法線，切線和副法線構成了所謂的切線空間（tangnet space），在Bump Mapping中，法線紋理中存儲的法線值就是在切線空間內的。
從網上找了一段求切線和副法線的代碼.
根據三個頂點的位置坐標和紋理坐標求面的副法線和正切
1
//let P = v1 - v0
2D3DXVECTOR3 P = v1.pos - v0.pos;
3//let Q = v2 - v0
4D3DXVECTOR3 Q = v2.pos - v0.pos;
5float s1 = v1.s - v0.s;
6float t1 = v1.t - v0.t;
7float s2 = v2.s - v0.s;
8float t2 = v2.t - v0.t;
9
10//we need to solve the equation
11// P = s1*T + t1*B
12// Q = s2*T + t2*B
13// for T and B
14
15
16//this is a linear system with six unknowns and six equatinos, for TxTyTz BxByBz
17//[px,py,pz] = [s1,t1] * [Tx,Ty,Tz]
18// qx,qy,qz s2,t2 Bx,By,Bz
19
20//multiplying both sides by the inverse of the s,t matrix gives
21//[Tx,Ty,Tz] = 1/(s1t2-s2t1) * [t2,-t1] * [px,py,pz]
22// Bx,By,Bz -s2,s1 qx,qy,qz
23
24//solve this for the unormalized T and B to get from tangent to object space
25
26
27float tmp = 0.0f;
28if(fabsf(s1*t2 - s2*t1) <= 0.0001f)
29
{
30
tmp = 1.0f;
31
}
32else
33{
34 tmp = 1.0f/(s1*t2 - s2*t1 );
35}
36
37tangent.x = (t2*P.x - t1*Q.x);
38tangent.y = (t2*P.y - t1*Q.y);
39tangent.z = (t2*P.z - t1*Q.z);
40
41tangent = tmp * tangent;
42
43binormal.x = (s1*Q.x - s2*P.x);
44binormal.y = (s1*Q.y - s2*P.y);
45binormal.z = (s1*Q.z - s2*P.z);
46
47binormal = tmp * binormal;
根據Maya里面的資料寫了一個求取tangent的函數，輸入為3個頂點的位置，法線和紋理坐標，輸出是切線值，副法線可以由切線和法線叉乘得到。
inline bool floatEqual(float a, float b)
{
return abs(a-b) < 0.00001f;
}
HRESULT ComputerTangent(D3DXVECTOR3 position[3], D3DXVECTOR3 normal[3], D3DXVECTOR2 texcoord[3],D3DXVECTOR3 oTangent[3])
{
D3DXVECTOR3 edge1;
D3DXVECTOR3 edge2;
D3DXVECTOR3 crossP;
//==============================================
// x, s, t
// S & T vectors get used several times in this vector,
// but are only computed once.
//==============================================
edge1.x = position[1].x - position[0].x;
edge1.y = texcoord[1].x - texcoord[0].x;// s-vector - don't need to compute this multiple times
edge1.z = texcoord[1].y - texcoord[0].y;// t-vector
edge2.x = position[2].x - position[0].x;
edge2.y = texcoord[2].x - texcoord[0].x;// another s-vector
edge2.z = texcoord[2].y - texcoord[0].y;// another t-vector
D3DXVec3Cross(&crossP,&edge1,&edge2);
D3DXVec3Normalize(&crossP,&crossP);
bool degnerateUVTangentPlane = floatEqual(crossP.x, 0.0f);
if(degnerateUVTangentPlane)
crossP.x = 1.0f;
float tanX = -crossP.y / crossP.x;
oTangent[0].x = tanX;
oTangent[1].x = tanX;
oTangent[2].x = tanX;
//--------------------------------------------------------
// y, s, t
//--------------------------------------------------------
edge1.x = position[1].y - position[0].y;
edge2.x = position[2].y - position[0].y;
edge2.y = texcoord[2].x - texcoord[0].x;// another s-vector
edge2.z = texcoord[2].y - texcoord[0].y;// another t-vector
D3DXVec3Cross(&crossP,&edge1,&edge2);
D3DXVec3Normalize(&crossP,&crossP);
degnerateUVTangentPlane = floatEqual(crossP.x, 0.0f);
if(degnerateUVTangentPlane)
crossP.x = 1.0f;
float tanY = -crossP.y / crossP.x;
oTangent[0].y = tanY;
oTangent[1].y = tanY;
oTangent[2].y = tanY;
//------------------------------------------------------
// z, s, t
//------------------------------------------------------
edge1.x = position[1].z - position[0].z;
edge2.x = position[2].z - position[0].z;
edge2.y = texcoord[2].x - texcoord[0].x;// another s-vector
edge2.z = texcoord[2].y - texcoord[0].y;// another t-vector
D3DXVec3Cross(&crossP,&edge1,&edge2);
D3DXVec3Normalize(&crossP,&crossP);
degnerateUVTangentPlane = floatEqual(crossP.x, 0.0f);
if(degnerateUVTangentPlane)
crossP.x = 1.0f;
float tanZ = -crossP.y / crossP.x;
oTangent[0].z = tanZ;
oTangent[1].z = tanZ;
oTangent[2].z = tanZ;
//------------------------------------------------------
for( int i = 0; i < 3; i++)
{
// Ortho-normalize to normal
float dot = D3DXVec3Dot(&oTangent[i],&normal[i]);
oTangent[i] -= normal[i] * dot;
// Normalize tangents
D3DXVec3Normalize(&oTangent[i],&oTangent[i]);
}
return S_OK;
}