source: trunk/src/math/vector.cpp @ 1804 Tweet

//
// Lol Engine
//
// Copyright: (c) 2010-2012 Sam Hocevar <sam@hocevar.net>
//   This program is free software; you can redistribute it and/or
//   modify it under the terms of the Do What The Fuck You Want To
//   Public License, Version 2, as published by Sam Hocevar. See
//   http://sam.zoy.org/projects/COPYING.WTFPL for more details.
//

#if defined HAVE_CONFIG_H
#   include "config.h"
#endif

#if defined _XBOX
#   define _USE_MATH_DEFINES /* for M_PI */
#   include <xtl.h>
#   undef near /* Fuck Microsoft */
#   undef far /* Fuck Microsoft again */
#elif defined _WIN32
#   define _USE_MATH_DEFINES /* for M_PI */
#   define WIN32_LEAN_AND_MEAN
#   include <windows.h>
#   undef near /* Fuck Microsoft */
#   undef far /* Fuck Microsoft again */
#endif

#include <cstdlib> /* free() */
#include <cstring> /* strdup() */

#include <ostream> /* std::ostream */

#include "core.h"

using namespace std;

namespace lol
{

static inline float det2(float a, float b,
                         float c, float d)
{
    return a * d - b * c;
}

static inline float det3(float a, float b, float c,
                         float d, float e, float f,
                         float g, float h, float i)
{
    return a * (e * i - h * f)
         + b * (f * g - i * d)
         + c * (d * h - g * e);
}

static inline float cofact(mat2 const &mat, int i, int j)
{
    return mat[(i + 1) & 1][(j + 1) & 1] * (((i + j) & 1) ? -1.0f : 1.0f);
}

static inline float cofact(mat3 const &mat, int i, int j)
{
    return det2(mat[(i + 1) % 3][(j + 1) % 3],
                mat[(i + 2) % 3][(j + 1) % 3],
                mat[(i + 1) % 3][(j + 2) % 3],
                mat[(i + 2) % 3][(j + 2) % 3]);
}

static inline float cofact(mat4 const &mat, int i, int j)
{
    return det3(mat[(i + 1) & 3][(j + 1) & 3],
                mat[(i + 2) & 3][(j + 1) & 3],
                mat[(i + 3) & 3][(j + 1) & 3],
                mat[(i + 1) & 3][(j + 2) & 3],
                mat[(i + 2) & 3][(j + 2) & 3],
                mat[(i + 3) & 3][(j + 2) & 3],
                mat[(i + 1) & 3][(j + 3) & 3],
                mat[(i + 2) & 3][(j + 3) & 3],
                mat[(i + 3) & 3][(j + 3) & 3]) * (((i + j) & 1) ? -1.0f : 1.0f);
}

template<> float determinant(mat2 const &mat)
{
    return mat[0][0] * mat[1][1] - mat[0][1] * mat[1][0];
}

template<> mat2 transpose(mat2 const &mat)
{
    mat2 ret;
    for (int j = 0; j < 2; j++)
        for (int i = 0; i < 2; i++)
            ret[j][i] = mat[i][j];
    return ret;
}

template<> mat2 inverse(mat2 const &mat)
{
    mat2 ret;
    float d = determinant(mat);
    if (d)
    {
        d = 1.0f / d;
        for (int j = 0; j < 2; j++)
            for (int i = 0; i < 2; i++)
                ret[j][i] = cofact(mat, i, j) * d;
    }
    return ret;
}

template<> float determinant(mat3 const &mat)
{
    return det3(mat[0][0], mat[0][1], mat[0][2],
                mat[1][0], mat[1][1], mat[1][2],
                mat[2][0], mat[2][1], mat[2][2]);
}

template<> mat3 transpose(mat3 const &mat)
{
    mat3 ret;
    for (int j = 0; j < 3; j++)
        for (int i = 0; i < 3; i++)
            ret[j][i] = mat[i][j];
    return ret;
}

template<> mat3 inverse(mat3 const &mat)
{
    mat3 ret;
    float d = determinant(mat);
    if (d)
    {
        d = 1.0f / d;
        for (int j = 0; j < 3; j++)
            for (int i = 0; i < 3; i++)
                ret[j][i] = cofact(mat, i, j) * d;
    }
    return ret;
}

template<> float determinant(mat4 const &mat)
{
    float ret = 0;
    for (int n = 0; n < 4; n++)
        ret += mat[n][0] * cofact(mat, n, 0);
    return ret;
}

template<> mat4 transpose(mat4 const &mat)
{
    mat4 ret;
    for (int j = 0; j < 4; j++)
        for (int i = 0; i < 4; i++)
            ret[j][i] = mat[i][j];
    return ret;
}

template<> mat4 inverse(mat4 const &mat)
{
    mat4 ret;
    float d = determinant(mat);
    if (d)
    {
        d = 1.0f / d;
        for (int j = 0; j < 4; j++)
            for (int i = 0; i < 4; i++)
                ret[j][i] = cofact(mat, i, j) * d;
    }
    return ret;
}

template<> void vec2::printf() const
{
    Log::Debug("[ %6.6f %6.6f ]\n", x, y);
}

template<> void ivec2::printf() const
{
    Log::Debug("[ %i %i ]\n", x, y);
}

template<> void cmplx::printf() const
{
    Log::Debug("[ %6.6f %6.6f ]\n", x, y);
}

template<> void vec3::printf() const
{
    Log::Debug("[ %6.6f %6.6f %6.6f ]\n", x, y, z);
}

template<> void ivec3::printf() const
{
    Log::Debug("[ %i %i %i ]\n", x, y, z);
}

template<> void vec4::printf() const
{
    Log::Debug("[ %6.6f %6.6f %6.6f %6.6f ]\n", x, y, z, w);
}

template<> void ivec4::printf() const
{
    Log::Debug("[ %i %i %i %i ]\n", x, y, z, w);
}

template<> void quat::printf() const
{
    Log::Debug("[ %6.6f %6.6f %6.6f %6.6f ]\n", w, x, y, z);
}

template<> void mat2::printf() const
{
    mat2 const &p = *this;

    Log::Debug("[ %6.6f %6.6f\n", p[0][0], p[1][0]);
    Log::Debug("  %6.6f %6.6f ]\n", p[0][1], p[1][1]);
}

template<> void mat3::printf() const
{
    mat3 const &p = *this;

    Log::Debug("[ %6.6f %6.6f %6.6f\n", p[0][0], p[1][0], p[2][0]);
    Log::Debug("  %6.6f %6.6f %6.6f\n", p[0][1], p[1][1], p[2][1]);
    Log::Debug("  %6.6f %6.6f %6.6f ]\n", p[0][2], p[1][2], p[2][2]);
}

template<> void mat4::printf() const
{
    mat4 const &p = *this;

    Log::Debug("[ %6.6f %6.6f %6.6f %6.6f\n",
               p[0][0], p[1][0], p[2][0], p[3][0]);
    Log::Debug("  %6.6f %6.6f %6.6f %6.6f\n",
               p[0][1], p[1][1], p[2][1], p[3][1]);
    Log::Debug("  %6.6f %6.6f %6.6f %6.6f\n",
               p[0][2], p[1][2], p[2][2], p[3][2]);
    Log::Debug("  %6.6f %6.6f %6.6f %6.6f ]\n",
               p[0][3], p[1][3], p[2][3], p[3][3]);
}

template<> std::ostream &operator<<(std::ostream &stream, ivec2 const &v)
{
    return stream << "(" << v.x << ", " << v.y << ")";
}

template<> std::ostream &operator<<(std::ostream &stream, icmplx const &v)
{
    return stream << "(" << v.x << ", " << v.y << ")";
}

template<> std::ostream &operator<<(std::ostream &stream, ivec3 const &v)
{
    return stream << "(" << v.x << ", " << v.y << ", " << v.z << ")";
}

template<> std::ostream &operator<<(std::ostream &stream, ivec4 const &v)
{
    return stream << "(" << v.x << ", " << v.y << ", "
                         << v.z << ", " << v.w << ")";
}

template<> std::ostream &operator<<(std::ostream &stream, iquat const &v)
{
    return stream << "(" << v.x << ", " << v.y << ", "
                         << v.z << ", " << v.w << ")";
}

template<> std::ostream &operator<<(std::ostream &stream, vec2 const &v)
{
    return stream << "(" << v.x << ", " << v.y << ")";
}

template<> std::ostream &operator<<(std::ostream &stream, cmplx const &v)
{
    return stream << "(" << v.x << ", " << v.y << ")";
}

template<> std::ostream &operator<<(std::ostream &stream, vec3 const &v)
{
    return stream << "(" << v.x << ", " << v.y << ", " << v.z << ")";
}

template<> std::ostream &operator<<(std::ostream &stream, vec4 const &v)
{
    return stream << "(" << v.x << ", " << v.y << ", "
                         << v.z << ", " << v.w << ")";
}

template<> std::ostream &operator<<(std::ostream &stream, quat const &v)
{
    return stream << "(" << v.x << ", " << v.y << ", "
                         << v.z << ", " << v.w << ")";
}

template<> std::ostream &operator<<(std::ostream &stream, mat4 const &m)
{
    stream << "((" << m[0][0] << ", " << m[1][0]
            << ", " << m[2][0] << ", " << m[3][0] << "), ";
    stream << "(" << m[0][1] << ", " << m[1][1]
           << ", " << m[2][1] << ", " << m[3][1] << "), ";
    stream << "(" << m[0][2] << ", " << m[1][2]
           << ", " << m[2][2] << ", " << m[3][2] << "), ";
    stream << "(" << m[0][3] << ", " << m[1][3]
           << ", " << m[2][3] << ", " << m[3][3] << "))";
    return stream;
}

template<> mat3 mat3::scale(float x, float y, float z)
{
    mat3 ret(1.0f);

    ret[0][0] = x;
    ret[1][1] = y;
    ret[2][2] = z;

    return ret;
}

template<> mat3 mat3::scale(vec3 v)
{
    return scale(v.x, v.y, v.z);
}

template<> mat4 mat4::translate(float x, float y, float z)
{
    mat4 ret(1.0f);
    ret[3][0] = x;
    ret[3][1] = y;
    ret[3][2] = z;
    return ret;
}

template<> mat4 mat4::translate(vec3 v)
{
    return translate(v.x, v.y, v.z);
}

template<> mat2 mat2::rotate(float angle)
{
    angle *= (M_PI / 180.0f);

    float st = sin(angle);
    float ct = cos(angle);

    mat2 ret;

    ret[0][0] = ct;
    ret[0][1] = st;

    ret[1][0] = -st;
    ret[1][1] = ct;

    return ret;
}

template<> mat3 mat3::rotate(float angle, float x, float y, float z)
{
    angle *= (M_PI / 180.0f);

    float st = sin(angle);
    float ct = cos(angle);

    float len = std::sqrt(x * x + y * y + z * z);
    float invlen = len ? 1.0f / len : 0.0f;
    x *= invlen;
    y *= invlen;
    z *= invlen;

    float mtx = (1.0f - ct) * x;
    float mty = (1.0f - ct) * y;
    float mtz = (1.0f - ct) * z;

    mat3 ret;

    ret[0][0] = x * mtx + ct;
    ret[0][1] = x * mty + st * z;
    ret[0][2] = x * mtz - st * y;

    ret[1][0] = y * mtx - st * z;
    ret[1][1] = y * mty + ct;
    ret[1][2] = y * mtz + st * x;

    ret[2][0] = z * mtx + st * y;
    ret[2][1] = z * mty - st * x;
    ret[2][2] = z * mtz + ct;

    return ret;
}

template<> mat3 mat3::rotate(float angle, vec3 v)
{
    return rotate(angle, v.x, v.y, v.z);
}

template<> mat3::Mat3(quat const &q)
{
    float n = norm(q);

    if (!n)
    {
        for (int j = 0; j < 3; j++)
            for (int i = 0; i < 3; i++)
                (*this)[i][j] = (i == j) ? 1.f : 0.f;
        return;
    }

    float s = 2.0f / n;

    v0[0] = 1.0f - s * (q.y * q.y + q.z * q.z);
    v0[1] = s * (q.x * q.y + q.z * q.w);
    v0[2] = s * (q.x * q.z - q.y * q.w);

    v1[0] = s * (q.x * q.y - q.z * q.w);
    v1[1] = 1.0f - s * (q.z * q.z + q.x * q.x);
    v1[2] = s * (q.y * q.z + q.x * q.w);

    v2[0] = s * (q.x * q.z + q.y * q.w);
    v2[1] = s * (q.y * q.z - q.x * q.w);
    v2[2] = 1.0f - s * (q.x * q.x + q.y * q.y);
}

template<> mat4::Mat4(quat const &q)
{
    *this = mat4(mat3(q), 1.f);
}

static inline void MatrixToQuat(quat &that, mat3 const &m)
{
    /* See http://www.euclideanspace.com/maths/geometry/rotations/conversions/matrixToQuaternion/christian.htm for a version with no branches */
    float t = m[0][0] + m[1][1] + m[2][2];
    if (t > 0)
    {
        that.w = 0.5f * std::sqrt(1.0f + t);
        float s = 0.25f / that.w;
        that.x = s * (m[1][2] - m[2][1]);
        that.y = s * (m[2][0] - m[0][2]);
        that.z = s * (m[0][1] - m[1][0]);
    }
    else if (m[0][0] > m[1][1] && m[0][0] > m[2][2])
    {
        that.x = 0.5f * std::sqrt(1.0f + m[0][0] - m[1][1] - m[2][2]);
        float s = 0.25f / that.x;
        that.y = s * (m[0][1] + m[1][0]);
        that.z = s * (m[2][0] + m[0][2]);
        that.w = s * (m[1][2] - m[2][1]);
    }
    else if (m[1][1] > m[2][2])
    {
        that.y = 0.5f * std::sqrt(1.0f - m[0][0] + m[1][1] - m[2][2]);
        float s = 0.25f / that.y;
        that.x = s * (m[0][1] + m[1][0]);
        that.z = s * (m[1][2] + m[2][1]);
        that.w = s * (m[2][0] - m[0][2]);
    }
    else
    {
        that.z = 0.5f * std::sqrt(1.0f - m[0][0] - m[1][1] + m[2][2]);
        float s = 0.25f / that.z;
        that.x = s * (m[2][0] + m[0][2]);
        that.y = s * (m[1][2] + m[2][1]);
        that.w = s * (m[0][1] - m[1][0]);
    }
}

template<> quat::Quat(mat3 const &m)
{
    MatrixToQuat(*this, m);
}

template<> quat::Quat(mat4 const &m)
{
    MatrixToQuat(*this, mat3(m));
}

template<> quat quat::rotate(float angle, vec3 const &v)
{
    angle *= (M_PI / 360.0f);

    vec3 tmp = normalize(v) * sin(angle);

    return quat(cos(angle), tmp.x, tmp.y, tmp.z);
}

template<> quat quat::rotate(float angle, float x, float y, float z)
{
    return quat::rotate(angle, vec3(x, y, z));
}

template<> quat quat::slerp(quat QuatA, quat QuatB, float const &Scalar)
{
        float magnitude = lol::sqrt(sqlength(QuatA) * sqlength(QuatB));
        //btAssert(magnitude > btScalar(0));

        float product = lol::dot(QuatA,QuatB) / magnitude;
        if (product > -1.0f && product < 1.0f)
        {
                // Take care of long angle case see http://en.wikipedia.org/wiki/Slerp
                const float sign = (product < 0.0f) ? -1.0f : 1.0f;

                const float theta = lol::acos(sign * product);
                const float s1 = lol::sin(sign * Scalar * theta);   
                const float d = 1.0f / lol::sin(theta);
                const float s0 = lol::sin((1.0f - Scalar) * theta);

                return quat(
                        (QuatA.w * s0 + QuatB.w * s1) * d,
                        (QuatA.x * s0 + QuatB.x * s1) * d,
                        (QuatA.y * s0 + QuatB.y * s1) * d,
                        (QuatA.z * s0 + QuatB.z * s1) * d);
        }
        else
        {
                return QuatA;
        }
}

template<> vec3 vec3::toeuler(quat const &q)
{
    float n = norm(q);

    if (!n)
        return vec3(0.f);

    vec3 ret(atan2(2.f * (q.w * q.x + q.y * q.z),
                   1.f - 2.f * (q.x * q.x + q.y * q.y)),
             asin(2.f * (q.w * q.y - q.z * q.x)),
             atan2(2.f * (q.w * q.z + q.y * q.x),
                   1.f - 2.f * (q.z * q.z + q.y * q.y)));

    return (180.0f / M_PI / n) * ret;
}

static inline mat3 mat3_fromeuler_generic(vec3 const &v, int i, int j, int k)
{
    mat3 ret;

    vec3 radians = (M_PI / 180.0f) * v;
    float s0 = sin(radians[0]), c0 = cos(radians[0]);
    float s1 = sin(radians[1]), c1 = cos(radians[1]);
    float s2 = sin(radians[2]), c2 = cos(radians[2]);

    if (k == i)
    {
        k = 3 - i - j;

        ret[i][i] =   c1;
        ret[j][i] =   s1 * s2;
        ret[i][j] =   s0 * s1;
        ret[j][j] =   c0 * c2 - s0 * c1 * s2;
        ret[k][k] = - s0 * s2 + c0 * c1 * c2;

        if ((2 + i - j) % 3)
        {
            ret[k][i] =   s1 * c2;
            ret[k][j] = - c0 * s2 - s0 * c1 * c2;
            ret[i][k] = - c0 * s1;
            ret[j][k] =   s0 * c2 + c0 * c1 * s2;
        }
        else
        {
            ret[k][i] = - s1 * c2;
            ret[k][j] =   c0 * s2 + s0 * c1 * c2;
            ret[i][k] =   c0 * s1;
            ret[j][k] = - s0 * c2 - c0 * c1 * s2;
        }
    }
    else
    {
        ret[i][i] =   c1 * c2;
        ret[k][k] =   c0 * c1;

        if ((2 + i - j) % 3)
        {
            ret[j][i] = - c1 * s2;
            ret[k][i] =   s1;

            ret[i][j] =   c0 * s2 + s0 * s1 * c2;
            ret[j][j] =   c0 * c2 - s0 * s1 * s2;
            ret[k][j] = - s0 * c1;

            ret[i][k] =   s0 * s2 - c0 * s1 * c2;
            ret[j][k] =   s0 * c2 + c0 * s1 * s2;
        }
        else
        {
            ret[j][i] =   c1 * s2;
            ret[k][i] = - s1;

            ret[i][j] = - c0 * s2 + s0 * s1 * c2;
            ret[j][j] =   c0 * c2 + s0 * s1 * s2;
            ret[k][j] =   s0 * c1;

            ret[i][k] =   s0 * s2 + c0 * s1 * c2;
            ret[j][k] = - s0 * c2 + c0 * s1 * s2;
        }
    }

    return ret;
}

static inline quat quat_fromeuler_generic(vec3 const &v, int i, int j, int k)
{
    vec3 half_angles = (M_PI / 360.0f) * v;
    float s0 = sin(half_angles[0]), c0 = cos(half_angles[0]);
    float s1 = sin(half_angles[1]), c1 = cos(half_angles[1]);
    float s2 = sin(half_angles[2]), c2 = cos(half_angles[2]);

    quat ret;

    if (k == i)
    {
        k = 3 - i - j;

        ret[0] = c1 * (c0 * c2 - s0 * s2);
        ret[1 + i] = c1 * (c0 * s2 + s0 * c2);
        ret[1 + j] = s1 * (c0 * c2 + s0 * s2);
        ret[1 + k] = ((2 + i - j) % 3) ? s1 * (s0 * c2 - c0 * s2)
                                       : s1 * (c0 * s2 - s0 * c2);
    }
    else
    {
        vec4 v1(c0 * c1 * c2,  s0 * c1 * c2, c0 * s1 * c2,  c0 * c1 * s2);
        vec4 v2(s0 * s1 * s2, -c0 * s1 * s2, s0 * c1 * s2, -s0 * s1 * c2);

        if ((2 + i - j) % 3)
            v1 -= v2;
        else
            v1 += v2;

        ret[0] = v1[0];
        ret[1 + i] = v1[1];
        ret[1 + j] = v1[2];
        ret[1 + k] = v1[3];
    }

    return ret;
}

#define DEFINE_FROMEULER_GENERIC(name, i, j, k) \
    template<> quat quat::fromeuler_##name(vec3 const &v) \
    { \
        return quat_fromeuler_generic(v, i, j, k); \
    } \
    \
    template<> quat quat::fromeuler_##name(float phi, float theta, float psi) \
    { \
        return quat::fromeuler_##name(vec3(phi, theta, psi)); \
    } \
    \
    template<> mat3 mat3::fromeuler_##name(vec3 const &v) \
    { \
        return mat3_fromeuler_generic(v, i, j, k); \
    } \
    \
    template<> mat3 mat3::fromeuler_##name(float phi, float theta, float psi) \
    { \
        return mat3::fromeuler_##name(vec3(phi, theta, psi)); \
    } \
    \
    template<> mat4 mat4::fromeuler_##name(vec3 const &v) \
    { \
        return mat4(mat3_fromeuler_generic(v, i, j, k), 1.f); \
    } \
    \
    template<> mat4 mat4::fromeuler_##name(float phi, float theta, float psi) \
    { \
        return mat4::fromeuler_##name(vec3(phi, theta, psi)); \
    }

DEFINE_FROMEULER_GENERIC(xyx, 0, 1, 0)
DEFINE_FROMEULER_GENERIC(xzx, 0, 2, 0)
DEFINE_FROMEULER_GENERIC(yxy, 1, 0, 1)
DEFINE_FROMEULER_GENERIC(yzy, 1, 2, 1)
DEFINE_FROMEULER_GENERIC(zxz, 2, 0, 2)
DEFINE_FROMEULER_GENERIC(zyz, 2, 1, 2)

DEFINE_FROMEULER_GENERIC(xyz, 0, 1, 2)
DEFINE_FROMEULER_GENERIC(xzy, 0, 2, 1)
DEFINE_FROMEULER_GENERIC(yxz, 1, 0, 2)
DEFINE_FROMEULER_GENERIC(yzx, 1, 2, 0)
DEFINE_FROMEULER_GENERIC(zxy, 2, 0, 1)
DEFINE_FROMEULER_GENERIC(zyx, 2, 1, 0)

#undef DEFINE_FROMEULER_GENERIC

template<> mat4 mat4::lookat(vec3 eye, vec3 center, vec3 up)
{
    vec3 v3 = normalize(eye - center);
    vec3 v2 = normalize(up);
    vec3 v1 = normalize(cross(v2, v3));
    v2 = cross(v3, v1);

    mat4 orient(1.0f);
    orient[0][0] = v1.x;
    orient[0][1] = v2.x;
    orient[0][2] = v3.x;
    orient[1][0] = v1.y;
    orient[1][1] = v2.y;
    orient[1][2] = v3.y;
    orient[2][0] = v1.z;
    orient[2][1] = v2.z;
    orient[2][2] = v3.z;

    return orient * mat4::translate(-eye);
}

template<> mat4 mat4::ortho(float left, float right, float bottom,
                            float top, float near, float far)
{
    float invrl = (right != left) ? 1.0f / (right - left) : 0.0f;
    float invtb = (top != bottom) ? 1.0f / (top - bottom) : 0.0f;
    float invfn = (far != near) ? 1.0f / (far - near) : 0.0f;

    mat4 ret(0.0f);
    ret[0][0] = 2.0f * invrl;
    ret[1][1] = 2.0f * invtb;
    ret[2][2] = -2.0f * invfn;
    ret[3][0] = - (right + left) * invrl;
    ret[3][1] = - (top + bottom) * invtb;
    ret[3][2] = - (far + near) * invfn;
    ret[3][3] = 1.0f;
    return ret;
}

template<> mat4 mat4::ortho(float width, float height,
                            float near, float far)
{
    return mat4::ortho(-0.5f * width, 0.5f * width,
                       -0.5f * height, 0.5f * height, near, far);
}

template<> mat4 mat4::frustum(float left, float right, float bottom,
                              float top, float near, float far)
{
    float invrl = (right != left) ? 1.0f / (right - left) : 0.0f;
    float invtb = (top != bottom) ? 1.0f / (top - bottom) : 0.0f;
    float invfn = (far != near) ? 1.0f / (far - near) : 0.0f;

    mat4 ret(0.0f);
    ret[0][0] = 2.0f * near * invrl;
    ret[1][1] = 2.0f * near * invtb;
    ret[2][0] = (right + left) * invrl;
    ret[2][1] = (top + bottom) * invtb;
    ret[2][2] = - (far + near) * invfn;
    ret[2][3] = -1.0f;
    ret[3][2] = -2.0f * far * near * invfn;
    return ret;
}

template<> mat4 mat4::perspective(float fov_y, float width,
                                  float height, float near, float far)
{
    fov_y *= (M_PI / 180.0f);

    float t2 = tanf(fov_y * 0.5f);
    float t1 = t2 * width / height;

    return frustum(-near * t1, near * t1, -near * t2, near * t2, near, far);
}

} /* namespace lol */