b2Math.h

/*
* 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>

inline bool b2IsValid(float32 x)
{
    int32 ix = *reinterpret_cast<int32*>(&x);
    return (ix & 0x7f800000) != 0x7f800000;
}

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)

struct b2Vec2
{
    b2Vec2() {}

    b2Vec2(float32 x, float32 y) : x(x), y(y) {}

    void SetZero() { x = 0.0f; y = 0.0f; }

    void Set(float32 x_, float32 y_) { x = x_; y = y_; }

    b2Vec2 operator -() const { b2Vec2 v; v.Set(-x, -y); return v; }
    
    float32 operator () (int32 i) const
    {
        return (&x)[i];
    }

    float32& operator () (int32 i)
    {
        return (&x)[i];
    }

    void operator += (const b2Vec2& v)
    {
        x += v.x; y += v.y;
    }
    
    void operator -= (const b2Vec2& v)
    {
        x -= v.x; y -= v.y;
    }

    void operator *= (float32 a)
    {
        x *= a; y *= a;
    }

    float32 Length() const
    {
        return b2Sqrt(x * x + y * y);
    }

    float32 LengthSquared() const
    {
        return x * x + y * y;
    }

    float32 Normalize()
    {
        float32 length = Length();
        if (length < b2_epsilon)
        {
            return 0.0f;
        }
        float32 invLength = 1.0f / length;
        x *= invLength;
        y *= invLength;

        return length;
    }

    bool IsValid() const
    {
        return b2IsValid(x) && b2IsValid(y);
    }

    b2Vec2 Skew() const
    {
        return b2Vec2(-y, x);
    }

    float32 x, y;
};

struct b2Vec3
{
    b2Vec3() {}

    b2Vec3(float32 x, float32 y, float32 z) : x(x), y(y), z(z) {}

    void SetZero() { x = 0.0f; y = 0.0f; z = 0.0f; }

    void Set(float32 x_, float32 y_, float32 z_) { x = x_; y = y_; z = z_; }

    b2Vec3 operator -() const { b2Vec3 v; v.Set(-x, -y, -z); return v; }

    void operator += (const b2Vec3& v)
    {
        x += v.x; y += v.y; z += v.z;
    }

    void operator -= (const b2Vec3& v)
    {
        x -= v.x; y -= v.y; z -= v.z;
    }

    void operator *= (float32 s)
    {
        x *= s; y *= s; z *= s;
    }

    float32 x, y, z;
};

struct b2Mat22
{
    b2Mat22() {}

    b2Mat22(const b2Vec2& c1, const b2Vec2& c2)
    {
        ex = c1;
        ey = c2;
    }

    b2Mat22(float32 a11, float32 a12, float32 a21, float32 a22)
    {
        ex.x = a11; ex.y = a21;
        ey.x = a12; ey.y = a22;
    }

    void Set(const b2Vec2& c1, const b2Vec2& c2)
    {
        ex = c1;
        ey = c2;
    }

    void SetIdentity()
    {
        ex.x = 1.0f; ey.x = 0.0f;
        ex.y = 0.0f; ey.y = 1.0f;
    }

    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;
    }

    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;
};

struct b2Mat33
{
    b2Mat33() {}

    b2Mat33(const b2Vec3& c1, const b2Vec3& c2, const b2Vec3& c3)
    {
        ex = c1;
        ey = c2;
        ez = c3;
    }

    void SetZero()
    {
        ex.SetZero();
        ey.SetZero();
        ez.SetZero();
    }

    b2Vec3 Solve33(const b2Vec3& b) const;

    b2Vec2 Solve22(const b2Vec2& b) const;

    void GetInverse22(b2Mat33* M) const;

    void GetSymInverse33(b2Mat33* M) const;

    b2Vec3 ex, ey, ez;
};

struct b2Rot
{
    b2Rot() {}

    explicit b2Rot(float32 angle)
    {
        s = sinf(angle);
        c = cosf(angle);
    }

    void Set(float32 angle)
    {
        s = sinf(angle);
        c = cosf(angle);
    }

    void SetIdentity()
    {
        s = 0.0f;
        c = 1.0f;
    }

    float32 GetAngle() const
    {
        return b2Atan2(s, c);
    }

    b2Vec2 GetXAxis() const
    {
        return b2Vec2(c, s);
    }

    b2Vec2 GetYAxis() const
    {
        return b2Vec2(-s, c);
    }

    float32 s, c;
};

struct b2Transform
{
    b2Transform() {}

    b2Transform(const b2Vec2& position, const b2Rot& rotation) : p(position), q(rotation) {}

    void SetIdentity()
    {
        p.SetZero();
        q.SetIdentity();
    }

    void Set(const b2Vec2& position, float32 angle)
    {
        p = position;
        q.Set(angle);
    }

    b2Vec2 p;
    b2Rot q;
};

struct b2Sweep
{
    void GetTransform(b2Transform* xfb, float32 beta) const;

    void Advance(float32 alpha);

    void Normalize();

    b2Vec2 localCenter; 
    b2Vec2 c0, c;       
    float32 a0, a;      

    float32 alpha0;
};

extern const b2Vec2 b2Vec2_zero;

inline float32 b2Dot(const b2Vec2& a, const b2Vec2& b)
{
    return a.x * b.x + a.y * b.y;
}

inline float32 b2Cross(const b2Vec2& a, const b2Vec2& b)
{
    return a.x * b.y - a.y * b.x;
}

inline b2Vec2 b2Cross(const b2Vec2& a, float32 s)
{
    return b2Vec2(s * a.y, -s * a.x);
}

inline b2Vec2 b2Cross(float32 s, const b2Vec2& a)
{
    return b2Vec2(-s * a.y, s * a.x);
}

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);
}

inline b2Vec2 b2MulT(const b2Mat22& A, const b2Vec2& v)
{
    return b2Vec2(b2Dot(v, A.ex), b2Dot(v, A.ey));
}

inline b2Vec2 operator + (const b2Vec2& a, const b2Vec2& b)
{
    return b2Vec2(a.x + b.x, a.y + b.y);
}

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);
}

inline b2Vec3 operator + (const b2Vec3& a, const b2Vec3& b)
{
    return b2Vec3(a.x + b.x, a.y + b.y, a.z + b.z);
}

inline b2Vec3 operator - (const b2Vec3& a, const b2Vec3& b)
{
    return b2Vec3(a.x - b.x, a.y - b.y, a.z - b.z);
}

inline float32 b2Dot(const b2Vec3& a, const b2Vec3& b)
{
    return a.x * b.x + a.y * b.y + a.z * b.z;
}

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);
}

inline b2Vec3 b2Mul(const b2Mat33& A, const b2Vec3& v)
{
    return v.x * A.ex + v.y * A.ey + v.z * A.ez;
}

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);
}

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;
}

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;
}

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);
}

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;
}

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;
}

inline void b2Sweep::Normalize()
{
    float32 twoPi = 2.0f * b2_pi;
    float32 d =  twoPi * floorf(a0 / twoPi);
    a0 -= d;
    a -= d;
}

#endif