Analysis of Subdivision Surface

Abstract

A mesh sequence is generated when a given mesh is recursively subdivided. Consequently, two questions are always asked: Is the mesh sequence convergent? How many is the continuity order of the limit surface if the mesh sequence is convergent? The two questions will be answered in this chapter.

Appendices

Remarks

Using the C-C subdivision surface as an example, this chapter presents discussions for the DFT in detail. Some different skills are probably used for different subdivision schemes when DFT is executed for subdivision matrices. Compared with the limit normal vector, the characteristic mapping is more precise to distinguish the $G^{1}$ continuity of subdivide surfaces. The characteristic mapping is very important to construct rules to distinguish $C^{k}$ continuity of subdivision surfaces [20, 102, 103]. Based on the characteristic mapping and $C^{k}$ continuity rules, Refs. [34, 84, 85] give subdivision rules that make subdivision surfaces G$^{2}$ continuous at the extraordinary point. Reference   [22, 105] also give some methods to adjust subdivision rules around extraordinary vertices. The analytic evaluation approach is discussed in the basis of analysis on the eigenstructures of subdivision matrices. By the analytic evaluation approach, we can also calculate the derivatives at any parameter. Peters et al.[21] give general principles to compute more geometric properties. Bolz et al. [71] sum up various evaluation methods as four types: (I) recursive evaluation; (II) direct evaluation; (III) reduction to the regular setting; and (IV) pretabulated basis function composition. The direct evaluation is the analytic evaluation. The last two types will be discussed in the subsequent chapters.

Exercises

• 1. Prove: The Catmull–Clark subdivision surface is the surface with the tangent plane continuity at least.

• 2. What is the characteristic mapping? Why do we define the characteristic mapping?

• 3. For the Catmull–Clark subdivision surface, Fig. 4.12 is the shape of the mesh formed by ${\varvec{t}}_{6}$ and ${\varvec{t}}_{7}$. Write program codes to compute ${\varvec{t}}_{6}$ and ${\varvec{t}}_{7}$. Draw the figure once more again using your code.