1.0f);
  //  input = so1.ApplyTo(input);
  //
  ScaleAndOffset Combine(const ScaleAndOffset& nextSO) const {
    ScaleAndOffset retValSO;
    retValSO.Offset = this->Offset * nextSO.Scale + nextSO.Offset;
    retValSO.Scale = nextSO.Scale * this->Scale;
    return retValSO;
  }
};

typedef ScaleAndOffset<Vector2f> ScaleAndOffset2D;
typedef ScaleAndOffset<Vector3f> ScaleAndOffset3D;

//-----------------------------------------------------------------------------------
// ***** FovPort

// FovPort describes Field Of View (FOV) of a viewport.
// This class has values for up, down, left and right, stored in
// tangent of the angle units to simplify calculations.
//
// As an example, for a standard 90 degree vertical FOV, we would
// have: { UpTan = tan(90 degrees / 2), DownTan = tan(90 degrees / 2) }.
//
// CreateFromRadians/Degrees helper functions can be used to
// access FOV in different units.

// ***** FovPort

struct FovPort {
  float UpTan;
  float DownTan;
  float LeftTan;
  float RightTan;

  FovPort(float sideTan = 0.0f)
      : UpTan(sideTan), DownTan(sideTan), LeftTan(sideTan), RightTan(sideTan) {}
  FovPort(float u, float d, float l, float r) : UpTan(u), DownTan(d), LeftTan(l), RightTan(r) {}

#ifndef OVR_EXCLUDE_CAPI_FROM_MATH
  // C-interop support.
  typedef CompatibleTypes<FovPort>::Type CompatibleType;

  FovPort(const CompatibleType& s)
      : UpTan(s.UpTan), DownTan(s.DownTan), LeftTan(s.LeftTan), RightTan(s.RightTan) {}

  operator const CompatibleType&() const {
    OVR_MATH_STATIC_ASSERT(sizeof(FovPort) == sizeof(CompatibleType), "sizeof(FovPort) failure");
    return reinterpret_cast<const CompatibleType&>(*this);
  }
#endif

  static FovPort CreateFromRadians(float horizontalFov, float verticalFov) {
    FovPort result;
    result.UpTan = tanf(verticalFov * 0.5f);
    result.DownTan = tanf(verticalFov * 0.5f);
    result.LeftTan = tanf(horizontalFov * 0.5f);
    result.RightTan = tanf(horizontalFov * 0.5f);
    return result;
  }

  static FovPort CreateFromDegrees(float horizontalFovDegrees, float verticalFovDegrees) {
    return CreateFromRadians(DegreeToRad(horizontalFovDegrees), DegreeToRad(verticalFovDegrees));
  }

  //  Get Horizontal/Vertical components of Fov in radians.
  float GetVerticalFovRadians() const {
    return atanf(UpTan) + atanf(DownTan);
  }
  float GetHorizontalFovRadians() const {
    return atanf(LeftTan) + atanf(RightTan);
  }
  //  Get Horizontal/Vertical components of Fov in degrees.
  float GetVerticalFovDegrees() const {
    return RadToDegree(GetVerticalFovRadians());
  }
  float GetHorizontalFovDegrees() const {
    return RadToDegree(GetHorizontalFovRadians());
  }

  // Compute maximum tangent value among all four sides.
  float GetMaxSideTan() const {
    return OVRMath_Max(OVRMath_Max(UpTan, DownTan), OVRMath_Max(LeftTan, RightTan));
  }

  static ScaleAndOffset2D CreateNDCScaleAndOffsetFromFov(FovPort tanHalfFov) {
    float projXScale = 2.0f / (tanHalfFov.LeftTan + tanHalfFov.RightTan);
    float projXOffset = (tanHalfFov.LeftTan - tanHalfFov.RightTan) * projXScale * 0.5f;
    float projYScale = 2.0f / (tanHalfFov.UpTan + tanHalfFov.DownTan);
    float projYOffset = (tanHalfFov.UpTan - tanHalfFov.DownTan) * projYScale * 0.5f;

    ScaleAndOffset2D result;
    result.Scale = Vector2f(projXScale, projYScale);
    result.Offset = Vector2f(projXOffset, projYOffset);
    // Hey - why is that Y.Offset negated?
    // It's because a projection matrix transforms from world coords with Y=up,
    // whereas this is from NDC which is Y=down.

    return result;
  }

  // Converts Fov Tan angle units to [-1,1] render target NDC space
  Vector2f TanAngleToRendertargetNDC(Vector2f const& tanEyeAngle) {
    ScaleAndOffset2D eyeToSourceNDC = CreateNDCScaleAndOffsetFromFov(*this);
    return tanEyeAngle * eyeToSourceNDC.Scale + eyeToSourceNDC.Offset;
  }

  // Compute per-channel minimum and maximum of Fov.
  static FovPort Min(const FovPort& a, const FovPort& b) {
    FovPort fov(
        OVRMath_Min(a.UpTan, b.UpTan),
        OVRMath_Min(a.DownTan, b.DownTan),
        OVRMath_Min(a.LeftTan, b.LeftTan),
        OVRMath_Min(a.RightTan, b.RightTan));
    return fov;
  }

  static FovPort Max(const FovPort& a, const FovPort& b) {
    FovPort fov(
        OVRMath_Max(a.UpTan, b.UpTan),
        OVRMath_Max(a.DownTan, b.DownTan),
        OVRMath_Max(a.LeftTan, b.LeftTan),
        OVRMath_Max(a.RightTan, b.RightTan));
    return fov;
  }

  static FovPort Uncant(const FovPort& cantedFov, Quatf canting) {
    FovPort uncantedFov = cantedFov;

    // make 3D vectors from the FovPorts projected to z=1 plane
    Vector3f leftUp = Vector3f(cantedFov.LeftTan, -cantedFov.UpTan, 1.0f);
    Vector3f rightUp = Vector3f(-cantedFov.RightTan, -cantedFov.UpTan, 1.0f);
    Vector3f leftDown = Vector3f(cantedFov.LeftTan, cantedFov.DownTan, 1.0f);
    Vector3f rightDown = Vector3f(-cantedFov.RightTan, cantedFov.DownTan, 1.0f);

    // rotate these vectors using the canting specified
    leftUp = canting.Rotate(leftUp);
    rightUp = canting.Rotate(rightUp);
    leftDown = canting.Rotate(leftDown);
    rightDown = canting.Rotate(rightDown);

    // If the z coordinates of any of the corners end up being really small or negative, then
    // projection will generate extremely large or inverted frustums and we don't really want that
    const float kMinValidZ = 0.01f;

    // re-project back to z=1 plane while making sure we don't generate gigantic values (hence max)
    leftUp /= OVRMath_Max(leftUp.z, kMinValidZ);
    rightUp /= OVRMath_Max(rightUp.z, kMinValidZ);
    leftDown /= OVRMath_Max(leftDown.z, kMinValidZ);
    rightDown /= OVRMath_Max(rightDown.z, kMinValidZ);

    // generate new FovTans as "bounding box" values
    uncantedFov.UpTan = OVRMath_Max(-leftUp.y, -rightUp.y);
    uncantedFov.DownTan = OVRMath_Max(leftDown.y, rightDown.y);
    uncantedFov.LeftTan = OVRMath_Max(leftUp.x, leftDown.x);
    uncantedFov.RightTan = OVRMath_Max(-rightDown.x, -rightUp.x);

    return uncantedFov;
  }

  // Widens a given FovPort by specified angle in each direction
  static FovPort Expand(const FovPort& inFov, float expandAngle) {
    auto ClampedExpand = [expandAngle](float t) -> float {
      // We don't want gigantic values coming out of this function, so we limit resulting FOV.
      // Limit before calling tanf() to avoid wrap around to negative values.
      const float limitFov = atanf(10.0f);
      return tanf(OVRMath_Min(OVRMath_Max(atanf(t) + expandAngle, -limitFov), limitFov));
    };

    FovPort modFov = FovPort(
        ClampedExpand(inFov.UpTan),
        ClampedExpand(inFov.DownTan),
        ClampedExpand(inFov.LeftTan),
        ClampedExpand(inFov.RightTan));

    return modFov;
  }

  template <class T>
  static FovPort ScaleFovPort(const FovPort& fov, OVR::Vector2<T> scaleFactors) {
    FovPort retFov = FovPort(fov);
    retFov.LeftTan *= ((scaleFactors.x != 0.0) ? scaleFactors.x : 1.0f);
    retFov.RightTan *= ((scaleFactors.x != 0.0) ? scaleFactors.x : 1.0f);
    retFov.UpTan *= ((scaleFactors.y != 0.0) ? scaleFactors.y : 1.0f);
    retFov.DownTan *= ((scaleFactors.y != 0.0) ? scaleFactors.y : 1.0f);
    return retFov;
  }
};

inline bool isNan(const Vector3d& v) {
  return v.IsNan();
}

inline bool isNan(const Vector3f& v) {
  return v.IsNan();
}

} // Namespace OVR

#if defined(_MSC_VER)
#pragma warning(pop)
#endif

#endif