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 3 comments
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
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 …
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:  空间曲线的 …
2011-04-24 Comments
☆.  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