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 …
We can compute a rotation matrix \( R \in SO(3) \) from an angle \(\theta\) and axis \(l\) (unit vector)
\[
R = e^{\theta C} = I + \sin(\theta)C + (1-\cos \theta)C^2
\]
where \(C\) is the antisymmetric matrix:
\[
C = \begin{bmat …
We can use it to find an orthonormal basis for Tangent, Bitangent and Normal:
\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{align}
If the …
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 …
☆. 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 …