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+/************************************************************************************
+
+PublicHeader: OVR_Kernel.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, LLC All Rights reserved.
+
+Licensed under the Oculus VR Rift SDK License Version 3.2 (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-3.2
+
+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
+
+
+// This file is intended to be independent of the rest of LibOVR and LibOVRKernel and thus
+// has no #include dependencies on either.
+
+#include <math.h>
+#include <stdint.h>
+#include <stdlib.h>
+#include <stdio.h>
+#include <string.h>
+#include <float.h>
+#include "../OVR_CAPI.h" // Currently required due to a dependence on the ovrFovPort_ declaration.
+
+#if defined(_MSC_VER)
+ #pragma warning(push)
+ #pragma warning(disable: 4127) // conditional expression is constant
+#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;
+
+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 Matrix3;
+template<class T> class Matrix4;
+template<class T> class Pose;
+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 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<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; };
+
+//------------------------------------------------------------------------------------//
+// ***** 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 4096*Epsilon;
+#define MATH_DOUBLE_SINGULARITYRADIUS 1e-12 // about 1-cos(.0001 degree), for gimbal lock numerical problems
+
+//------------------------------------------------------------------------------------//
+// ***** 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 64*EPSILON;
+#define MATH_FLOAT_SINGULARITYRADIUS 1e-7f // about 1-cos(.025 degree), for gimbal lock numerical problems
+
+
+
+// Single-precision Math constants class.
+template<>
+class Math<float>
+{
+public:
+ typedef double OtherFloatType;
+
+ 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
+};
+
+// 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
+};
+
+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; }
+
+// 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 : acos(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 : asin(x); }
+inline double Asin(double x) { return (x > 1.0) ? MATH_DOUBLE_PIOVER2 : (x < -1.0) ? -MATH_DOUBLE_PIOVER2 : asin(x); }
+
+#if defined(_MSC_VER)
+ inline int isnan(double x) { return ::_isnan(x); }
+#elif !defined(isnan) // Some libraries #define isnan.
+ 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:
+ 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); }
+
+ // Compare two vectors for equality with tolerance. Returns true if vectors match withing tolerance.
+ bool Compare(const Vector2&b, T tolerance =Math<T>::Tolerance())
+ {
+ return (fabs(b.x-x) < tolerance) && (fabs(b.y-y) < 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 != 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 != 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);
+ }
+
+ // Compare two vectors for equality with tolerance. Returns true if vectors match withing tolerance.
+ bool IsEqual(const Vector3&b, T tolerance = Math<T>::Tolerance())
+ {
+ return (fabs(b.x-x) < tolerance) &&
+ (fabs(b.y-y) < tolerance) &&
+ (fabs(b.z-z) < 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 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 != 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 != 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); }
+};
+
+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 float w_=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);
+ }
+
+ // Compare two vectors for equality with tolerance. Returns true if vectors match withing tolerance.
+ bool Compare(const Vector4&b, T tolerance = Math<T>::Tolerance())
+ {
+ return (fabs(b.x-x) < tolerance) &&
+ (fabs(b.y-y) < tolerance) &&
+ (fabs(b.z-z) < tolerance) &&
+ (fabs(b.w-w) < 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);}
+
+ // 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 != 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 != 0)
+ s = T(1) / s;
+ return *this * s;
+ }
+};
+
+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()
+ {
+ }
+
+ 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);
+ }
+
+ 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;
+
+
+
+//-----------------------------------------------------------------------------------
+// ***** 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) { }
+
+ 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() == 0)
+ {
+ // Assert if the axis is zero, but the angle isn't
+ OVR_MATH_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];
+ }
+
+ // Explicit identity for clairity, saves ambiguity at callsite about Quat()
+ // being initialized or not.
+ static Quat<T> Identity() { return Quat<T>(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= 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
+ {
+ 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
+ {
+ T s;
+ T sinHalfSquared = x*x + y*y + z*z;
+ if (sinHalfSquared < T(.0037)) // =~ sin(7/2 degrees)^2
+ {
+ // Max rotation magnitude error is (1 - 2*halfAngle/sin(halfAngle)) = -.0044 radians (.06%) at 7 degrees rotation
+ s = T(2);
+ }
+ 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 .0031 degrees at 5 degrees, or .06%
+ if (normalize && angleSquared > 0)
+ {
+ T invLen = T(1) / sqrt(s * angleSquared + c); // 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;
+ }
+ }
+
+ // 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; }
+
+ // Compare two vectors for equality with tolerance. Returns true if vectors match withing tolerance.
+ bool IsEqual(const Quat&b, T tolerance = Math<T>::Tolerance())
+ {
+ return Abs(Dot(b)) >= 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 2 * Acos(Abs(Dot(q)));
+ }
+
+ // Normalize
+ bool IsNormalized() const { return fabs(LengthSq() - T(1)) < Math<T>::Tolerance(); }
+
+ void Normalize()
+ {
+ T s = Length();
+ if (s != 0)
+ s = T(1) / s;
+ *this *= s;
+ }
+
+ Quat Normalized() const
+ {
+ T s = Length();
+ if (s != 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)
+ {
+ 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);
+ }
+ }
+ }
+
+ // Normalized linear interpolation of quaternions
+ Quat Lerp(const Quat& b, T s) const
+ {
+ return (*this * (T(1) - s) + b * (Dot(b) < 0 ? -s : s)).Normalized();
+ }
+
+ // Normalized linear interpolation of quaternions
+ // WARNING, POSSIBLE BUG: This function doesn't work like Lerp() for other classes!
+ // If f is 1, then full value is given to *this, not "other" like other Lerp() functions
+ 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
+ {
+ // 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
+ {
+ // 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(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(Vector3<T> angularVelocity, Vector3<T> angularAcceleration, T dt, T dtAcceleration) const
+ {
+ (void)dtAcceleration;
+
+ const Vector3<T>& omega = angularVelocity;
+ const Vector3<T>& omegaDot = angularAcceleration;
+
+ Vector3<T> omega1 = (omega + omegaDot * dtAcceleration);
+ Vector3<T> W = T(0.5)*dt*((omega + omega1) / T(2) + omega.Cross(omega1)*(dt * T(1)/T(12)));
+
+ // 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(yaw, pitch, roll, Rotate_CCW, Handed_R);
+ }
+
+ // 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_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_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;
+ if (c2 < T(-1) + Math<T>::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) - Math<T>::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]);
+ }
+ }
+};
+
+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.
+
+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) { }
+
+ 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); }
+
+ bool IsEqual(const Pose&b, T tolerance = Math<T>::Tolerance())
+ {
+ return Translation.IsEqual(b.Translation, tolerance) && Rotation.IsEqual(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]);
+ return Pose<T>(rotation, 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> 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));
+ }
+
+ 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, T dtAcceleration) const
+ {
+ return Pose(Rotation.TimeIntegrate(angularVelocity, angularAcceleration, dt, dtAcceleration),
+ Translation + linearVelocity*dt + linearAcceleration*dt*dt / T(2));
+ }
+};
+
+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] = 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;
+ }
+
+ 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 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] = 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] = 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] = 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 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] = 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;
+ }
+
+ void SetXBasis(const Vector3f & v)
+ {
+ M[0][0] = v.x;
+ M[1][0] = v.y;
+ M[2][0] = v.z;
+ }
+ Vector3f GetXBasis() const
+ {
+ return Vector3f(M[0][0], M[1][0], M[2][0]);
+ }
+
+ void SetYBasis(const Vector3f & v)
+ {
+ M[0][1] = v.x;
+ M[1][1] = v.y;
+ M[2][1] = v.z;
+ }
+ Vector3f GetYBasis() const
+ {
+ return Vector3f(M[0][1], M[1][1], M[2][1]);
+ }
+
+ void SetZBasis(const Vector3f & v)
+ {
+ M[0][2] = v.x;
+ M[1][2] = v.y;
+ M[2][2] = v.z;
+ }
+ Vector3f GetZBasis() const
+ {
+ return Vector3f(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;
+ }
+
+ Vector3<T> Transform(const Vector3<T>& v) const
+ {
+ const T rcpW = 1.0f / (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() * (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) const
+ {
+ OVR_MATH_STATIC_ASSERT((A1 != A2) && (A2 != A3) && (A1 != A3), "(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*((T)MATH_DOUBLE_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*((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] );
+ }
+
+ 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) const
+ {
+ OVR_MATH_STATIC_ASSERT(A1 != A2, "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*((T)MATH_DOUBLE_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 / (znear - zfar);
+ 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.
+ // 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 = tan(yfov * 0.5f);
+
+ m.M[0][0] = 1. / (aspect * tanHalfFov);
+ m.M[1][1] = 1. / tanHalfFov;
+ //m.M[2][2] = zfar / (znear - zfar);
+ 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.
+ // This is the case even for RHS coordinate 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<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] = 1;
+ M[0][1] = M[1][0] = M[2][0] = 0;
+ M[0][2] = M[1][2] = M[2][1] = 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)
+ {
+ 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;
+ }
+
+ Matrix3(T m11, T m22, T m33)
+ {
+ M[0][0] = m11; M[0][1] = 0; M[0][2] = 0;
+ M[1][0] = 0; M[1][1] = m22; M[1][2] = 0;
+ M[2][0] = 0; M[2][1] = 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] = 1;
+ M[0][1] = M[1][0] = M[2][0] = 0;
+ M[0][2] = M[1][2] = M[2][1] = 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 float rcpZ = 1.0f / (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);
+ }
+
+ // Angle in radians between two rotation matrices
+ // Uses identity trace(a*b') = 2*cos(theta) + 1
+ T Angle(const Matrix3& b) const
+ {
+ const Matrix3& a = *this;
+ return Acos(((a * b.Transposed()).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] = 1;
+ M[0][1] = M[1][0] = 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] = 0;
+ }
+
+ Matrix2(T m11, T m22)
+ {
+ M[0][0] = m11; M[0][1] = 0;
+ M[1][0] = 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] = 1;
+ M[0][1] = M[1][0] = 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;
+ }
+
+ 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] = 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; }
+
+ T Distance(const Angle& b) { T c = fabs(a - b.a); return (c <= ((T)MATH_DOUBLE_PI)) ? c : ((T)MATH_DOUBLE_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 < -((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 * 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;
+
+
+
+
+//-----------------------------------------------------------------------------------
+// ***** ScaleAndOffset2D
+
+struct ScaleAndOffset2D
+{
+ Vector2f Scale;
+ Vector2f Offset;
+
+ ScaleAndOffset2D(float sx = 0.0f, float sy = 0.0f, float ox = 0.0f, float oy = 0.0f)
+ : Scale(sx, sy), Offset(ox, oy)
+ { }
+};
+
+
+//-----------------------------------------------------------------------------------
+// ***** 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) { }
+
+ // C-interop support: FovPort <-> ovrFovPort (implementation in OVR_CAPI.cpp).
+ FovPort(const ovrFovPort &src)
+ : UpTan(src.UpTan), DownTan(src.DownTan), LeftTan(src.LeftTan), RightTan(src.RightTan)
+ { }
+
+ operator ovrFovPort () const
+ {
+ ovrFovPort result;
+ result.LeftTan = LeftTan;
+ result.RightTan = RightTan;
+ result.UpTan = UpTan;
+ result.DownTan = DownTan;
+ return result;
+ }
+
+ 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;
+ }
+};
+
+
+} // Namespace OVR
+
+
+#if defined(_MSC_VER)
+ #pragma warning(pop)
+#endif
+
+
+#endif
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