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im3d_math.h
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im3d_math.h
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#pragma once
// im3d_math.h is optional - include only if you want to use the Im3d math types directly
#include "im3d.h"
#include <cmath>
namespace Im3d {
// Vec2
inline Vec2 operator+(const Vec2& _lhs, const Vec2& _rhs) { return Vec2(_lhs.x + _rhs.x, _lhs.y + _rhs.y); }
inline Vec2 operator-(const Vec2& _lhs, const Vec2& _rhs) { return Vec2(_lhs.x - _rhs.x, _lhs.y - _rhs.y); }
inline Vec2 operator*(const Vec2& _lhs, const Vec2& _rhs) { return Vec2(_lhs.x * _rhs.x, _lhs.y * _rhs.y); }
inline Vec2 operator/(const Vec2& _lhs, const Vec2& _rhs) { return Vec2(_lhs.x / _rhs.x, _lhs.y / _rhs.y); }
inline Vec2 operator*(const Vec2& _lhs, float _rhs) { return Vec2(_lhs.x * _rhs, _lhs.y * _rhs); }
inline Vec2 operator/(const Vec2& _lhs, float _rhs) { return Vec2(_lhs.x / _rhs, _lhs.y / _rhs); }
inline Vec2 operator-(const Vec2& _v) { return Vec2(-_v.x, -_v.y); }
inline float Dot(const Vec2& _lhs, const Vec2& _rhs) { return _lhs.x * _rhs.x + _lhs.y * _rhs.y; }
inline float Length(const Vec2& _v) { return sqrtf(Dot(_v, _v)); }
inline float Length2(const Vec2& _v) { return Dot(_v, _v); }
inline Vec2 Abs(const Vec2& _v) { return Vec2(fabs(_v.x), fabs(_v.y)); }
inline Vec2 Normalize(const Vec2& _v) { return _v / Length(_v); }
// Vec3
inline Vec3 operator+(const Vec3& _lhs, const Vec3& _rhs) { return Vec3(_lhs.x + _rhs.x, _lhs.y + _rhs.y, _lhs.z + _rhs.z); }
inline Vec3 operator-(const Vec3& _lhs, const Vec3& _rhs) { return Vec3(_lhs.x - _rhs.x, _lhs.y - _rhs.y, _lhs.z - _rhs.z); }
inline Vec3 operator*(const Vec3& _lhs, const Vec3& _rhs) { return Vec3(_lhs.x * _rhs.x, _lhs.y * _rhs.y, _lhs.z * _rhs.z); }
inline Vec3 operator/(const Vec3& _lhs, const Vec3& _rhs) { return Vec3(_lhs.x / _rhs.x, _lhs.y / _rhs.y, _lhs.z / _rhs.z); }
inline Vec3 operator*(const Vec3& _lhs, float _rhs) { return Vec3(_lhs.x * _rhs, _lhs.y * _rhs, _lhs.z * _rhs); }
inline Vec3 operator/(const Vec3& _lhs, float _rhs) { return Vec3(_lhs.x / _rhs, _lhs.y / _rhs, _lhs.z / _rhs); }
inline Vec3 operator-(const Vec3& _v) { return Vec3(-_v.x, -_v.y, -_v.z); }
inline float Dot(const Vec3& _lhs, const Vec3& _rhs) { return _lhs.x * _rhs.x + _lhs.y * _rhs.y + _lhs.z * _rhs.z; }
inline float Length(const Vec3& _v) { return sqrtf(Dot(_v, _v)); }
inline float Length2(const Vec3& _v) { return Dot(_v, _v); }
inline Vec3 Abs(const Vec3& _v) { return Vec3(fabs(_v.x), fabs(_v.y), fabs(_v.z)); }
inline Vec3 Normalize(const Vec3& _v) { return _v / Length(_v); }
inline Vec3 Cross(const Vec3& _a, const Vec3& _b)
{
return Vec3(
_a.y * _b.z - _b.y * _a.z,
_a.z * _b.x - _b.z * _a.x,
_a.x * _b.y - _b.x * _a.y
);
}
// Vec4
inline Vec4 operator+(const Vec4& _lhs, const Vec4& _rhs) { return Vec4(_lhs.x + _rhs.x, _lhs.y + _rhs.y, _lhs.z + _rhs.z, _lhs.w + _rhs.w); }
inline Vec4 operator-(const Vec4& _lhs, const Vec4& _rhs) { return Vec4(_lhs.x - _rhs.x, _lhs.y - _rhs.y, _lhs.z - _rhs.z, _lhs.w - _rhs.w); }
inline Vec4 operator*(const Vec4& _lhs, const Vec4& _rhs) { return Vec4(_lhs.x * _rhs.x, _lhs.y * _rhs.y, _lhs.z * _rhs.z, _lhs.w * _rhs.w); }
inline Vec4 operator/(const Vec4& _lhs, const Vec4& _rhs) { return Vec4(_lhs.x / _rhs.x, _lhs.y / _rhs.y, _lhs.z / _rhs.z, _lhs.w / _rhs.w); }
inline Vec4 operator*(const Vec4& _lhs, float _rhs) { return Vec4(_lhs.x * _rhs, _lhs.y * _rhs, _lhs.z * _rhs, _lhs.w * _rhs); }
inline Vec4 operator/(const Vec4& _lhs, float _rhs) { return Vec4(_lhs.x / _rhs, _lhs.y / _rhs, _lhs.z / _rhs, _lhs.w / _rhs); }
inline Vec4 operator-(const Vec4& _v) { return Vec4(-_v.x, -_v.y, -_v.z, -_v.w); }
inline float Dot(const Vec4& _lhs, const Vec4& _rhs) { return _lhs.x * _rhs.x + _lhs.y * _rhs.y + _lhs.z * _rhs.z + _lhs.w * _rhs.w; }
inline float Length(const Vec4& _v) { return sqrtf(Dot(_v, _v)); }
inline float Length2(const Vec4& _v) { return Dot(_v, _v); }
inline Vec4 Abs(const Vec4& _v) { return Vec4(fabs(_v.x), fabs(_v.y), fabs(_v.z), fabs(_v.w)); }
inline Vec4 Normalize(const Vec4& _v) { return _v / Length(_v); }
// Mat3
inline Mat3 operator*(const Mat3& _lhs, const Mat3& _rhs)
{
Mat3 ret;
ret(0, 0) = _lhs(0, 0) * _rhs(0, 0) + _lhs(0, 1) * _rhs(1, 0) + _lhs(0, 2) * _rhs(2, 0);
ret(0, 1) = _lhs(0, 0) * _rhs(0, 1) + _lhs(0, 1) * _rhs(1, 1) + _lhs(0, 2) * _rhs(2, 1);
ret(0, 2) = _lhs(0, 0) * _rhs(0, 2) + _lhs(0, 1) * _rhs(1, 2) + _lhs(0, 2) * _rhs(2, 2);
ret(1, 0) = _lhs(1, 0) * _rhs(0, 0) + _lhs(1, 1) * _rhs(1, 0) + _lhs(1, 2) * _rhs(2, 0);
ret(1, 1) = _lhs(1, 0) * _rhs(0, 1) + _lhs(1, 1) * _rhs(1, 1) + _lhs(1, 2) * _rhs(2, 1);
ret(1, 2) = _lhs(1, 0) * _rhs(0, 2) + _lhs(1, 1) * _rhs(1, 2) + _lhs(1, 2) * _rhs(2, 2);
ret(2, 0) = _lhs(2, 0) * _rhs(0, 0) + _lhs(2, 1) * _rhs(1, 0) + _lhs(2, 2) * _rhs(2, 0);
ret(2, 1) = _lhs(2, 0) * _rhs(0, 1) + _lhs(2, 1) * _rhs(1, 1) + _lhs(2, 2) * _rhs(2, 1);
ret(2, 2) = _lhs(2, 0) * _rhs(0, 2) + _lhs(2, 1) * _rhs(1, 2) + _lhs(2, 2) * _rhs(2, 2);
return ret;
}
inline Vec3 operator*(const Mat3& _m, const Vec3& _v)
{
return Vec3(
_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
);
}
inline Vec4 operator*(const Mat3& _m, const Vec4& _v)
{
return Vec4(
_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,
_v.w
);
}
Mat3 Transpose(const Mat3& _m);
Vec3 ToEulerXYZ(const Mat3& _m);
Mat3 FromEulerXYZ(Vec3& _xyz);
Mat3 Rotation(const Vec3& _axis, float _rads); // _axis must be unit length
Mat3 Scale(const Vec3& _s);
// Mat4
inline Mat4 operator*(const Mat4& _lhs, const Mat4& _rhs)
{
Mat4 ret;
ret(0, 0) = _lhs(0, 0) * _rhs(0, 0) + _lhs(0, 1) * _rhs(1, 0) + _lhs(0, 2) * _rhs(2, 0) + _lhs(0, 3) * _rhs(3, 0);
ret(0, 1) = _lhs(0, 0) * _rhs(0, 1) + _lhs(0, 1) * _rhs(1, 1) + _lhs(0, 2) * _rhs(2, 1) + _lhs(0, 3) * _rhs(3, 1);
ret(0, 2) = _lhs(0, 0) * _rhs(0, 2) + _lhs(0, 1) * _rhs(1, 2) + _lhs(0, 2) * _rhs(2, 2) + _lhs(0, 3) * _rhs(3, 2);
ret(0, 3) = _lhs(0, 0) * _rhs(0, 3) + _lhs(0, 1) * _rhs(1, 3) + _lhs(0, 2) * _rhs(2, 3) + _lhs(0, 3) * _rhs(3, 3);
ret(1, 0) = _lhs(1, 0) * _rhs(0, 0) + _lhs(1, 1) * _rhs(1, 0) + _lhs(1, 2) * _rhs(2, 0) + _lhs(1, 3) * _rhs(3, 0);
ret(1, 1) = _lhs(1, 0) * _rhs(0, 1) + _lhs(1, 1) * _rhs(1, 1) + _lhs(1, 2) * _rhs(2, 1) + _lhs(1, 3) * _rhs(3, 1);
ret(1, 2) = _lhs(1, 0) * _rhs(0, 2) + _lhs(1, 1) * _rhs(1, 2) + _lhs(1, 2) * _rhs(2, 2) + _lhs(1, 3) * _rhs(3, 2);
ret(1, 3) = _lhs(1, 0) * _rhs(0, 3) + _lhs(1, 1) * _rhs(1, 3) + _lhs(1, 2) * _rhs(2, 3) + _lhs(1, 3) * _rhs(3, 3);
ret(2, 0) = _lhs(2, 0) * _rhs(0, 0) + _lhs(2, 1) * _rhs(1, 0) + _lhs(2, 2) * _rhs(2, 0) + _lhs(2, 3) * _rhs(3, 0);
ret(2, 1) = _lhs(2, 0) * _rhs(0, 1) + _lhs(2, 1) * _rhs(1, 1) + _lhs(2, 2) * _rhs(2, 1) + _lhs(2, 3) * _rhs(3, 1);
ret(2, 2) = _lhs(2, 0) * _rhs(0, 2) + _lhs(2, 1) * _rhs(1, 2) + _lhs(2, 2) * _rhs(2, 2) + _lhs(2, 3) * _rhs(3, 2);
ret(2, 3) = _lhs(2, 0) * _rhs(0, 3) + _lhs(2, 1) * _rhs(1, 3) + _lhs(2, 2) * _rhs(2, 3) + _lhs(2, 3) * _rhs(3, 3);
ret(3, 0) = _lhs(3, 0) * _rhs(0, 0) + _lhs(3, 1) * _rhs(1, 0) + _lhs(3, 2) * _rhs(2, 0) + _lhs(3, 3) * _rhs(3, 0);
ret(3, 1) = _lhs(3, 0) * _rhs(0, 1) + _lhs(3, 1) * _rhs(1, 1) + _lhs(3, 2) * _rhs(2, 1) + _lhs(3, 3) * _rhs(3, 1);
ret(3, 2) = _lhs(3, 0) * _rhs(0, 2) + _lhs(3, 1) * _rhs(1, 2) + _lhs(3, 2) * _rhs(2, 2) + _lhs(3, 3) * _rhs(3, 2);
ret(3, 3) = _lhs(3, 0) * _rhs(0, 3) + _lhs(3, 1) * _rhs(1, 3) + _lhs(3, 2) * _rhs(2, 3) + _lhs(3, 3) * _rhs(3, 3);
return ret;
}
inline Vec3 operator*(const Mat4& _m, const Vec3& _pos)
{
return Vec3(
_m(0, 0) * _pos.x + _m(0, 1) * _pos.y + _m(0, 2) * _pos.z + _m(0, 3),
_m(1, 0) * _pos.x + _m(1, 1) * _pos.y + _m(1, 2) * _pos.z + _m(1, 3),
_m(2, 0) * _pos.x + _m(2, 1) * _pos.y + _m(2, 2) * _pos.z + _m(2, 3)
);
}
inline Vec4 operator*(const Mat4& _m, const Vec4& _v)
{
return Vec4(
_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
);
}
Mat4 Inverse(const Mat4& _m);
Mat4 Transpose(const Mat4& _m);
Mat4 Translation(const Vec3& _t);
Mat4 AlignZ(const Vec3& _axis, const Vec3& _up = Vec3(0.0f, 1.0f, 0.0f)); // generate an orthonormal bases with +z as _axis, which must be unit length
Mat4 LookAt(const Vec3& _from, const Vec3& _to, const Vec3& _up = Vec3(0.0f, 1.0f, 0.0f)); // align _z with (_to - _from), set _from as translation
struct Line
{
Vec3 m_origin;
Vec3 m_direction; // unit length
Line() {}
Line(const Vec3& _origin, const Vec3& _direction);
};
struct Ray
{
Vec3 m_origin;
Vec3 m_direction; // unit length
Ray() {}
Ray(const Vec3& _origin, const Vec3& _direction);
};
struct LineSegment
{
Vec3 m_start;
Vec3 m_end;
LineSegment() {}
LineSegment(const Vec3& _start, const Vec3& _end);
};
struct Sphere
{
Vec3 m_origin;
float m_radius;
Sphere() {}
Sphere(const Vec3& _origin, float _radius);
};
struct Plane
{
Vec3 m_normal;
float m_offset;
Plane() {}
Plane(const Vec3& _normal, float _offset);
Plane(const Vec3& _normal, const Vec3& _origin);
};
struct Capsule
{
Vec3 m_start;
Vec3 m_end;
float m_radius;
Capsule() {}
Capsule(const Vec3& _start, const Vec3& _end, float _radius);
};
// Ray-primitive intersections. Use Intersects() when you don't need t.
bool Intersects(const Ray& _ray, const Plane& _plane );
bool Intersect (const Ray& _ray, const Plane& _plane, float& t0_ );
bool Intersects(const Ray& _ray, const Sphere& _sphere );
bool Intersect (const Ray& _ray, const Sphere& _sphere, float& t0_, float& t1_);
bool Intersects(const Ray& _ray, const Capsule& _capsule );
bool Intersect (const Ray& _ray, const Capsule& _capsule, float& t0_, float& t1_);
// Find point t0_ along _line0 nearest to _line1 and t1_ along _line1 nearest to _line0.
void Nearest(const Line& _line0, const Line& _line1, float& t0_, float& t1_);
// Find point tr_ along _ray nearest to _line and tl_ along _line nearest to _ray.
void Nearest(const Ray& _ray, const Line& _line, float& tr_, float& tl_);
// Find point tr_ along _ray nearest to _segment, return point on segment nearest to _ray.
Vec3 Nearest(const Ray& _ray, const LineSegment& _segment, float& tr_);
float Distance2(const Ray& _ray, const LineSegment& _segment);
inline float Distance(const Vec4& _plane, const Vec3& _point){ return _plane.x * _point.x + _plane.y * _point.y + _plane.z * _point.z - _plane.w; }
constexpr float Pi = 3.14159265359f;
constexpr float TwoPi = 2.0f * Pi;
constexpr float HalfPi = 0.5f * Pi;
inline float Radians(float _degrees) { return _degrees * (Pi / 180.0f); }
inline float Degrees(float _radians) { return _radians * (180.0f / Pi); }
namespace internal {
struct ScalarT {};
struct FloatT: public ScalarT {};
struct IntT: public ScalarT {};
struct CompositeT {};
struct VecT: public CompositeT {};
struct MatT: public CompositeT {};
template <typename T>
struct TypeTraits { typedef typename T::Type Type; enum { kSize = T::kSize }; };
template<> struct TypeTraits<int> { typedef IntT Type; enum { kSize = 1 }; };
template<> struct TypeTraits<float> { typedef FloatT Type; enum { kSize = 1 }; };
template<> struct TypeTraits<Vec2> { typedef VecT Type; enum { kSize = 2 }; };
template<> struct TypeTraits<Vec3> { typedef VecT Type; enum { kSize = 3 }; };
template<> struct TypeTraits<Vec4> { typedef VecT Type; enum { kSize = 4 }; };
template<> struct TypeTraits<Mat4> { typedef MatT Type; enum { kSize = 16 }; };
template <typename T>
inline bool _AllLess(const T& _a, const T& _b, ScalarT)
{
return _a < _b;
}
template <typename T>
inline bool _AllLess(const T& _a, const T& _b, CompositeT)
{
for (int i = 0, n = TypeTraits<T>::kSize; i < n; ++i)
{
if (_a[i] > _b[i])
{
return false;
}
}
return true;
}
template <typename T>
inline T _Max(const T& _a, const T& _b, ScalarT)
{
return _a < _b ? _b : _a;
}
template <typename T>
inline T _Max(const T& _a, const T& _b, CompositeT)
{
T ret;
for (int i = 0, n = TypeTraits<T>::kSize; i < n; ++i) {
ret[i] = _Max(_a[i], _b[i], ScalarT());
}
return ret;
}
template <typename T>
inline T _Min(const T& _a, const T& _b, ScalarT)
{
return _a < _b ? _a : _b;
}
template <typename T>
inline T _Min(const T& _a, const T& _b, CompositeT)
{
T ret;
for (int i = 0, n = TypeTraits<T>::kSize; i < n; ++i)
{
ret[i] = _Min(_a[i], _b[i], ScalarT());
}
return ret;
}
} // namespace internal
template <typename T>
inline bool AllLess(const T& _a, const T& _b) { return internal::_AllLess(_a, _b, typename internal::TypeTraits<T>::Type()); }
template <typename T>
inline T Max(T _a, T _b) { return internal::_Max(_a, _b, typename internal::TypeTraits<T>::Type()); }
template <typename T>
inline T Min(T _a, T _b) { return internal::_Min(_a, _b, typename internal::TypeTraits<T>::Type()); }
template <typename T>
inline T Clamp(T _a, T _min, T _max) { return Max(Min(_a, _max), _min); }
// Remap _x in [_start,_end] to [0,1].
inline float Remap(float _x, float _start, float _end) { return Clamp(_x * (1.0f / (_end - _start)) + (-_start / (_end - _start)), 0.0f, 1.0f); }
} // namespace Im3d