Consider a discrete cuve with vertices $\{...x_{i-1}, x_i, x_{i+1}, ...\}\in \mathbb{R}^{3n}$, and denote the edge $e^i=x_ix_{i+1} $ and tangent $t_i=x_{i+1} - x_i$. When perform discrete parallel transport, I know the steps:
- Compute the binormal: $b_i = t_{i-1} \times t_i$
- Compute the normals: $n_{i-1} = t_{i-1} \times b_i$, $n_{i} = t_{i} \times b_i$
- Compute the projection of any vector $\mathbb{v_{i-1}}$ of $e^{i-1}$ on the $b_i$ and $n_{i-1}$, denote the lengths are $w_1, w_2$
- Assume the length is the same, then the vector $\mathbb{v_i}=w_{1}b_i + w_{2}n_i$
But when I read the paper Discrete Elastic Rods in siggraph 2008, the discrete parallel transport can be performed directly by a rotation matrix. I'm wondering how to construct such rotation matrix. Thanks.