D3D中四元数的表示

1、定义。

DIRECTX9文档中定义,令q为一四元数，theta为绕轴axis旋转的角度，则：
q.x = sin(theta/2) * axis.x
q.y = sin(theta/2) * axis.y
q.z = sin(theta/2) * axis.z
q.w = cos(theta/2)

可以简单写为：q = [sin(theta/2)axis, sin(theta/2)w] 或者 [x,y,z,w] 或者[v,w]

2、负四元数-q 。
-q = [-q.x, -q.y, -q.z, -q.w]

3D几何意义：轴变换了方向，角度也换了方向，刚好 -q = q。这个特性比较郁闷，在3D中任意方位，都有2种不同的四元数表示方法，竟然不是唯一的。

3、单位四元数q = [0,0,0,1].
3D几何意义：好像还看不出有什么几何意义。补充：用处是为了表示旋转0角度，一般用在变量初始化，和矩阵的单位初始话一个道理。
D3DXQuaternionIdentity

D3DXQuaternionIsIdentity

4、四元数的长度或者模 ||q||。
D3DXQuaternionLengthSq 长度的平方 = x*x + y*y + z*z + w*w;
D3DXQuaternionLength 长度，上式的平方根。
根据3D中四元数的定义，只有单位长度的四元数才有意义。尽管在DIRECTX中很多公式并不要求单位四元数，但是我们必须明白，只有单位四元数才有意义。补充：单位四元数的长度为1.

D3DXQuaternionNormalize,这个公式是求标准四元数，类似矢量的正则化。

5、共轭: [-x, -y, -z, w]。逆：共轭除于长度,记为q-1 。
共轭: D3DXQuaternionConjugate
逆：D3DXQuaternionInverse

3D几何意义：相当于转了一个相反的角。由于3D中只有单位长度的四元数才有意义，共轭和逆是一样的。

6、四元数的点积。
D3DXQuaternionDot
q1.q2 = [v1,w1].[v2,w2]=v1.v2 + w1.w2 = x1x2+y1y2+z1z2+w1w2;

3D几何意义：绝对值越大，2个方位越近似，夹角的余弦。补充：对于单位四元数，有-1 <= a.b <= +1,我们一般只关心其绝对值，和矢量点乘类似，绝对值越大，表示2个角度的角位移越小（即相似）。

7、四元数的叉积，一般我们说四元数的乘，指的就是这个叉积。
D3DXQuaternionMultiply
[v1,w1] X [v2,w2] = [w1v2 + w2v1 + v1 X v2, w1w2 - v1.v2];
=[ w1x2 + x1w2 + z1y2 - y1z2,
w1y2 + y1w2 + x1z2 - z1x2,
w1z2 + z1w2 + y1x2 - x1y2,
w1w2 - x1x2 - y1y2 - z1z2 ]
特性：
a、结合律 (ab)c = a(bc). 交换律不满足ab != ba
b、 ||q1 q2|| = ||q1|| ||q2|| 这个说明：单位四元数相乘的结果还是单位四元数。
c、 (ab)-1 = b-1a-1. 这个和矩阵的逆是一样的。同理：

(Q1Q2...Q(n-1)Qn)-1 = Qn-1Q(n-1)-1...Q2-1Q1-1

8、四元数的差：一个方位到另一个方位的角位移。方位a旋转到b的角位移为d，则ad = b. 经过推导：d = a-1b

这个差找不到相应的DIRECTX公式。

9、四元数的对数、指数运算。

D3DXQuaternionLn，描述为：
A unit quaternion, is defined by:
Q == (cos(theta), sin(theta) * v) where |v| = 1
The natural logarithm of Q is, ln(Q) = (0, theta * v)

D3DXQuaternionExp，描述为：
Given a pure quaternion defined by:
Q = (0, theta * v);
This method calculates the exponential result.

exp(Q) = (cos(theta), sin(theta) * v)

10、四元数的插值－－slerp
D3DXQuaternionSlerp
四元数的精华就是可以线性平滑的进行插值。slerp__ Spherical Liear Interpolation.
令矢量v0 v1之间的夹角为a, vt = k0v0 + k1v1;
则：k0 = sin((1-t)a)/sin(a); k1 = sin(ta)/sin(a);

其中，二四元数之间的夹角余弦可以用点积来计算：x1x2+y1y2+z1z2+w1w2;

11、四个四元数之间的插值__Squad。很显然是为了插值平滑平保证过渡点之间连续（DIRECTX 文档说是保证过渡点间的切线连续，可以理解为矢量插值的二阶连续）。用到如下2个公式：
D3DXQuaternionSquadSetup
D3DXQuaternionSquad
DIRCTX 文档中的例子：
The following example shows how to use a set of quaternion keys (Q0, Q1, Q2, Q3) to compute the inner quadrangle points (A, B, C). This ensures that the tangents are continuous across adjacent segments.
A B Q0 Q1 Q2 Q3
The following code example demonstrates how you can interpolate between Q1 and Q2.
// Rotation about the z-axis
D3DXQUATERNION Q0 = D3DXQUATERNION(0, 0, 0.707f, -.707f);
D3DXQUATERNION Q1 = D3DXQUATERNION(0, 0, 0.000f, 1.000f);
D3DXQUATERNION Q2 = D3DXQUATERNION(0, 0, 0.707f, 0.707f);
D3DXQUATERNION Q3 = D3DXQUATERNION(0, 0, 1.000f, 0.000f);
D3DXQUATERNION A, B, C, Qt;
FLOAT time = 0.5f;
D3DXQuaternionSquadSetup(&A, &B, &C, &Q0, &Q1, &Q2, &Q3);
D3DXQuaternionSquad(&Qt, &Q1, &A, &B, &C, time);
注意:
1 C is +/- Q2 depending on the result of the function. （这点我还是不太明白哦？）
2 Qt is the result of the function.
The result is a rotation of 45 degrees around the z-axis for time = 0.5.

DIRCTX9 的例子：SkinnedMesh Sample.

12、四元数和矩阵的变换
a. 矩阵变换到四元数，D3DXQuaternionRotationMatrix
b. 更一般的，矩阵分解为比例，旋转和位移三个分量：HRESULT WINAPI D3DXMatrixDecompose
( D3DXVECTOR3 *pOutScale, D3DXQUATERNION *pOutRotation,
D3DXVECTOR3 *pOutTranslation, CONST D3DXMATRIX *pM );
c. 四元数变换到矩阵
D3DXMATRIX* WINAPI D3DXMatrixRotationQuaternion
( D3DXMATRIX *pOut, CONST D3DXQUATERNION *pQ);
13、四元数和欧拉角的变换
a. 欧拉角到四元数
// Yaw around the Y axis, a pitch around the X axis,
// and a roll around the Z axis.
D3DXQUATERNION* WINAPI D3DXQuaternionRotationYawPitchRoll
( D3DXQUATERNION *pOut, FLOAT Yaw, FLOAT Pitch, FLOAT Roll );
b.绕轴旋转
// Rotation about arbitrary axis.
D3DXQUATERNION* WINAPI D3DXQuaternionRotationAxis

( D3DXQUATERNION *pOut, CONST D3DXVECTOR3 *pV, FLOAT Angle );

插值的例子：

bool AnimationModelClass::UpdateMD5Model(float deltaTime, int animation)
{   MD5Model.m_animations[animation].currAnimTime += deltaTime;         // Update the current animation timeif(MD5Model.m_animations[animation].currAnimTime > MD5Model.m_animations[animation].totalAnimTime)MD5Model.m_animations[animation].currAnimTime = 0.0f;// Which frame are we onfloat currentFrame = MD5Model.m_animations[animation].currAnimTime * MD5Model.m_animations[animation].frameRate;    int frame0 = floorf( currentFrame );int frame1 = frame0 + 1;// Make sure we don't go over the number of frames  if(frame0 == MD5Model.m_animations[animation].numFrames-1)frame1 = 0;float interpolation = currentFrame - frame0;    // Get the remainder (in time) between frame0 and frame1 to use as interpolation factorstd::vector<Joint> interpolatedSkeleton;        // Create a frame skeleton to store the interpolated skeletons in// Compute the interpolated skeletonfor( int i = 0; i < MD5Model.m_animations[animation].numJoints; i++){Joint tempJoint;Joint joint0 = MD5Model.m_animations[animation].frameSkeleton[frame0][i];       // Get the i'th joint of frame0's skeletonJoint joint1 = MD5Model.m_animations[animation].frameSkeleton[frame1][i];       // Get the i'th joint of frame1's skeletontempJoint.parentID = joint0.parentID;                                           // Set the tempJoints parent id// Turn the two quaternions into XMVECTORs for easy computationsD3DXQUATERNION joint0Orient = D3DXQUATERNION(joint0.orientation.x, joint0.orientation.y, joint0.orientation.z, joint0.orientation.w);D3DXQUATERNION joint1Orient = D3DXQUATERNION(joint1.orientation.x, joint1.orientation.y, joint1.orientation.z, joint1.orientation.w);// Interpolate positionstempJoint.pos.x = joint0.pos.x + (interpolation * (joint1.pos.x - joint0.pos.x));tempJoint.pos.y = joint0.pos.y + (interpolation * (joint1.pos.y - joint0.pos.y));tempJoint.pos.z = joint0.pos.z + (interpolation * (joint1.pos.z - joint0.pos.z));// Interpolate orientations using spherical interpolation (Slerp)D3DXQUATERNION qtemp;D3DXQuaternionSlerp(&qtemp, &joint0Orient, &joint1Orient, interpolation);tempJoint.orientation.x = qtemp.x;tempJoint.orientation.y = qtemp.y;tempJoint.orientation.z = qtemp.z;tempJoint.orientation.w = qtemp.w;// Push the joint back into our interpolated skeletoninterpolatedSkeleton.push_back(tempJoint);      }for ( int k = 0; k < MD5Model.numSubsets; k++){for ( int i = 0; i < MD5Model.m_subsets[k].numVertices; ++i ){Vertex tempVert = MD5Model.m_subsets[k].m_vertices[i];// Make sure the vertex's pos is cleared firsttempVert.x = 0;tempVert.y = 0;tempVert.z = 0;// Clear vertices normaltempVert.nx = 0;tempVert.ny = 0;tempVert.nz = 0;// Sum up the joints and weights information to get vertex's position and normalfor ( int j = 0; j < tempVert.WeightCount; ++j ){Weight tempWeight = MD5Model.m_subsets[k].m_weights[tempVert.StartWeight + j];Joint tempJoint = interpolatedSkeleton[tempWeight.jointID];// Convert joint orientation and weight pos to vectors for easier computationD3DXQUATERNION tempJointOrientation = D3DXQUATERNION(tempJoint.orientation.x, tempJoint.orientation.y, tempJoint.orientation.z, tempJoint.orientation.w);D3DXQUATERNION tempWeightPos = D3DXQUATERNION(tempWeight.pos.x, tempWeight.pos.y, tempWeight.pos.z, 0.0f);// We will need to use the conjugate of the joint orientation quaternionD3DXQUATERNION tempJointOrientationConjugate;D3DXQuaternionInverse(&tempJointOrientationConjugate, &tempJointOrientation);// Calculate vertex position (in joint space, eg. rotate the point around (0,0,0)) for this weight using the joint orientation quaternion and its conjugate// We can rotate a point using a quaternion with the equation "rotatedPoint = quaternion * point * quaternionConjugate"D3DXVECTOR3 rotatedPoint;D3DXQUATERNION qqtemp;D3DXQuaternionMultiply(&qqtemp, &tempJointOrientation, &tempWeightPos);D3DXQuaternionMultiply(&qqtemp, &qqtemp, &tempJointOrientationConjugate);rotatedPoint.x = qqtemp.x;rotatedPoint.y = qqtemp.y;rotatedPoint.z = qqtemp.z;// Now move the verices position from joint space (0,0,0) to the joints position in world space, taking the weights bias into accounttempVert.x += ( tempJoint.pos.x + rotatedPoint.x ) * tempWeight.bias;tempVert.y += ( tempJoint.pos.y + rotatedPoint.y ) * tempWeight.bias;tempVert.z += ( tempJoint.pos.z + rotatedPoint.z ) * tempWeight.bias;// Compute the normals for this frames skeleton using the weight normals from before// We can comput the normals the same way we compute the vertices position, only we don't have to translate them (just rotate)D3DXQUATERNION tempWeightNormal = D3DXQUATERNION(tempWeight.normal.x, tempWeight.normal.y, tempWeight.normal.z, 0.0f);D3DXQuaternionMultiply(&qqtemp, &tempJointOrientation, &tempWeightNormal);D3DXQuaternionMultiply(&qqtemp, &qqtemp, &tempJointOrientationConjugate);// Rotate the normalrotatedPoint.x = qqtemp.x;rotatedPoint.y = qqtemp.y;rotatedPoint.z = qqtemp.z;  // Add to vertices normal and ake weight bias into accounttempVert.nx -= rotatedPoint.x * tempWeight.bias;tempVert.ny -= rotatedPoint.y * tempWeight.bias;tempVert.nz -= rotatedPoint.z * tempWeight.bias;}// Store the vertices position in the position vector instead of straight into the vertex vectorMD5Model.m_subsets[k].m_positions[i].x = tempVert.x;    MD5Model.m_subsets[k].m_positions[i].y = tempVert.y;    MD5Model.m_subsets[k].m_positions[i].z = tempVert.z;    // Store the vertices normalMD5Model.m_subsets[k].m_vertices[i].nx = tempVert.nx;   MD5Model.m_subsets[k].m_vertices[i].ny = tempVert.ny;   MD5Model.m_subsets[k].m_vertices[i].nz = tempVert.nz;   // Create the temp D3DXVECTOR3 for normalizeD3DXVECTOR3 dtemp = D3DXVECTOR3(0,0,0);dtemp.x = MD5Model.m_subsets[k].m_vertices[i].nx;dtemp.y = MD5Model.m_subsets[k].m_vertices[i].ny;dtemp.z = MD5Model.m_subsets[k].m_vertices[i].nz;D3DXVec3Normalize(&dtemp, &dtemp);MD5Model.m_subsets[k].m_vertices[i].nx = dtemp.x;MD5Model.m_subsets[k].m_vertices[i].ny = dtemp.y;MD5Model.m_subsets[k].m_vertices[i].nz = dtemp.z;// Put the positions into the vertices for this subsetMD5Model.m_subsets[k].m_vertices[i].x = MD5Model.m_subsets[k].m_positions[i].x;MD5Model.m_subsets[k].m_vertices[i].y = MD5Model.m_subsets[k].m_positions[i].y;MD5Model.m_subsets[k].m_vertices[i].z = MD5Model.m_subsets[k].m_positions[i].z;}// Update the subsets vertex buffer// First lock the buffervoid* mappedVertBuff;bool result;result = MD5Model.m_subsets[k].vertBuff->Map(D3D10_MAP_WRITE_DISCARD, 0, &mappedVertBuff);if(FAILED(result)){return false;}// Copy the data into the vertex buffer.memcpy(mappedVertBuff, &MD5Model.m_subsets[k].m_vertices[0], (sizeof(Vertex) * MD5Model.m_subsets[k].numVertices));MD5Model.m_subsets[k].vertBuff->Unmap();}return true;
}

相关的函数：

D3DXQUATERNION* WINAPI D3DXQuaternionBaryCentric//三个四元数的重心
D3DXQUATERNION* D3DXQuaternionConjugate //求共轭四元数
FLOAT D3DXQuaternionDot //四元数点乘
D3DXQUATERNION* WINAPI D3DXQuaternionExp //四元数指数
D3DXQUATERNION* WINAPI D3DXQuaternionInverse//共轭变换并规格化四元数
D3DXQUATERNION* D3DXQuaternionIdentity //单位四元数
BOOL D3DXQuaternionIsIdentity //判断单位四元数
FLOAT D3DXQuaternionLength //返回四元数长度
FLOAT D3DXQuaternionLengthSq //返回四元数长度平方
D3DXQUATERNION* WINAPI D3DXQuaternionLn //计算自然对数
D3DXQUATERNION* WINAPI D3DXQuaternionMultiply //两个四元数相乘
D3DXQUATERNION* WINAPI D3DXQuaternionNormalize //单位长度的四元数
D3DXQUATERNION* WINAPI D3DXQuaternionRotationAxis//绕轴旋转四元数
D3DXQUATERNION* WINAPI D3DXQuaternionRotationMatrix//旋转矩阵创建四元数
D3DXQUATERNION* WINAPI D3DXQuaternionRotationYawPitchRoll//绕XYZ旋转创建的四元数
D3DXQUATERNION* WINAPI D3DXQuaternionSlerp //两个四元数球面插值
D3DXQUATERNION* WINAPI D3DXQuaternionSquad //球面四边形插四元数
void WINAPI D3DXQuaternionSquadSetup //创建球面四边形插值的控制点
void WINAPI D3DXQuaternionToAxisAngle//向量绕自身旋转角度计算四元数

四元数旋转物品：

首先我们需要定义一些变量来储存所需信息
D3DXQUATERNION g_qNow, g_qDown; //分别是现在的旋转用四元数按下鼠标时的旋转用四元数
D3DXVECTOR3 g_vDownPt, g_vCurrentPt; //分别是按下鼠标时的球面上的位置向量当前的位置向量
bool g_bDrag = false; //是否按下鼠标左键用来判断按键拖拽

然后初始化这些变量
D3DXQuaternionIdentity(&g_qNow);
D3DXQuaternionIdentity(&g_qDown);
g_bDrag = false;
好了，初始化工作做完了，然后具体讲下算法。
在鼠标按下处理过程

case WM_LBUTTONDOWN:
   {
    int iMouseX = ( short )LOWORD( lParam );
    int iMouseY = ( short )HIWORD( lParam );    //得到屏幕坐标
    g_bDrag = true;
    g_qDown = g_qNow;
    g_vDownPt = ScreenToVector( ( float )iMouseX, ( float )iMouseY );   //得到投影窗口下的坐标
   }
在鼠标移动处理过程

case WM_MOUSEMOVE:
   {
    if( g_bDrag )
    {
     int iMouseX = ( short )LOWORD( lParam );
     int iMouseY = ( short )HIWORD( lParam );  //得到屏幕坐标
     g_vCurrentPt = ScreenToVector( ( float )iMouseX, ( float )iMouseY );//得到投影窗口下的坐标
     g_qNow = g_qDown * QuatFromBallPoints( g_vDownPt, g_vCurrentPt );
    }
   }

//看看QuatFromBallPoints函数

D3DXQUATERNION QuatFromBallPoints( const D3DXVECTOR3& vFrom, const D3DXVECTOR3& vTo )
{
D3DXVECTOR3 vPart;

float fDot = D3DXVec3Dot( &vFrom, &vTo ); //取得两向量的点乘，因为两个都是单位向量，所以fDot等于cos theta

D3DXVec3Cross( &vPart, &vFrom, &vTo );//叉乘，获得的是垂直于两个向量的一个向量，即旋转轴。其模等于|a||b|sin theta等于sin theta

return D3DXQUATERNION( vPart.x, vPart.y, vPart.z, fDot );//正好构成一个旋转2*theta角度的四元数

}

首先说一下旋转四元数，一个(x*sin theta, y*sin theta, z*sin theta, cos theta)的四元数被用来旋转2*theta角度。在上面的代码中通过两个单位向量得到了一个旋转四元数。从效果上来说就是鼠标在球体上做过theta角度，物体就旋转2*theta角度。这是一个很方便的解决方法，不仅省却了换算，也使得物体能够在一次拖拽中旋转360度。现在大部分事情都清楚了，接下来只要在绘制之前把世界矩阵按所得的旋转四元数旋转之后绘制物体就可以了。
我们使用D3DXMatrixRotationQuaternion这个函数从四元数得到一个旋转矩阵作为世界矩阵。
D3DXMATRIXA16 matWorld;
D3DXMatrixRotationQuaternion( &matWorld, &g_qNow );
g_pd3dDevice->SetTransform( D3DTS_WORLD, &matWorld );
至此完成，我们只需要在每一次绘制的时候按当前的旋转四元数即g_qNow来设置世界矩阵即可。

//
for (unsigned int i = 0; i < bones->size(); i++) {// unsigned long t0, t1;D3DXQUATERNION q0, q1;D3DXVECTOR3 p0, p1;//========================================//========================================if (ite_keyFrames[i] != keyFrames[i].end()) {//t0 = (*ite_keyFrames[i]).frameNo;//boneRot[i] = q0 = (*ite_keyFrames[i]).rotation;//bonePos[i] = p0 = (*ite_keyFrames[i]).position;//if (++ite_keyFrames[i] != keyFrames[i].end()) {//KeyFrame k = *ite_keyFrames[i];//t1 = k.frameNo;//q1 = k.rotation;//p1 = k.position;//========================================//(25 - 10) / 30 - 10 = 3 / 4 = 0.75 → 75%//========================================float s = (float)(time - t0) / (float)(t1 - t0);//========================================//========================================D3DXQuaternionSlerp(&boneRot[i], &q0, &q1, k.bezie_r->GetY(s));bonePos[i].x = p0.x + (p1.x - p0.x)*k.bezie_x->GetY(s);bonePos[i].y = p0.y + (p1.y - p0.y)*k.bezie_y->GetY(s);bonePos[i].z = p0.z + (p1.z - p0.z)*k.bezie_z->GetY(s);if (time != t1) --ite_keyFrames[i];}}//========================================//========================================D3DXMATRIX rot, trans;D3DXMatrixRotationQuaternion(&rot, &boneRot[i]);D3DXMatrixTranslation(&trans, bonePos[i].x, bonePos[i].y, bonePos[i].z);//========================================//========================================(*bones)[i].boneMatBL = rot*trans*(*bones)[i].initMatBL;}

参考：

http://blog.csdn.net/silangquan/article/details/50390570
http://blog.csdn.net/ycl295644/article/details/50961201

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