四元数的插值

今天我们和大家分享的是四元数的插值。这里的插值指的是球面线性插值。例如，我们要模拟一下地球绕着太阳，从P1到P2。这中间的每一个位置都要用球面线性插值来做。

首先我们聊一聊线性插值：

x=x1-x2,t是插值系数，则lerp(x1,x2,t)=x1+t*x表示x1到x2的插值。

四元数的插值

q=(p-1)*p1,插值系数为t，则p到p1的插值为：slerp(p,p1,t)=P*((p-1)p1)t，表示p的逆乘以p1，他们乘积的t次方，乘以p

但是上面的公式在编程的时候用的挺麻烦的，所以我们使用下面的公式：

旋转插值，想开头所说的，这里说的插值是球面插值，是在3D空间中旋转的，因此我们可以将它等价于旋转插值。

例如：向量V1，V0

W=v1-v0

vt=v0+tw

通过上图可知，VT=K0V0+K1V1.在这里，V1，V0，VT都是单位向量，V1和K1V1平行。

因此，我们可以求得sinw=sintw/k1,则k1=sintw/sinw,同样可得 k0=sin(1-t)w/sinw.

所以代入VT，就可以求出VT了。

四元数也是一个道理，注意slerp这个函数返回的是一个四元数。

以上就是四元数插值的公式，在编程实现的时候我们要注意

使用点成来纠结夹角W，还有就是放夹角W很小的时候，sinw很小，但是cosw趋于1，因此，就变成了线性插值。

在这里，我们贴出cocos2dx 3.6中四元数的实现代码，大家学习学习。

    /**
     * Interpolates between two quaternions using spherical linear interpolation.
     *
     * Spherical linear interpolation provides smooth transitions between different
     * orientations and is often useful for animating models or cameras in 3D.
     *
     * Note: For accurate interpolation,the input quaternions must be at (or close to) unit length.
     * This method does not automatically normalize the input quaternions,so it is up to the
     * caller to ensure they call normalize beforehand,if necessary.
     *
     * @param q1x The x component of the first quaternion.
     * @param q1y The y component of the first quaternion.
     * @param q1z The z component of the first quaternion.
     * @param q1w The w component of the first quaternion.
     * @param q2x The x component of the second quaternion.
     * @param q2y The y component of the second quaternion.
     * @param q2z The z component of the second quaternion.
     * @param q2w The w component of the second quaternion.
     * @param t The interpolation coefficient.
     * @param dstx A pointer to store the x component of the slerp in.
     * @param dsty A pointer to store the y component of the slerp in.
     * @param dstz A pointer to store the z component of the slerp in.
     * @param dstw A pointer to store the w component of the slerp in.
     */
    static void slerp(float q1x,float q1y,float q1z,float q1w,float q2x,float q2y,float q2z,float q2w,float t,float* dstx,float* dsty,float* dstz,float* dstw);

void Quaternion::slerp(float q1x,float* dstw)
{
    // Fast slerp implementation by kwhatmough:
    // It contains no division operations,no trig,no inverse trig
    // and no sqrt. Not only does this code tolerate small constraint
    // errors in the input quaternions,it actually corrects for them.
    GP_ASSERT(dstx && dsty && dstz && dstw);
    GP_ASSERT(!(t < 0.0f || t > 1.0f));

    if (t == 0.0f)
    {
        *dstx = q1x;
        *dsty = q1y;
        *dstz = q1z;
        *dstw = q1w;
        return;
    }
    else if (t == 1.0f)
    {
        *dstx = q2x;
        *dsty = q2y;
        *dstz = q2z;
        *dstw = q2w;
        return;
    }

    if (q1x == q2x && q1y == q2y && q1z == q2z && q1w == q2w)
    {
        *dstx = q1x;
        *dsty = q1y;
        *dstz = q1z;
        *dstw = q1w;
        return;
    }

    float halfY,alpha,beta;
    float u,f1,f2a,f2b;
    float ratio1,ratio2;
    float halfSecHalfTheta,versHalfTheta;
    float sqNotU,sqU;

    float cosTheta = q1w * q2w + q1x * q2x + q1y * q2y + q1z * q2z;

    // As usual in all slerp implementations,we fold theta.
    alpha = cosTheta >= 0 ? 1.0f : -1.0f;
    halfY = 1.0f + alpha * cosTheta;

    // Here we bisect the interval,so we need to fold t as well.
    f2b = t - 0.5f;
    u = f2b >= 0 ? f2b : -f2b;
    f2a = u - f2b;
    f2b += u;
    u += u;
    f1 = 1.0f - u;

    // One iteration of Newton to get 1-cos(theta / 2) to good accuracy.
    halfSecHalfTheta = 1.09f - (0.476537f - 0.0903321f * halfY) * halfY;
    halfSecHalfTheta *= 1.5f - halfY * halfSecHalfTheta * halfSecHalfTheta;
    versHalfTheta = 1.0f - halfY * halfSecHalfTheta;

    // Evaluate series expansions of the coefficients.
    sqNotU = f1 * f1;
    ratio2 = 0.0000440917108f * versHalfTheta;
    ratio1 = -0.00158730159f + (sqNotU - 16.0f) * ratio2;
    ratio1 = 0.0333333333f + ratio1 * (sqNotU - 9.0f) * versHalfTheta;
    ratio1 = -0.333333333f + ratio1 * (sqNotU - 4.0f) * versHalfTheta;
    ratio1 = 1.0f + ratio1 * (sqNotU - 1.0f) * versHalfTheta;

    sqU = u * u;
    ratio2 = -0.00158730159f + (sqU - 16.0f) * ratio2;
    ratio2 = 0.0333333333f + ratio2 * (sqU - 9.0f) * versHalfTheta;
    ratio2 = -0.333333333f + ratio2 * (sqU - 4.0f) * versHalfTheta;
    ratio2 = 1.0f + ratio2 * (sqU - 1.0f) * versHalfTheta;

    // Perform the bisection and resolve the folding done earlier.
    f1 *= ratio1 * halfSecHalfTheta;
    f2a *= ratio2;
    f2b *= ratio2;
    alpha *= f1 + f2a;
    beta = f1 + f2b;

    // Apply final coefficients to a and b as usual.
    float w = alpha * q1w + beta * q2w;
    float x = alpha * q1x + beta * q2x;
    float y = alpha * q1y + beta * q2y;
    float z = alpha * q1z + beta * q2z;

    // This final adjustment to the quaternion's length corrects for
    // any small constraint error in the inputs q1 and q2 But as you
    // can see,it comes at the cost of 9 additional multiplication
    // operations. If this error-correcting feature is not required,// the following code may be removed.
    f1 = 1.5f - 0.5f * (w * w + x * x + y * y + z * z);
    *dstw = w * f1;
    *dstx = x * f1;
    *dsty = y * f1;
    *dstz = z * f1;
}