Rotations with Quaternions

A quaternion is a 4 dimensional complex-like number, it has four components, three of which are the “imaginary” part.

$$q = a + b\textrm{i} + c\textrm{j} + d\textrm{k} \\ q = (b,c,d, a) \\ \textrm{i}^{2}=\textrm{j}^{2}=\textrm{k}^{2}=\textrm{i}\textrm{j}\textrm{k}=−1$$

We represent a quaternion with this data structure:

四元数长度

$$\| q \|^2 = \sqrt{x^2 + y^2 + z^2 + w^2}$$

四元数单位化

单位四元数表示三维空间中的一个旋转。

恒等四元数

四元数缩放

四元数乘法

$$q_1 = a + b \textrm{i} + c \textrm{j} + d \textrm{k} \\ q_2 = e + f \textrm{i} + g \textrm{j} + h \textrm{k} \\ q_{1}\cdot q_{2} = (ae-bf-cg-dh) + (af+be+ch-dg)\textrm{i} + \\ (ag-bh+ce+df)\textrm{j}+(ah+bg-cf+de)\textrm{k}$$

四元数到旋转矩阵

$$q = (x, y, z, w)$$ $$M_{\textit{rotate}} = \begin{bmatrix} 1-2yy-2zz && 2xy-2zw && 2xz+2yw && 0 \\ 2xy+2zw && 1-2xx-2zz && 2yz-2xw && 0 \\ 2xz-2yw && 2yz+2xw && 1-2xx-2yy && 0 \\ 0 && 0 && 0 && 1 \end{bmatrix}$$

四元数共轭

$$q^* = a - b\textrm{i} - c\textrm{j} - d\textrm{k}$$

四元数取逆

$$q^{-1} = \frac{q^*}{\| q \|^2}$$

四元数差

$$q_3=q_1^{−1} \cdot q_2$$

四元数 Exp & Log

$$\exp(q) = \exp(w) \begin{pmatrix} \frac{v_{x}}{||v||}\sin(||v||) \\ \frac{v_{y}}{||v||}\sin(||v||) \\ \frac{v_{z}}{||v||}\sin(||v||) \\ \cos(||v||) \end{pmatrix}$$ $$\log(q) = \begin{pmatrix} \frac{v_{x}}{||v||}\arccos(\frac{w}{||q||}) \\ \frac{v_{y}}{||v||}\arccos(\frac{w}{||q||}) \\ \frac{v_{z}}{||v||}\arccos(\frac{w}{||q||}) \\ \log(||q||) \end{pmatrix}$$

四元数指数

$$q^n = \exp(n \log(q))$$

四元数球面插值

$$\textrm{slerp}(q_{1}, q_{2}, t) = q_{1}(q_{1}^{-1}q_{2})^{t}$$

参考

• Rotations with quaternions