OVR_Math.h

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/************************************************************************************

PublicHeader:   OVR.h
Filename    :   OVR_Math.h
Content     :   Implementation of 3D primitives such as vectors, matrices.
Created     :   September 4, 2012
Authors     :   Andrew Reisse, Michael Antonov, Steve LaValle, 
                Anna Yershova, Max Katsev, Dov Katz

Copyright   :   Copyright 2014 Oculus VR, Inc. All Rights reserved.

Licensed under the Oculus VR Rift SDK License Version 3.1 (the "License"); 
you may not use the Oculus VR Rift SDK except in compliance with the License, 
which is provided at the time of installation or download, or which 
otherwise accompanies this software in either electronic or hard copy form.

You may obtain a copy of the License at

http://www.oculusvr.com/licenses/LICENSE-3.1 

Unless required by applicable law or agreed to in writing, the Oculus VR SDK 
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.

*************************************************************************************/

#ifndef OVR_Math_h
#define OVR_Math_h

#include <assert.h>
#include <stdlib.h>
#include <math.h>

#include "OVR_Types.h"
#include "OVR_RefCount.h"
#include "OVR_Std.h"
#include "OVR_Alg.h"


namespace OVR {

//-------------------------------------------------------------------------------------
// ***** 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_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 ovrRecti_ ovrRecti;
typedef struct ovrVector2i_ ovrVector2i;
typedef struct ovrVector2f_ ovrVector2f;
typedef struct ovrVector3f_ ovrVector3f;
typedef struct ovrVector3d_ ovrVector3d;
typedef struct ovrMatrix3d_ ovrMatrix3d;
typedef struct ovrMatrix4f_ ovrMatrix4f;
typedef struct ovrPosef_ ovrPosef;
typedef struct ovrPosed_ ovrPosed;
typedef struct ovrPoseStatef_ ovrPoseStatef;
typedef struct ovrPoseStated_ ovrPoseStated;

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 Matrix3;
template<class T> class Matrix4;
template<class T> class Transform;
template<class T> class PoseState;

// 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 float 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<Matrix3<double> > { typedef ovrMatrix3d Type; };
template<> struct CompatibleTypes<Matrix4<float> >  { typedef ovrMatrix4f Type; };
template<> struct CompatibleTypes<Size<int> >       { typedef ovrSizei Type; };
template<> struct CompatibleTypes<Size<float> >     { typedef ovrSizef 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<Vector3<float> >  { typedef ovrVector3f Type; };
template<> struct CompatibleTypes<Vector3<double> > { typedef ovrVector3d Type; };

template<> struct CompatibleTypes<Transform<float> > { typedef ovrPosef Type; };
template<> struct CompatibleTypes<PoseState<float> > { typedef ovrPoseStatef Type; };

template<> struct CompatibleTypes<Transform<double> > { typedef ovrPosed Type; };
template<> struct CompatibleTypes<PoseState<double> > { typedef ovrPoseStated 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 Type>
class Math
{  
public:
    // By default, support explicit conversion to float. This allows Vector2<int> to
    // compile, for example.
    typedef float OtherFloatType;
};

// Single-precision Math constants class.
template<>
class Math<float>
{
public:
    static const float Pi;
    static const float TwoPi;
    static const float PiOver2;
    static const float PiOver4;
    static const float E;

    static const float MaxValue;            // Largest positive float Value
    static const float MinPositiveValue;    // Smallest possible positive value

    static const float RadToDegreeFactor;
    static const float DegreeToRadFactor;

    static const float Tolerance;           // 0.00001f;
    static const float SingularityRadius;   // 0.0000001f for Gimbal lock numerical problems

    // Used to support direct conversions in template classes.
    typedef double OtherFloatType;
};

// Double-precision Math constants class.
template<>
class Math<double>
{
public:
    static const double Pi;
    static const double TwoPi;
    static const double PiOver2;
    static const double PiOver4;
    static const double E;

    static const double MaxValue;           // Largest positive double Value
    static const double MinPositiveValue;   // Smallest possible positive value

    static const double RadToDegreeFactor;
    static const double DegreeToRadFactor;

    static const double Tolerance;          // 0.00001;
    static const double SingularityRadius;  // 0.000000000001 for Gimbal lock numerical problems    

    typedef float OtherFloatType;
};


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

// Conversion functions between degrees and radians
template<class T>
T RadToDegree(T rads) { return rads * Math<T>::RadToDegreeFactor; }
template<class T>
T DegreeToRad(T rads) { return rads * Math<T>::DegreeToRadFactor; }

// Numerically stable acos function
template<class T>
T Acos(T val) { 
        if (val > T(1))             return T(0);
        else if (val < T(-1))       return Math<T>::Pi;
        else                        return acos(val); 
};

// Numerically stable asin function
template<class T>
T Asin(T val) { 
    if (val > T(1))             return Math<T>::PiOver2;
    else if (val < T(-1))       return Math<T>::PiOver2 * T(3);
    else                        return asin(val); 
};

#ifdef OVR_CC_MSVC
inline int isnan(double x) { return _isnan(x); };
#endif

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:
    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) { }


    // 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_COMPILER_ASSERT(sizeof(Vector2<T>) == sizeof(CompatibleType));
        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); }

    // Compare two vectors for equality with tolerance. Returns true if vectors match withing tolerance.
    bool    Compare(const Vector2&b, T tolerance = Mathf::Tolerance)
    {
        return (fabs(b.x-x) < tolerance) && (fabs(b.y-y) < tolerance);
    }
    
    // Entrywise 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_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(Vector2& b) const         { return (*this - b).LengthSq(); }

    // Returns distance between two points represented by vectors.
    T       Distance(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 l = Length();
        OVR_ASSERT(l != T(0));
        *this /= l; 
    }
    // Returns normalized (unit) version of the vector without modifying itself.
    Vector2 Normalized() const                   
    { 
        T l = Length();
        OVR_ASSERT(l != T(0));
        return *this / l; 
    }

    // 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_ASSERT(l2 != T(0));
        return b * ( Dot(b) / l2 ); 
    }
};


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

//-------------------------------------------------------------------------------------
// ***** 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:
    T x, y, z;

    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) { }


    // 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_COMPILER_ASSERT(sizeof(Vector3<T>) == sizeof(CompatibleType));
        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);
    }        

    // Compare two vectors for equality with tolerance. Returns true if vectors match withing tolerance.
    bool      Compare(const Vector3&b, T tolerance = Mathf::Tolerance)
    {
        return (fabs(b.x-x) < tolerance) && 
               (fabs(b.y-y) < tolerance) && 
               (fabs(b.z-z) < tolerance);
    }
    
    T& operator[] (int idx)
    {
        OVR_ASSERT(0 <= idx && idx < 3);
        return *(&x + idx);
    }

    const T& operator[] (int idx) const
    {
        OVR_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_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 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(); }
    
    // 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 l = Length();
        OVR_ASSERT(l != T(0));
        *this /= l; 
    }

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

    // 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_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); }
};


typedef Vector3<float>  Vector3f;
typedef Vector3<double> Vector3d;
typedef Vector3<SInt32>  Vector3i;


// JDC: this was defined in Render_Device.h, I moved it here, but it
// needs to be fleshed out like the other Vector types.
//
// A vector with a dummy w component for alignment in uniform buffers (and for float colors).
// The w component is not used in any calculations.

struct Vector4f : public Vector3f
{
    float w;

    Vector4f() : w(1) {}
    Vector4f(const Vector3f& v) : Vector3f(v), w(1) {}
    Vector4f(float r, float g, float b, float a) : Vector3f(r,g,b), w(a) {}
};



//-------------------------------------------------------------------------------------
// ***** 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_COMPILER_ASSERT(sizeof(Size<T>) == sizeof(CompatibleType));
        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;



//-----------------------------------------------------------------------------------
// ***** 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_COMPILER_ASSERT(sizeof(Rect<T>) == sizeof(CompatibleType));
        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:
    // w + Xi + Yj + Zk
    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) { }

    // C-interop support.
    Quat(const typename CompatibleTypes<Quat<T> >::Type& s) : x(s.x), y(s.y), z(s.z), w(s.w) { }

    operator typename CompatibleTypes<Quat<T> >::Type () const
    {
        typename CompatibleTypes<Quat<T> >::Type 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() == 0)
        {
            // Assert if the axis is zero, but the angle isn't
            OVR_ASSERT(angle == 0);
            x = 0; y = 0; z = 0; w = 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];
    }

    // 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 > Math<T>::Pi) // Reduce the magnitude of the angle, if necessary
            {
                *angle = Math<T>::TwoPi - *angle;
                *axis = *axis * (-1);
            }
        }
        else 
        {
            *axis = Vector3<T>(1, 0, 0);
            *angle= 0;
        }
    }

    // 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;
        }
    }

    // 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;
        }
    }

    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; }


    // 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 2 * Acos(Alg::Abs(Dot(q)));
    }

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

    void    Normalize()                     
    {
        T l = Length();
        OVR_ASSERT(l != T(0));
        *this /= l; 
    }

    Quat    Normalized() const              
    { 
        T l = Length();
        OVR_ASSERT(l != T(0));
        return *this / l; 
    }

    // 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); }

    // 
    // 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);
    }

    // Normalized linear interpolation of quaternions
    Quat Nlerp(const Quat& other, T a)
    {
        T sign = (Dot(other) >= 0) ? 1 : -1;
        return (*this * sign * a + other * (1-a)).Normalized();
    }
    
    // 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
    {
        return ((*this * Quat<T>(v.x, v.y, v.z, T(0))) * Inverted()).Imag();
    }
    
    // Inversed quaternion rotates in the opposite direction.
    Quat        Inverted() 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);
    }
    
    // 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 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 GetEulerAngles(T *a, T *b, T *c) const
    {
        OVR_COMPILER_ASSERT((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]);

        if (s2 < T(-1) + Math<T>::SingularityRadius)
        { // South pole singularity
            *a = T(0);
            *b = -S*D*Math<T>::PiOver2;
            *c = S*D*atan2(T(2)*(psign*Q[A1]*Q[A2] + w*Q[A3]),
                           ww + Q22 - Q11 - Q33 );
        }
        else if (s2 > T(1) - Math<T>::SingularityRadius)
        {  // North pole singularity
            *a = T(0);
            *b = S*D*Math<T>::PiOver2;
            *c = S*D*atan2(T(2)*(psign*Q[A1]*Q[A2] + w*Q[A3]),
                           ww + Q22 - Q11 - Q33);
        }
        else
        {
            *a = -S*D*atan2(T(-2)*(w*Q[A1] - psign*Q[A2]*Q[A3]),
                            ww + Q33 - Q11 - Q22);
            *b = S*D*asin(s2);
            *c = S*D*atan2(T(2)*(w*Q[A3] - psign*Q[A1]*Q[A2]),
                           ww + Q11 - Q22 - Q33);
        }      
        return;
    }

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

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

//-------------------------------------------------------------------------------------
// ***** Pose

// Position and orientation combined.

template<class T>
class Transform
{
public:

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

    Transform() { }
    Transform(const Quat<T>& orientation, const Vector3<T>& pos)
        : Rotation(orientation), Translation(pos) {  }
    Transform(const Transform& s)
        : Rotation(s.Rotation), Translation(s.Translation) {  }
    Transform(const CompatibleType& s)
        : Rotation(s.Orientation), Translation(s.Position) {  }
    explicit Transform(const Transform<typename Math<T>::OtherFloatType> &s)
        : Rotation(s.Rotation), Translation(s.Translation) {  }

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

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

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

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

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

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

    PoseState<T> operator*(const PoseState<T>& poseState) const   
    {
        PoseState<T> result;
        result.Pose                = (*this) * poseState.Pose;
        result.LinearVelocity      = this->Rotate(poseState.LinearVelocity);
        result.LinearAcceleration  = this->Rotate(poseState.LinearAcceleration);
        result.AngularVelocity     = this->Rotate(poseState.AngularVelocity);
        result.AngularAcceleration = this->Rotate(poseState.AngularAcceleration);
        return result;
    }

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

typedef Transform<float>  Transformf;
typedef Transform<double> Transformd;


//-------------------------------------------------------------------------------------
// ***** 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
{
    static const Matrix4 IdentityValue;

public:
    T M[4][4];    

    enum NoInitType { NoInit };

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

    // By default, we construct identity matrix.
    Matrix4()
    {
        SetIdentity();        
    }

    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] = 0;
        M[1][0] = m21; M[1][1] = m22; M[1][2] = m23; M[1][3] = 0;
        M[2][0] = m31; M[2][1] = m32; M[2][2] = m33; M[2][3] = 0;
        M[3][0] = 0;   M[3][1] = 0;   M[3][2] = 0;   M[3][3] = 1;
    }

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

    explicit Matrix4(const Transform<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_COMPILER_ASSERT(sizeof(s) == sizeof(Matrix4));
        memcpy(M, s.M, sizeof(M));
    }

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

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

    static Matrix4 FromString(const char* src)
    {
        Matrix4 result;
        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 const Matrix4& Identity()  { return IdentityValue; }

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

    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_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;
    }

    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[0][3],
                          M[1][0] * v.x + M[1][1] * v.y + M[1][2] * v.z + M[1][3],
                          M[2][0] * v.x + M[2][1] * v.y + M[2][2] * v.z + M[2][3]);
    }

    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 UPInt* rows, const UPInt* 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(UPInt I, UPInt J) const
    {
        const UPInt 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();
        assert(det != 0);
        return Adjugated() * (1.0f/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] = 0.0f;
        // 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)
    {
        OVR_COMPILER_ASSERT((A1 != A2) && (A2 != A3) && (A1 != A3));

        T psign = -1;
        if (((A1 + 1) % 3 == A2) && ((A2 + 1) % 3 == A3)) // Determine whether even permutation
        psign = 1;
        
        T pm = psign*M[A1][A3];
        if (pm < -1.0f + Math<T>::SingularityRadius)
        { // South pole singularity
            *a = 0;
            *b = -S*D*Math<T>::PiOver2;
            *c = S*D*atan2( psign*M[A2][A1], M[A2][A2] );
        }
        else if (pm > 1.0f - Math<T>::SingularityRadius)
        { // North pole singularity
            *a = 0;
            *b = S*D*Math<T>::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] );
        }

        return;
    }

    // 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)
    {        
         OVR_COMPILER_ASSERT(A1 != A2);
  
        // Determine the axis that was not supplied
        int m = 3 - A1 - A2;

        T psign = -1;
        if ((A1 + 1) % 3 == A2) // Determine whether even permutation
            psign = 1.0f;

        T c2 = M[A1][A1];
        if (c2 < -1 + Math<T>::SingularityRadius)
        { // South pole singularity
            *a = 0;
            *b = S*D*Math<T>::Pi;
            *c = S*D*atan2( -psign*M[A2][m],M[A2][A2]);
        }
        else if (c2 > 1.0f - Math<T>::SingularityRadius)
        { // North pole singularity
            *a = 0;
            *b = 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]);
        }
        return;
    }
  
    // 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 = 0.0f)
    {
        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);
        }
    }


    // 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 = sin(angle);
        T cosa = 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 = tan(yfov * 0.5f);

        m.M[0][0] = 1 / (aspect * tanHalfFov);
        m.M[1][1] = 1 / tanHalfFov;
        m.M[2][2] = zfar / (zfar - znear);
        m.M[3][2] = 1;
        m.M[2][3] = (zfar * znear) / (znear - zfar);
        m.M[3][3] = 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.
        return m;
    }
    
    // PerspectiveRH 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 = tan(yfov * 0.5f);

        m.M[0][0] = 1.0 / (aspect * tanHalfFov);
        m.M[1][1] = 1.0 / tanHalfFov;
        m.M[2][2] = zfar / (znear - zfar);
        // m.M[2][2] = zfar / (zfar - znear);
        m.M[3][2] = -1.0;
        m.M[2][3] = (zfar * znear) / (znear - zfar);
        m.M[3][3] = 0.0;

        // 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 cooridnate input.       
        return m;
    }

    static Matrix4 Ortho2D(T w, T h)
    {
        Matrix4 m;
        m.M[0][0] = 2.0/w;
        m.M[1][1] = -2.0/h;
        m.M[0][3] = -1.0;
        m.M[1][3] = 1.0;
        m.M[2][2] = 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<typename T>
class SymMat3;

template<class T>
class Matrix3
{
    static const Matrix3 IdentityValue;

public:
    T M[3][3];    

    enum NoInitType { NoInit };

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

    // By default, we construct identity matrix.
    Matrix3()
    {
        SetIdentity();        
    }

    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;
    }
    
    /*
    explicit Matrix3(const Quat<T>& q)
    {
        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[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[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;      
    }
    */
    
    explicit Matrix3(const Quat<T>& q)
    {
        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] = 0;
    }

    explicit Matrix3(const Transform<T>& p)
    {
        Matrix3 result(p.Rotation);
        result.SetTranslation(p.Translation);
        *this = result;
    }

    // C-interop support
    explicit Matrix3(const Matrix4<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_COMPILER_ASSERT(sizeof(s) == sizeof(Matrix3));
        memcpy(M, s.M, sizeof(M));
    }

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

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

    static Matrix3 FromString(const char* src)
    {
        Matrix3 result;
        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 const Matrix3& Identity()  { return IdentityValue; }

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

    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
    {
        Matrix4<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];
        return;
    }

    void operator= (const SymMat3<T>& b)
    {
        for (int i = 0; i < 3; i++)
            for (int j = 0; j < 3; j++)
                M[i][j] = 0;

        M[0][0] = b.v[0];
        M[0][1] = b.v[1];
        M[0][2] = b.v[2];
        M[1][1] = b.v[3];
        M[1][2] = b.v[4];
        M[2][2] = b.v[5];

        return;
    }

    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_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;
    }

    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 UPInt* rows, const UPInt* 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 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();

        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;
    }
    
};

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

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

template<typename 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] = 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;
    }

    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)]; }

    template<typename U>
    inline SymMat3<U> CastTo() const
    {
        return SymMat3<U>(static_cast<U>(v[0]), static_cast<U>(v[1]), static_cast<U>(v[2]),
                          static_cast<U>(v[3]), static_cast<U>(v[4]), static_cast<U>(v[5]));
    }

    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();

        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<typename 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<typename 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_*Math<T>::DegreeToRadFactor) { FixRange(); }

    T    Get(AngularUnits u = Radians) const       { return (u == Radians) ? a : a*Math<T>::RadToDegreeFactor; }
    void Set(const T& x, AngularUnits u = Radians) { a = (u == Radians) ? x : x*Math<T>::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; }
    
    T   Distance(const Angle& b)              { T c = fabs(a - b.a); return (c <= Math<T>::Pi) ? c : Math<T>::TwoPi - c; }

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 < -Math<T>::Pi)
            a += Math<T>::TwoPi;
        else if (a > Math<T>::Pi)
            a -= Math<T>::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 >= -Math<T>::Pi && a <= Math<T>::Pi)
            return;
        a = fmod(a,Math<T>::TwoPi);
        if (a < -Math<T>::Pi)
            a += Math<T>::TwoPi;
        else if (a > Math<T>::Pi)
            a -= Math<T>::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 RefCountBase<Plane<T> >
{
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 * 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;

} // Namespace OVR

#endif