Introduction

Abstract

Surface modeling is a fundamental realm of CAD and greatly affects the development of CAD compared with NURBS.

Appendices

Remarks

This chapter gives a review for the development of subdivision surfaces. This review focuses on the stationary subdivision scheme though there are also some literatures on the non-stationary subdivision [82, 83, 90]. Stationary subdivision schemes currently are the most frequently applied subdivision schemes. Discussions in this book will also focus on stationary subdivision schemes. As far as expressions of geometric shapes are concerned, there are three forms: univariate, bivariate, and trivariate which are, respectively, fits for curves, surfaces, and volumes. There are probably expressions of higher dimensions [16]. However, our discussions are also limited to bivariate subdivision schemes, i.e., subdivision schemes to construct surfaces. Surfaces are the most frequently applied geometric shapes. A majority of researches on subdivision modeling are researches of subdivision surfaces. Topics involved by researches of subdivision surfaces are very extensive. These topics discussed in this book are only several ones. However, we try to enumerate other topics and literatures so that the readers can know them. For meshes, there are manifold meshes and non-manifold meshes. This book also refers to two-manifold meshes because a single surface is a two-manifold in the differential geometric.

In this chapter, there are probably some concepts that are not known by beginners. We have given simple descriptions for some elementary concepts such as mesh, subdivision scheme, triangle subdivision. Strict definitions will be given in later chapters. It is a good idea for a beginner to directly read the second chapter if he or she does not have a clear understanding of those elementary concepts. After reading the second chapter, you may read the introduction again. We want to give readers a comprehensive cognition for subdivision surfaces by the introduction.

Note: In some literatures, $C^{k}$ continuity of subdivision surfaces is discussed, while $G^{k}$ continuity of subdivision surfaces is discussed in other literatures. In the view of derivations, $C^{k}$ continuity and $G^{k}$ continuity are different. That is to say, $C^{k}$ continuity is a concept related to expressions of curves and surfaces, while $G^{k}$ continuity is related to geometric variables. However, it is easy to know that a surface must be $G^{k}$ continuous if it is $C^{k}$ continuous. Consequently, for geometric shapes, we do not use the concept of $G^{ k}$ continuity except special requirements. For functions whose values are scalar, we do use the concept of $C^{ k}$ continuity.

Exercises

• 1. What is the polygon mesh? Why is a polygon mesh able to represent a complex shape?

• 2. What is the subdivision surface? Why is a single subdivision surface able to represent a complex shape but why a single NURBS surface hasn’t the ability?

• 3. Why do most of the polygons of a mesh have the similar shape after the mesh is subdivided several times?