Surfaces and Beyond¶

Surface Modeling¶

Parametric Surface: Bezier Surface $$ S(u,v)=\sum_{i=0}^{n}\sum_{j=0}^{m}P_{ij}B_{i,n}(u)B_{j,m}(v) , $$
- $B_{i,n}(u),B_{j,m}(v)$分别是n维和m维的Bernsterin basis
- $(n+1)\times(m+1)P_{i,j}$ constructs the control mesh
- $S(u,v)=\sum_{i=0}^n[\sum_{j=0}^mP_{ij}B_{j,m}(v)]B_{i,n}(u)=\sum_{i=0}^nQ_i(v)B_{i,n}(u) ,$
Normal on a Bezier Surface $$ \begin{aligned} \frac{\partial}{\partial u}S(u,v)&=\frac{\partial}{\partial u}\sum_{i=0}^{n}\sum_{j=0}^{m}P_{ij}B_{i,n}(u)B_{j,m}(v) \newline
&=n\sum_{i=0}^{n-1}\sum_{j=0}^{m}(P_{i+1,j}-P_{ij})B_{i,n-1}(u)B_{j,m}(v) \newline \frac{\partial}{\partial v}S(u,v)&=\frac{\partial}{\partial v}\sum_{i=0}^{n}\sum_{j=0}^{m}P_{ij}B_{i,n}(u)B_{j,n}(v) \newline
&=m\sum_{i=0}^{n}\sum_{j=0}^{m-1}(P_{i,j+1}-P_{ij})B_{i,n}(u)B_{j,m-1}(v) \end{aligned} $$

Normal $N(u,v)$ $$ N(u,v)=\frac{\partial S(u,v)}{\partial u}\times\frac{\partial S(u,v)}{\partial v} $$

B-spline Surface $$ S(u,v)=\sum_{i=0}^{n}\sum_{j=0}^{m}N_{i,p}(u)N_{j,q}(v)P_{i,j} $$
NURBS Surface $$ S(u,v)=\frac{\sum_{i=0}^{n}\sum_{j=0}^{m}N_{i,p}(u)N_{j,q}(v)\omega_{i,j}P_{i,j}}{\sum_{i=0}^{n}\sum_{j=0}^{m}N_{i,p}(u)N_{j,q}(v)\omega_{i,j}} $$

Algorithms
Starting from a set of vertices
Generate new vertices according to certain rules to replace old ones
- Geometric rule
Connect new vertices according to certain rules
- Topological rule
Repeat a few times or until convergence
Advantages
topologically complex shape
easy to implement
easy to edit by manipulating the control mesh