各大引擎矩阵的矩阵存储方式 ----行矩阵 or 列矩阵
OpenGL 里面的矩阵
float m[16];
OpenGL中的矩阵是这样的
m[0] m[4] m[8] m[12] m[1] m[5] m[9] m[13] m[2] m[6] m[10] m[14] m[3] m[7] m[11] m[15]
对它定义的新坐标系，OpenGL是这么说的：
x轴向量(m[0], m[1], m[2]); y轴向量(m[4], m[5], m[6]); z轴向量(m[8], m9], m[10]); 原点(m[12], m[13], m[14]);
乘法是这样的
x' = m[0]x + m[4]y +m[8]z + m[12]w; y' = m[1]x + m[5]y +m[9]z + m[13]w; z' = m[2]x + m[6]y +m[10]z + m[14]w; w' = m[3]x + m[7]y +m[11]z + m[15]w;
分割线 补充
id 的idmat4
//===============================================================
//
// idMat4 - 4x4 matrix
//
//===============================================================
class idMat4 {
public:
idMat4( void );
explicit idMat4( const idVec4 &x, const idVec4 &y, const idVec4 &z, const idVec4 &w );
explicit idMat4(const float xx, const float xy, const float xz, const float xw,
const float yx, const float yy, const float yz, const float yw,
const float zx, const float zy, const float zz, const float zw,
const float wx, const float wy, const float wz, const float ww );
explicit idMat4( const idMat3 &rotation, const idVec3 &translation );
explicit idMat4( const float src[ 4 ][ 4 ] );
const idVec4 & operator[]( int index ) const;
idVec4 & operator[]( int index );
idMat4 operator*( const float a ) const;
idVec4 operator*( const idVec4 &vec ) const;
idVec3 operator*( const idVec3 &vec ) const;
idMat4 operator*( const idMat4 &a ) const;
idMat4 operator+( const idMat4 &a ) const;
idMat4 operator-( const idMat4 &a ) const;
idMat4 & operator*=( const float a );
idMat4 & operator*=( const idMat4 &a );
idMat4 & operator+=( const idMat4 &a );
idMat4 & operator-=( const idMat4 &a );
friend idMat4 operator*( const float a, const idMat4 &mat );
friend idVec4 operator*( const idVec4 &vec, const idMat4 &mat );
friend idVec3 operator*( const idVec3 &vec, const idMat4 &mat );
friend idVec4 & operator*=( idVec4 &vec, const idMat4 &mat );
friend idVec3 & operator*=( idVec3 &vec, const idMat4 &mat );
bool Compare( const idMat4 &a ) const; // exact compare, no epsilon
bool Compare( const idMat4 &a, const float epsilon ) const; // compare with epsilon
bool operator==( const idMat4 &a ) const; // exact compare, no epsilon
bool operator!=( const idMat4 &a ) const; // exact compare, no epsilon
void Zero( void );
void Identity( void );
bool IsIdentity( const float epsilon = MATRIX_EPSILON ) const;
bool IsSymmetric( const float epsilon = MATRIX_EPSILON ) const;
bool IsDiagonal( const float epsilon = MATRIX_EPSILON ) const;
bool IsRotated( void ) const;
void ProjectVector( const idVec4 &src, idVec4 &dst ) const;
void UnprojectVector( const idVec4 &src, idVec4 &dst ) const;
float Trace( void ) const;
float Determinant( void ) const;
idMat4 Transpose( void ) const; // returns transpose
idMat4 & TransposeSelf( void );
idMat4 Inverse( void ) const; // returns the inverse ( m * m.Inverse() = identity )
bool InverseSelf( void ); // returns false if determinant is zero
idMat4 InverseFast( void ) const; // returns the inverse ( m * m.Inverse() = identity )
bool InverseFastSelf( void ); // returns false if determinant is zero
idMat4 TransposeMultiply( const idMat4 &b ) const;
int GetDimension( void ) const;
const float * ToFloatPtr( void ) const;
float * ToFloatPtr( void );
const char * ToString( int precision = 2 ) const;
private:
idVec4 mat[ 4 ];
};
ID_INLINE idVec4 idMat4::operator*( const idVec4 &vec ) const {
return idVec4(
mat[ 0 ].x * vec.x + mat[ 0 ].y * vec.y + mat[ 0 ].z * vec.z + mat[ 0 ].w * vec.w,
mat[ 1 ].x * vec.x + mat[ 1 ].y * vec.y + mat[ 1 ].z * vec.z + mat[ 1 ].w * vec.w,
mat[ 2 ].x * vec.x + mat[ 2 ].y * vec.y + mat[ 2 ].z * vec.z + mat[ 2 ].w * vec.w,
mat[ 3 ].x * vec.x + mat[ 3 ].y * vec.y + mat[ 3 ].z * vec.z + mat[ 3 ].w * vec.w );
}
ID_INLINE idVec3 idMat4::operator*( const idVec3 &vec ) const {
float s = mat[ 3 ].x * vec.x + mat[ 3 ].y * vec.y + mat[ 3 ].z * vec.z + mat[ 3 ].w;
if ( s == 0.0f ) {
return idVec3( 0.0f, 0.0f, 0.0f );
}
if ( s == 1.0f ) {
return idVec3(
mat[ 0 ].x * vec.x + mat[ 0 ].y * vec.y + mat[ 0 ].z * vec.z + mat[ 0 ].w,
mat[ 1 ].x * vec.x + mat[ 1 ].y * vec.y + mat[ 1 ].z * vec.z + mat[ 1 ].w,
mat[ 2 ].x * vec.x + mat[ 2 ].y * vec.y + mat[ 2 ].z * vec.z + mat[ 2 ].w );
}
else {
float invS = 1.0f / s;
return idVec3(
(mat[ 0 ].x * vec.x + mat[ 0 ].y * vec.y + mat[ 0 ].z * vec.z + mat[ 0 ].w) * invS,
(mat[ 1 ].x * vec.x + mat[ 1 ].y * vec.y + mat[ 1 ].z * vec.z + mat[ 1 ].w) * invS,
(mat[ 2 ].x * vec.x + mat[ 2 ].y * vec.y + mat[ 2 ].z * vec.z + mat[ 2 ].w) * invS );
}
idsoft的并不完全是OpenGL的矩阵相匹配,他是行矩阵
Ogre的
/** \addtogroup Math
* @{
*/
/** Class encapsulating a standard 4x4 homogeneous matrix.
@remarks
OGRE uses column vectors when applying matrix multiplications,
This means a vector is represented as a single column, 4-row
matrix. This has the effect that the transformations implemented
by the matrices happens right-to-left e.g. if vector V is to be
transformed by M1 then M2 then M3, the calculation would be
M3 * M2 * M1 * V. The order that matrices are concatenated is
vital since matrix multiplication is not commutative, i.e. you
can get a different result if you concatenate in the wrong order.
@par
The use of column vectors and right-to-left ordering is the
standard in most mathematical texts, and is the same as used in
OpenGL. It is, however, the opposite of Direct3D, which has
inexplicably chosen to differ from the accepted standard and uses
row vectors and left-to-right matrix multiplication.
@par
OGRE deals with the differences between D3D and OpenGL etc.
internally when operating through different render systems. OGRE
users only need to conform to standard maths conventions, i.e.
right-to-left matrix multiplication, (OGRE transposes matrices it
passes to D3D to compensate).
@par
The generic form M * V which shows the layout of the matrix
entries is shown below:
<pre>
[ m[0][0] m[0][1] m[0][2] m[0][3] ] {x}
| m[1][0] m[1][1] m[1][2] m[1][3] | * {y}
| m[2][0] m[2][1] m[2][2] m[2][3] | {z}
[ m[3][0] m[3][1] m[3][2] m[3][3] ] {1}
</pre>
*/
class _OgreExport Matrix4
{
protected:
/// The matrix entries, indexed by [row][col].
union {
Real m[4][4];
Real _m[16];
};
public:
inline Vector4 operator * (const Vector4& v) const
{
return Vector4(
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
);
}
};
Ogre也是一样，行矩阵
NVIDIA 的
template<class T>
class matrix4
{
public:
// dst = M * src
vec4<T> operator *( const vec4<T> &src) const
{
vec4<T> r;
for ( int32_t i = 0; i < 4; i++)
r[i] = ( src[0] * element(i,0) + src[1] * element(i,1) +
src[2] * element(i,2) + src[3] * element(i,3));
return r;
}
T & element (int32_t row, int32_t col)
{
return _array[row | (col<<2)];
}
const T & element (int32_t row, int32_t col) const
{
return _array[row | (col<<2)];
}
union
{
struct
{
T _11, _12, _13, _14; // standard names for components
T _21, _22, _23, _24; // standard names for components
T _31, _32, _33, _34; // standard names for components
T _41, _42, _43, _44; // standard names for components
};
T _array[16]; // array access
};
};
这里我翻译下
vec4<T> operator *( const vec4<T> &src) const
{
vec4<T> ret;
ret[0] = _array[0] * src[0] + _array[4]* src[1] + _array[8] *src[2] + _array[12] *src[3];
ret[1] = _array[1] * src[0] + _array[5]* src[1] + _array[9] *src[2] + _array[13] *src[3];
ret[2] = _array[2] * src[0] + _array[6]* src[1] + _array[10] *src[2] + _array[14] *src[3];
ret[3] = _array[3] * src[0] + _array[7]* src[1] + _array[11] *src[2] + _array[15] *src[3];
return ret;
}
所以和OpenGL一模一样 ，列矩阵
gkEngine
///////////////////////////////////////////////////////////////////////////////
///////////////////////////////////////////////////////////////////////////////
///////////////////////////////////////////////////////////////////////////////
// struct Matrix44_tpl
///////////////////////////////////////////////////////////////////////////////
///////////////////////////////////////////////////////////////////////////////
///////////////////////////////////////////////////////////////////////////////
template<typename F, class A=XMVec4A> struct Matrix44_tpl
{
#ifndef XENON_INTRINSICS
F m00,m01,m02,m03;
F m10,m11,m12,m13;
F m20,m21,m22,m23;
F m30,m31,m32,m33;
//! transform a vector
Vec3 TransformVector(const Vec3 &b) const {
assert(b.IsValid());
Vec3 v;
v.x = m00*b.x + m01*b.y + m02*b.z;
v.y = m10*b.x + m11*b.y + m12*b.z;
v.z = m20*b.x + m21*b.y + m22*b.z;
return v;
}
//! transform a point
Vec3 TransformPoint(const Vec3 &b) const {
assert(b.IsValid());
Vec3 v;
v.x = m00*b.x + m01*b.y + m02* b.z + m03;
v.y = m10*b.x + m11*b.y + m12* b.z + m13;
v.z = m20*b.x + m21*b.y + m22* b.z + m23;
return v;
}
};
他是行矩阵
再来一个gameplay的
/**
* Defines a 4 x 4 floating point matrix representing a 3D transformation.
*
* Vectors are treated as columns, resulting in a matrix that is represented as follows,
* where x, y and z are the translation components of the matrix:
*
* 1 0 0 x
* 0 1 0 y
* 0 0 1 z
* 0 0 0 1
*
* This matrix class is directly compatible with OpenGL since its elements are
* laid out in memory exactly as they are expected by OpenGL.
* The matrix uses column-major format such that array indices increase down column first.
* Since matrix multiplication is not commutative, multiplication must be done in the
* correct order when combining transformations. Suppose we have a translation
* matrix T and a rotation matrix R. To first rotate an object around the origin
* and then translate it, you would multiply the two matrices as TR.
*
* Likewise, to first translate the object and then rotate it, you would do RT.
* So generally, matrices must be multiplied in the reverse order in which you
* want the transformations to take place (this also applies to
* the scale, rotate, and translate methods below; these methods are convenience
* methods for post-multiplying by a matrix representing a scale, rotation, or translation).
*
* In the case of repeated local transformations (i.e. rotate around the Z-axis by 0.76 radians,
* then translate by 2.1 along the X-axis, then ...), it is better to use the Transform class
* (which is optimized for that kind of usage).
*
* @see Transform
*/
class Matrix
{
public:
/**
* Stores the columns of this 4x4 matrix.
* */
float m[16];
}
inline void MathUtil::transformVector4(const float* m, float x, float y, float z, float w, float* dst)
{
dst[0] = x * m[0] + y * m[4] + z * m[8] + w * m[12];
dst[1] = x * m[1] + y * m[5] + z * m[9] + w * m[13];
dst[2] = x * m[2] + y * m[6] + z * m[10] + w * m[14];
}
inline void MathUtil::transformVector4(const float* m, const float* v, float* dst)
{
// Handle case where v == dst.
float x = v[0] * m[0] + v[1] * m[4] + v[2] * m[8] + v[3] * m[12];
float y = v[0] * m[1] + v[1] * m[5] + v[2] * m[9] + v[3] * m[13];
float z = v[0] * m[2] + v[1] * m[6] + v[2] * m[10] + v[3] * m[14];
float w = v[0] * m[3] + v[1] * m[7] + v[2] * m[11] + v[3] * m[15];
dst[0] = x;
dst[1] = y;
dst[2] = z;
dst[3] = w;
}
当然这个说明里已经写了是Opengl的了，那么自然是列矩阵。
等回家看看cryEngine 和 sourceEngine的
----------------------------------我是分割线------------------------------------------------
cryEngine的
///////////////////////////////////////////////////////////////////////////////
///////////////////////////////////////////////////////////////////////////////
///////////////////////////////////////////////////////////////////////////////
// struct Matrix44_tpl
///////////////////////////////////////////////////////////////////////////////
///////////////////////////////////////////////////////////////////////////////
///////////////////////////////////////////////////////////////////////////////
template<typename F, class A=XMVec4A> struct Matrix44_tpl
{
#ifndef XENON_INTRINSICS
F m00,m01,m02,m03;
F m10,m11,m12,m13;
F m20,m21,m22,m23;
F m30,m31,m32,m33;
}；
template<class F1, class F2, class B>
ILINE Vec4_tpl<F1> operator*(const Matrix44_tpl<F2, B> &m, const Vec4_tpl<F1> &v)
{
CHECK_SIMD_ALIGNMENT_P(&v);
assert(m.IsValid());
assert(v.IsValid());
return Vec4_tpl<F1>(v.x*m.m00 + v.y*m.m01 + v.z*m.m02 + v.w*m.m03,
v.x*m.m10 + v.y*m.m11 + v.z*m.m12 + v.w*m.m13,
v.x*m.m20 + v.y*m.m21 + v.z*m.m22 + v.w*m.m23,
v.x*m.m30 + v.y*m.m31 + v.z*m.m32 + v.w*m.m33);
}
可见CryEngine是行矩阵，CryEngine只有Dx的渲染方式，不可能用Opengl的列矩阵吧，呵呵
下面是SourceEngine
//=============================================================================//
//
// VMatrix always postmultiply vectors as in Ax = b.
// Given a set of basis vectors ((F)orward, (L)eft, (U)p), and a (T)ranslation,
// a matrix to transform a vector into that space looks like this:
// Fx Lx Ux Tx
// Fy Ly Uy Ty
// Fz Lz Uz Tz
// 0 0 0 1
// Note that concatenating matrices needs to multiply them in reverse order.
// ie: if I want to apply matrix A, B, then C, the equation needs to look like this:
// C * B * A * v
// ie:
// v = A * v;
// v = B * v;
// v = C * v;
class VMatrix
{
public:
// The matrix.
vec_t m[4][4];
};
inline Vector VMatrix::operator*(const Vector &vVec) const
{
Vector vRet;
vRet.x = m[0][0]*vVec.x + m[0][1]*vVec.y + m[0][2]*vVec.z + m[0][3];
vRet.y = m[1][0]*vVec.x + m[1][1]*vVec.y + m[1][2]*vVec.z + m[1][3];
vRet.z = m[2][0]*vVec.x + m[2][1]*vVec.y + m[2][2]*vVec.z + m[2][3];
return vRet;
}
inline Vector VMatrix::VMul3x3(const Vector &vVec) const
{
return Vector(
m[0][0]*vVec.x + m[0][1]*vVec.y + m[0][2]*vVec.z,
m[1][0]*vVec.x + m[1][1]*vVec.y + m[1][2]*vVec.z,
m[2][0]*vVec.x + m[2][1]*vVec.y + m[2][2]*vVec.z
);
}
SourceEngine也不是列矩阵，而是行矩阵
2016 -6 -17 添加 unreal3
// Homogeneous transform. FORCEINLINE FVector4 FMatrix::TransformFVector4(const FVector4 &P) const { FVector4 Result; #if ASM_X86 #ifdef _MSC_VER __asm { mov eax,[P] mov ecx,[this] movups xmm4,[ecx] // M[0][0] movups xmm5,[ecx+16] // M[1][0] movups xmm6,[ecx+32] // M[2][0] movups xmm7,[ecx+48] // M[3][0] movss xmm0,[eax]FVector.X shufps xmm0,xmm0,0 mulps xmm0,xmm4 movss xmm1,[eax]FVector.Y shufps xmm1,xmm1,0 mulps xmm1,xmm5 movss xmm2,[eax]FVector.Z shufps xmm2,xmm2,0 mulps xmm2,xmm6 addps xmm0,xmm1 movss xmm3,[eax]FVector4.W shufps xmm3,xmm3,0 mulps xmm3,xmm7 // stall lea eax,[Result] addps xmm2,xmm3 // stall addps xmm0,xmm2 movups [eax],xmm0 } #else #error Please implement for your compiler. #endif #else Result.X = P.X * M[0][0] + P.Y * M[1][0] + P.Z * M[2][0] + P.W * M[3][0]; Result.Y = P.X * M[0][1] + P.Y * M[1][1] + P.Z * M[2][1] + P.W * M[3][1]; Result.Z = P.X * M[0][2] + P.Y * M[1][2] + P.Z * M[2][2] + P.W * M[3][2]; Result.W = P.X * M[0][3] + P.Y * M[1][3] + P.Z * M[2][3] + P.W * M[3][3]; #endif return Result; } // Regular transform. /** Transform a location - will take into account translation part of the FMatrix. */ FORCEINLINE FVector4 FMatrix::TransformFVector(const FVector &V) const { return TransformFVector4(FVector4(V.X,V.Y,V.Z,1.0f)); }
unreal 是列矩阵
大部分都是按照个人的习惯吧
