Idea: Approximation function

• Bezier, BSpline, NURBS, etc.

• Reminder for 1D curve \(c\) approximating points \((p_i)_{i=0..N}\) (control polygon)
• • \(\displaystyle c(u)=\sum_{i=0}^{N} b_i(u)\, p_i \)
  • \( b_i(u)={N \choose i} \, u^i \, (1-u)^{N-i}\;,\;\;u\in[0,1]\) (Bernstein Polynomial)

• For a 2D surface \(S\) approximating points \(p_{ij}\)
• • \(\displaystyle S(u,v)=\sum_i \sum_j b_i(u) \, b_j(v)\; p_{ij}\) (tensor-product)

• For a 3D volume \(V\) approximating points on the lattice \(p_{ijk}\)
• • \(\displaystyle V(u,v,w)=\sum_i \sum_j \sum_k b_i(u) b_j(v) b_k(w)\; p_{ijk}\)

• • \(\Rightarrow\) Use V as a spatial deformation on vertex coordinates \((u,v,w)\), use \(p_{ijk}\) as grid.