mvg:lecture_03:start
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Lecture 3
Comments
Preserve length, preserve cross product $\Leftrightarrow$ rigid body motion
$\hat{u} \in so(3)$
- $g(v) = |v|$, length preserved
- $g(u) \times g(v) = g(u \times v)$, cross product preserved
$\left<u,v\right> = \frac{1}{4}(|u+v|^2-|u-v|^2)$
$\Rightarrow \left<g(u), g(v) \times g(w)\right> = \left<u, v\times w\right> $, volume preserving
Representation of Rigid-Body Motion
The RBM preserves length and cross product.
$r_i = g_t(e_i)$
$R = (r_1, r_2, r_3)$
$r_i^Tr_j = g_t(e_i)^T g_t(e_j) = e_i^T e_j = \delta_{ij}$
$\quad r_1 \times r_2 = r_3 \Rightarrow \det(R) =1$
$R^T R = RR^T = I$
$R$ is in the Special Orthogonal Group $SO(3)$
mvg/lecture_03/start.1543497790.txt.gz · Last modified: 2018/11/29 13:23 by admin