In Clifford algebra all units forms a group, so we can construct a unit dual-quaternion from two quaternions q and t where q is a unit rotation quaternion and t is a pure quaternion representing the translation: $d = (1 + \fra … 2016-07-09 5 comments ## Rodrigues’ Rotation Formula We can compute a rotation matrix $$R \in SO(3)$$ from an angle $$\theta$$ and axis $$l$$ (unit vector) \[ R = I + \sin(\theta)C + (1-\cos \theta)C^2$ where $$C$$ is the antisymmetric matrix: C = \begin{bmatrix} 0 & -l … 2016-07-08 Comments ## Gram-Schmidt Orthonormalization We can use it to find an orthonormal basis for Tangent, Bitangent and Normal ( Vector3 ): \begin{align} T &= T-\frac{(T \cdot N)N}{N \cdot N} \\ B &= B-\frac{(B \cdot N)N}{N \cdot N}-\frac{(B \cdot T)T}{T \cdot T} \end{a … 2013-04-03 Comments ## 曲线的标架运动习题 今天发现一道有趣的习题: 设{r(t); e1(t), e2(t), e3(t)}是沿曲线r(t)定义的一个单位正交标架场, 假定1≤i≤3 \[ e’_{i}(t)=\sum_{j=1}^{3}a_{ij}(t)e_{j}(t) 证明: $a_{ij}(t)+a_{ji}(t)=0$ 证: 其实就是证明一个重要的结论:  空间 …
2011-04-24 Comments

## Rotation

☆.  Rotation about the x, y, z axis \begin{align} R_x(\theta) &= \begin{bmatrix} 1 & 0 & 0\\ 0 & \cos\theta & -\sin\theta\\ 0 & \sin\theta & \cos\theta \end{bmatrix} = exp \left (\theta \begin{bmatrix …
2010-07-15 Comments