text: /*
  * Copyright (c) 2006-2009 Erin Catto http://www.box2d.org
  *
  * This software is provided 'as-is', without any express or implied
  * warranty.  In no event will the authors be held liable for any damages
  * arising from the use of this software.
  * Permission is granted to anyone to use this software for any purpose,
  * including commercial applications, and to alter it and redistribute it
  * freely, subject to the following restrictions:
  * 1. The origin of this software must not be misrepresented; you must not
  * claim that you wrote the original software. If you use this software
  * in a product, an acknowledgment in the product documentation would be
  * appreciated but is not required.
  * 2. Altered source versions must be plainly marked as such, and must not be
  * misrepresented as being the original software.
  * 3. This notice may not be removed or altered from any source distribution.
  */
  
  #ifndef B2_MATH_H
  #define B2_MATH_H
  
  #include <Box2D/Common/b2Settings.h>
  #include <math.h>
  
  /// This function is used to ensure that a floating point number is not a NaN or infinity.
  inline bool b2IsValid(float32 x)
  {
  	return std::isfinite(x);
  }
  
  /// This is a approximate yet fast inverse square-root.
  inline float32 b2InvSqrt(float32 x)
  {
  	union
  	{
  		float32 x;
  		int32 i;
  	} convert;
  
  	convert.x = x;
  	float32 xhalf = 0.5f * x;
  	convert.i = 0x5f3759df - (convert.i >> 1);
  	x = convert.x;
  	x = x * (1.5f - xhalf * x * x);
  	return x;
  }
  
  #define	b2Sqrt(x)	sqrtf(x)
  #define	b2Atan2(y, x)	atan2f(y, x)
  
  /// A 2D column vector.
  struct b2Vec2
  {
  	/// Default constructor does nothing (for performance).
  	b2Vec2() {}
  
  	/// Construct using coordinates.
  	b2Vec2(float32 x, float32 y) : x(x), y(y) {}
  
  	/// Set this vector to all zeros.
  	void SetZero() { x = 0.0f; y = 0.0f; }
  
  	/// Set this vector to some specified coordinates.
  	void Set(float32 x_, float32 y_) { x = x_; y = y_; }
  
  	/// Negate this vector.
  	b2Vec2 operator -() const { b2Vec2 v; v.Set(-x, -y); return v; }
  	
  	/// Read from and indexed element.
  	float32 operator () (int32 i) const
  	{
  		return (&x)[i];
  	}
  
  	/// Write to an indexed element.
  	float32& operator () (int32 i)
  	{
  		return (&x)[i];
  	}
  
  	/// Add a vector to this vector.
  	void operator += (const b2Vec2& v)
  	{
  		x += v.x; y += v.y;
  	}
  	
  	/// Subtract a vector from this vector.
  	void operator -= (const b2Vec2& v)
  	{
  		x -= v.x; y -= v.y;
  	}
  
  	/// Multiply this vector by a scalar.
  	void operator *= (float32 a)
  	{
  		x *= a; y *= a;
  	}
  
  	/// Get the length of this vector (the norm).
  	float32 Length() const
  	{
  		return b2Sqrt(x * x + y * y);
  	}
  
  	/// Get the length squared. For performance, use this instead of
  	/// b2Vec2::Length (if possible).
  	float32 LengthSquared() const
  	{
  		return x * x + y * y;
  	}
  
  	/// Convert this vector into a unit vector. Returns the length.
  	float32 Normalize()
  	{
  		float32 length = Length();
  		if (length < b2_epsilon)
  		{
  			return 0.0f;
  		}
  		float32 invLength = 1.0f / length;
  		x *= invLength;
  		y *= invLength;
  
  		return length;
  	}
  
  	/// Does this vector contain finite coordinates?
  	bool IsValid() const
  	{
  		return b2IsValid(x) && b2IsValid(y);
  	}
  
  	/// Get the skew vector such that dot(skew_vec, other) == cross(vec, other)
  	b2Vec2 Skew() const
  	{
  		return b2Vec2(-y, x);
  	}
  
  	float32 x, y;
  };
  
  /// A 2D column vector with 3 elements.
  struct b2Vec3
  {
  	/// Default constructor does nothing (for performance).
  	b2Vec3() {}
  
  	/// Construct using coordinates.
  	b2Vec3(float32 x, float32 y, float32 z) : x(x), y(y), z(z) {}
  
  	/// Set this vector to all zeros.
  	void SetZero() { x = 0.0f; y = 0.0f; z = 0.0f; }
  
  	/// Set this vector to some specified coordinates.
  	void Set(float32 x_, float32 y_, float32 z_) { x = x_; y = y_; z = z_; }
  
  	/// Negate this vector.
  	b2Vec3 operator -() const { b2Vec3 v; v.Set(-x, -y, -z); return v; }
  
  	/// Add a vector to this vector.
  	void operator += (const b2Vec3& v)
  	{
  		x += v.x; y += v.y; z += v.z;
  	}
  
  	/// Subtract a vector from this vector.
  	void operator -= (const b2Vec3& v)
  	{
  		x -= v.x; y -= v.y; z -= v.z;
  	}
  
  	/// Multiply this vector by a scalar.
  	void operator *= (float32 s)
  	{
  		x *= s; y *= s; z *= s;
  	}
  
  	float32 x, y, z;
  };
  
  /// A 2-by-2 matrix. Stored in column-major order.
  struct b2Mat22
  {
  	/// The default constructor does nothing (for performance).
  	b2Mat22() {}
  
  	/// Construct this matrix using columns.
  	b2Mat22(const b2Vec2& c1, const b2Vec2& c2)
  	{
  		ex = c1;
  		ey = c2;
  	}
  
  	/// Construct this matrix using scalars.
  	b2Mat22(float32 a11, float32 a12, float32 a21, float32 a22)
  	{
  		ex.x = a11; ex.y = a21;
  		ey.x = a12; ey.y = a22;
  	}
  
  	/// Initialize this matrix using columns.
  	void Set(const b2Vec2& c1, const b2Vec2& c2)
  	{
  		ex = c1;
  		ey = c2;
  	}
  
  	/// Set this to the identity matrix.
  	void SetIdentity()
  	{
  		ex.x = 1.0f; ey.x = 0.0f;
  		ex.y = 0.0f; ey.y = 1.0f;
  	}
  
  	/// Set this matrix to all zeros.
  	void SetZero()
  	{
  		ex.x = 0.0f; ey.x = 0.0f;
  		ex.y = 0.0f; ey.y = 0.0f;
  	}
  
  	b2Mat22 GetInverse() const
  	{
  		float32 a = ex.x, b = ey.x, c = ex.y, d = ey.y;
  		b2Mat22 B;
  		float32 det = a * d - b * c;
  		if (det != 0.0f)
  		{
  			det = 1.0f / det;
  		}
  		B.ex.x =  det * d;	B.ey.x = -det * b;
  		B.ex.y = -det * c;	B.ey.y =  det * a;
  		return B;
  	}
  
  	/// Solve A * x = b, where b is a column vector. This is more efficient
  	/// than computing the inverse in one-shot cases.
  	b2Vec2 Solve(const b2Vec2& b) const
  	{
  		float32 a11 = ex.x, a12 = ey.x, a21 = ex.y, a22 = ey.y;
  		float32 det = a11 * a22 - a12 * a21;
  		if (det != 0.0f)
  		{
  			det = 1.0f / det;
  		}
  		b2Vec2 x;
  		x.x = det * (a22 * b.x - a12 * b.y);
  		x.y = det * (a11 * b.y - a21 * b.x);
  		return x;
  	}
  
  	b2Vec2 ex, ey;
  };
  
  /// A 3-by-3 matrix. Stored in column-major order.
  struct b2Mat33
  {
  	/// The default constructor does nothing (for performance).
  	b2Mat33() {}
  
  	/// Construct this matrix using columns.
  	b2Mat33(const b2Vec3& c1, const b2Vec3& c2, const b2Vec3& c3)
  	{
  		ex = c1;
  		ey = c2;
  		ez = c3;
  	}
  
  	/// Set this matrix to all zeros.
  	void SetZero()
  	{
  		ex.SetZero();
  		ey.SetZero();
  		ez.SetZero();
  	}
  
  	/// Solve A * x = b, where b is a column vector. This is more efficient
  	/// than computing the inverse in one-shot cases.
  	b2Vec3 Solve33(const b2Vec3& b) const;
  
  	/// Solve A * x = b, where b is a column vector. This is more efficient
  	/// than computing the inverse in one-shot cases. Solve only the upper
  	/// 2-by-2 matrix equation.
  	b2Vec2 Solve22(const b2Vec2& b) const;
  
  	/// Get the inverse of this matrix as a 2-by-2.
  	/// Returns the zero matrix if singular.
  	void GetInverse22(b2Mat33* M) const;
  
  	/// Get the symmetric inverse of this matrix as a 3-by-3.
  	/// Returns the zero matrix if singular.
  	void GetSymInverse33(b2Mat33* M) const;
  
  	b2Vec3 ex, ey, ez;
  };
  
  /// Rotation
  struct b2Rot
  {
  	b2Rot() {}
  
  	/// Initialize from an angle in radians
  	explicit b2Rot(float32 angle)
  	{
  		/// TODO_ERIN optimize
  		s = sinf(angle);
  		c = cosf(angle);
  	}
  
  	/// Set using an angle in radians.
  	void Set(float32 angle)
  	{
  		/// TODO_ERIN optimize
  		s = sinf(angle);
  		c = cosf(angle);
  	}
  
  	/// Set to the identity rotation
  	void SetIdentity()
  	{
  		s = 0.0f;
  		c = 1.0f;
  	}
  
  	/// Get the angle in radians
  	float32 GetAngle() const
  	{
  		return b2Atan2(s, c);
  	}
  
  	/// Get the x-axis
  	b2Vec2 GetXAxis() const
  	{
  		return b2Vec2(c, s);
  	}
  
  	/// Get the u-axis
  	b2Vec2 GetYAxis() const
  	{
  		return b2Vec2(-s, c);
  	}
  
  	/// Sine and cosine
  	float32 s, c;
  };
  
  /// A transform contains translation and rotation. It is used to represent
  /// the position and orientation of rigid frames.
  struct b2Transform
  {
  	/// The default constructor does nothing.
  	b2Transform() {}
  
  	/// Initialize using a position vector and a rotation.
  	b2Transform(const b2Vec2& position, const b2Rot& rotation) : p(position), q(rotation) {}
  
  	/// Set this to the identity transform.
  	void SetIdentity()
  	{
  		p.SetZero();
  		q.SetIdentity();
  	}
  
  	/// Set this based on the position and angle.
  	void Set(const b2Vec2& position, float32 angle)
  	{
  		p = position;
  		q.Set(angle);
  	}
  
  	b2Vec2 p;
  	b2Rot q;
  };
  
  /// This describes the motion of a body/shape for TOI computation.
  /// Shapes are defined with respect to the body origin, which may
  /// no coincide with the center of mass. However, to support dynamics
  /// we must interpolate the center of mass position.
  struct b2Sweep
  {
  	/// Get the interpolated transform at a specific time.
  	/// @param beta is a factor in [0,1], where 0 indicates alpha0.
  	void GetTransform(b2Transform* xfb, float32 beta) const;
  
  	/// Advance the sweep forward, yielding a new initial state.
  	/// @param alpha the new initial time.
  	void Advance(float32 alpha);
  
  	/// Normalize the angles.
  	void Normalize();
  
  	b2Vec2 localCenter;	///< local center of mass position
  	b2Vec2 c0, c;		///< center world positions
  	float32 a0, a;		///< world angles
  
  	/// Fraction of the current time step in the range [0,1]
  	/// c0 and a0 are the positions at alpha0.
  	float32 alpha0;
  };
  
  /// Useful constant
  extern const b2Vec2 b2Vec2_zero;
  
  /// Perform the dot product on two vectors.
  inline float32 b2Dot(const b2Vec2& a, const b2Vec2& b)
  {
  	return a.x * b.x + a.y * b.y;
  }
  
  /// Perform the cross product on two vectors. In 2D this produces a scalar.
  inline float32 b2Cross(const b2Vec2& a, const b2Vec2& b)
  {
  	return a.x * b.y - a.y * b.x;
  }
  
  /// Perform the cross product on a vector and a scalar. In 2D this produces
  /// a vector.
  inline b2Vec2 b2Cross(const b2Vec2& a, float32 s)
  {
  	return b2Vec2(s * a.y, -s * a.x);
  }
  
  /// Perform the cross product on a scalar and a vector. In 2D this produces
  /// a vector.
  inline b2Vec2 b2Cross(float32 s, const b2Vec2& a)
  {
  	return b2Vec2(-s * a.y, s * a.x);
  }
  
  /// Multiply a matrix times a vector. If a rotation matrix is provided,
  /// then this transforms the vector from one frame to another.
  inline b2Vec2 b2Mul(const b2Mat22& A, const b2Vec2& v)
  {
  	return b2Vec2(A.ex.x * v.x + A.ey.x * v.y, A.ex.y * v.x + A.ey.y * v.y);
  }
  
  /// Multiply a matrix transpose times a vector. If a rotation matrix is provided,
  /// then this transforms the vector from one frame to another (inverse transform).
  inline b2Vec2 b2MulT(const b2Mat22& A, const b2Vec2& v)
  {
  	return b2Vec2(b2Dot(v, A.ex), b2Dot(v, A.ey));
  }
  
  /// Add two vectors component-wise.
  inline b2Vec2 operator + (const b2Vec2& a, const b2Vec2& b)
  {
  	return b2Vec2(a.x + b.x, a.y + b.y);
  }
  
  /// Subtract two vectors component-wise.
  inline b2Vec2 operator - (const b2Vec2& a, const b2Vec2& b)
  {
  	return b2Vec2(a.x - b.x, a.y - b.y);
  }
  
  inline b2Vec2 operator * (float32 s, const b2Vec2& a)
  {
  	return b2Vec2(s * a.x, s * a.y);
  }
  
  inline bool operator == (const b2Vec2& a, const b2Vec2& b)
  {
  	return a.x == b.x && a.y == b.y;
  }
  
  inline float32 b2Distance(const b2Vec2& a, const b2Vec2& b)
  {
  	b2Vec2 c = a - b;
  	return c.Length();
  }
  
  inline float32 b2DistanceSquared(const b2Vec2& a, const b2Vec2& b)
  {
  	b2Vec2 c = a - b;
  	return b2Dot(c, c);
  }
  
  inline b2Vec3 operator * (float32 s, const b2Vec3& a)
  {
  	return b2Vec3(s * a.x, s * a.y, s * a.z);
  }
  
  /// Add two vectors component-wise.
  inline b2Vec3 operator + (const b2Vec3& a, const b2Vec3& b)
  {
  	return b2Vec3(a.x + b.x, a.y + b.y, a.z + b.z);
  }
  
  /// Subtract two vectors component-wise.
  inline b2Vec3 operator - (const b2Vec3& a, const b2Vec3& b)
  {
  	return b2Vec3(a.x - b.x, a.y - b.y, a.z - b.z);
  }
  
  /// Perform the dot product on two vectors.
  inline float32 b2Dot(const b2Vec3& a, const b2Vec3& b)
  {
  	return a.x * b.x + a.y * b.y + a.z * b.z;
  }
  
  /// Perform the cross product on two vectors.
  inline b2Vec3 b2Cross(const b2Vec3& a, const b2Vec3& b)
  {
  	return b2Vec3(a.y * b.z - a.z * b.y, a.z * b.x - a.x * b.z, a.x * b.y - a.y * b.x);
  }
  
  inline b2Mat22 operator + (const b2Mat22& A, const b2Mat22& B)
  {
  	return b2Mat22(A.ex + B.ex, A.ey + B.ey);
  }
  
  // A * B
  inline b2Mat22 b2Mul(const b2Mat22& A, const b2Mat22& B)
  {
  	return b2Mat22(b2Mul(A, B.ex), b2Mul(A, B.ey));
  }
  
  // A^T * B
  inline b2Mat22 b2MulT(const b2Mat22& A, const b2Mat22& B)
  {
  	b2Vec2 c1(b2Dot(A.ex, B.ex), b2Dot(A.ey, B.ex));
  	b2Vec2 c2(b2Dot(A.ex, B.ey), b2Dot(A.ey, B.ey));
  	return b2Mat22(c1, c2);
  }
  
  /// Multiply a matrix times a vector.
  inline b2Vec3 b2Mul(const b2Mat33& A, const b2Vec3& v)
  {
  	return v.x * A.ex + v.y * A.ey + v.z * A.ez;
  }
  
  /// Multiply a matrix times a vector.
  inline b2Vec2 b2Mul22(const b2Mat33& A, const b2Vec2& v)
  {
  	return b2Vec2(A.ex.x * v.x + A.ey.x * v.y, A.ex.y * v.x + A.ey.y * v.y);
  }
  
  /// Multiply two rotations: q * r
  inline b2Rot b2Mul(const b2Rot& q, const b2Rot& r)
  {
  	// [qc -qs] * [rc -rs] = [qc*rc-qs*rs -qc*rs-qs*rc]
  	// [qs  qc]   [rs  rc]   [qs*rc+qc*rs -qs*rs+qc*rc]
  	// s = qs * rc + qc * rs
  	// c = qc * rc - qs * rs
  	b2Rot qr;
  	qr.s = q.s * r.c + q.c * r.s;
  	qr.c = q.c * r.c - q.s * r.s;
  	return qr;
  }
  
  /// Transpose multiply two rotations: qT * r
  inline b2Rot b2MulT(const b2Rot& q, const b2Rot& r)
  {
  	// [ qc qs] * [rc -rs] = [qc*rc+qs*rs -qc*rs+qs*rc]
  	// [-qs qc]   [rs  rc]   [-qs*rc+qc*rs qs*rs+qc*rc]
  	// s = qc * rs - qs * rc
  	// c = qc * rc + qs * rs
  	b2Rot qr;
  	qr.s = q.c * r.s - q.s * r.c;
  	qr.c = q.c * r.c + q.s * r.s;
  	return qr;
  }
  
  /// Rotate a vector
  inline b2Vec2 b2Mul(const b2Rot& q, const b2Vec2& v)
  {
  	return b2Vec2(q.c * v.x - q.s * v.y, q.s * v.x + q.c * v.y);
  }
  
  /// Inverse rotate a vector
  inline b2Vec2 b2MulT(const b2Rot& q, const b2Vec2& v)
  {
  	return b2Vec2(q.c * v.x + q.s * v.y, -q.s * v.x + q.c * v.y);
  }
  
  inline b2Vec2 b2Mul(const b2Transform& T, const b2Vec2& v)
  {
  	float32 x = (T.q.c * v.x - T.q.s * v.y) + T.p.x;
  	float32 y = (T.q.s * v.x + T.q.c * v.y) + T.p.y;
  
  	return b2Vec2(x, y);
  }
  
  inline b2Vec2 b2MulT(const b2Transform& T, const b2Vec2& v)
  {
  	float32 px = v.x - T.p.x;
  	float32 py = v.y - T.p.y;
  	float32 x = (T.q.c * px + T.q.s * py);
  	float32 y = (-T.q.s * px + T.q.c * py);
  
  	return b2Vec2(x, y);
  }
  
  // v2 = A.q.Rot(B.q.Rot(v1) + B.p) + A.p
  //    = (A.q * B.q).Rot(v1) + A.q.Rot(B.p) + A.p
  inline b2Transform b2Mul(const b2Transform& A, const b2Transform& B)
  {
  	b2Transform C;
  	C.q = b2Mul(A.q, B.q);
  	C.p = b2Mul(A.q, B.p) + A.p;
  	return C;
  }
  
  // v2 = A.q' * (B.q * v1 + B.p - A.p)
  //    = A.q' * B.q * v1 + A.q' * (B.p - A.p)
  inline b2Transform b2MulT(const b2Transform& A, const b2Transform& B)
  {
  	b2Transform C;
  	C.q = b2MulT(A.q, B.q);
  	C.p = b2MulT(A.q, B.p - A.p);
  	return C;
  }
  
  template <typename T>
  inline T b2Abs(T a)
  {
  	return a > T(0) ? a : -a;
  }
  
  inline b2Vec2 b2Abs(const b2Vec2& a)
  {
  	return b2Vec2(b2Abs(a.x), b2Abs(a.y));
  }
  
  inline b2Mat22 b2Abs(const b2Mat22& A)
  {
  	return b2Mat22(b2Abs(A.ex), b2Abs(A.ey));
  }
  
  template <typename T>
  inline T b2Min(T a, T b)
  {
  	return a < b ? a : b;
  }
  
  inline b2Vec2 b2Min(const b2Vec2& a, const b2Vec2& b)
  {
  	return b2Vec2(b2Min(a.x, b.x), b2Min(a.y, b.y));
  }
  
  template <typename T>
  inline T b2Max(T a, T b)
  {
  	return a > b ? a : b;
  }
  
  inline b2Vec2 b2Max(const b2Vec2& a, const b2Vec2& b)
  {
  	return b2Vec2(b2Max(a.x, b.x), b2Max(a.y, b.y));
  }
  
  template <typename T>
  inline T b2Clamp(T a, T low, T high)
  {
  	return b2Max(low, b2Min(a, high));
  }
  
  inline b2Vec2 b2Clamp(const b2Vec2& a, const b2Vec2& low, const b2Vec2& high)
  {
  	return b2Max(low, b2Min(a, high));
  }
  
  template<typename T> inline void b2Swap(T& a, T& b)
  {
  	T tmp = a;
  	a = b;
  	b = tmp;
  }
  
  /// "Next Largest Power of 2
  /// Given a binary integer value x, the next largest power of 2 can be computed by a SWAR algorithm
  /// that recursively "folds" the upper bits into the lower bits. This process yields a bit vector with
  /// the same most significant 1 as x, but all 1's below it. Adding 1 to that value yields the next
  /// largest power of 2. For a 32-bit value:"
  inline uint32 b2NextPowerOfTwo(uint32 x)
  {
  	x |= (x >> 1);
  	x |= (x >> 2);
  	x |= (x >> 4);
  	x |= (x >> 8);
  	x |= (x >> 16);
  	return x + 1;
  }
  
  inline bool b2IsPowerOfTwo(uint32 x)
  {
  	bool result = x > 0 && (x & (x - 1)) == 0;
  	return result;
  }
  
  inline void b2Sweep::GetTransform(b2Transform* xf, float32 beta) const
  {
  	xf->p = (1.0f - beta) * c0 + beta * c;
  	float32 angle = (1.0f - beta) * a0 + beta * a;
  	xf->q.Set(angle);
  
  	// Shift to origin
  	xf->p -= b2Mul(xf->q, localCenter);
  }
  
  inline void b2Sweep::Advance(float32 alpha)
  {
  	b2Assert(alpha0 < 1.0f);
  	float32 beta = (alpha - alpha0) / (1.0f - alpha0);
  	c0 += beta * (c - c0);
  	a0 += beta * (a - a0);
  	alpha0 = alpha;
  }
  
  /// Normalize an angle in radians to be between -pi and pi
  inline void b2Sweep::Normalize()
  {
  	float32 twoPi = 2.0f * b2_pi;
  	float32 d =  twoPi * floorf(a0 / twoPi);
  	a0 -= d;
  	a -= d;
  }
  
  #endif
  