Splines and Subdivision

Abstract

Most subdivision schemes are derived from refinement methods of control meshes of spline curves and surfaces.

Appendices

Remarks

This chapter gives refinement rules for control meshes of spline surfaces. The refinement has two meanings: (1) double knot intervals and obtain a new spline function space; (2) densify vertices of old control meshes and obtain new control meshes. The refinement of control meshes of spline surfaces is also called subdivision of spline surfaces. We firstly obtain refinement rules of basic functions, and then deduce subdivision rules of spline surfaces by refinement rules of basic functions. For four-directional box spline surfaces, we decompose subdivision masks to obtain smaller subdivision masks because of subdivision masks directly derived from refinement rules of basic functions are large. There are several methods to deduce refinement rules of basic functions: the de-Boor algorithm and the convolution definition for B-spline basic functions, the recursive method, and the generating function for box spline basic functions. It should be noticed that B-spline is a unique case of box splines.

Exercises

• 1. For the knot vector $U = [{\ldots },-3,-2,-1,0,1,2,3,{\ldots }]$, please write program codes to compute the B-splines: $N_{0,2}(u), N_{0,3}(u), N_{0,4}(u), N_{0,5}(u)$ using the formula (2.1) and render the curves of these spline functions.

• 2. Assume that ${\varvec{P}}_0 =[0,0], {\varvec{P}}_1=[1,1],{\varvec{P}}_3=[3,0]$. The cubic spline curve is ${\varvec{p}}(u)=\displaystyle \sum _{i=0}^3 {\varvec{P}}_i N_{i, 3} (u)$, where $U = [0~0~0~0~1~1~1~1]$. Compute its point using the de-Boor algorithm (see Formula (2.7)) and render the curve.

• 3. For the curve in the above item, double the knot interval [01] as [0 0.51] and then obtain the knot vector $U^{1} =[0~0~0~0~0.5~1~1~1~1]$. Please compute the new control vertices for the curve.

• 4. For the curve in the item 2, double knot intervals with nonzero length k times and then the knot vector $U^{k} =[0, 0, 0, 0, u_{1}, u_{2}, {\ldots }, u_{n}, 1, 1, 1, 1]$. Please compute the new control vertices after every doubling step.

• 5. Using Formula (2.11), write program codes to subdivide the control polygon of a non-uniform cubic B-spline curve.

• 6. Using Formula (2.13)$\sim$(2.15), write program codes to subdivide the control mesh of a non-uniform bi-cubic B-spline surface.

• 7. For uniform cubic B-spline, when double knots, please deduce the formula:

  $$ N_{i, 3}(u)=\dfrac{1}{8}N^1_{2i-2,3}(u)+\dfrac{4}{8}N^1_{2i-1,3}(u)+\dfrac{6}{8}N^1_{2i, 3}(u)+\dfrac{4}{8}N^1_{2i+1,3}(u)+\dfrac{1}{8}N^1_{2i+2,3}(u). $$

• 8. See Definition 2.4. Write program codes to construct the grid $G_{D}^{3}$ according to vectors in $D^{3}$.