3D Shape Analysis

2Basic Elements of 3D Geometry and Topology

This chapter introduces some of the fundamental concepts of 3D geometry and 3D geometry processing. Since this is a very large topic, we only focus in this chapter on the concepts that are relevant to the 3D shape analysis tasks covered in the subsequent chapters. We structure this chapter into two main parts. The first part (Section 2.1) covers the elements of differential geometry that are used to describe the local properties of 3D shapes. The second part (Section 2.2) defines the concept of shape, the transformations that preserve the shape of a 3D object, and the deformations that affect the shape of 3D objects. It also describes some preprocessing algorithms, e.g. alignment, which are used to prepare 3D models for shape analysis tasks.

2.1 Elements of Differential Geometry

2.1.1 Parametric Curves

A one‐dimensional curve in can be represented in a parametric form by a vector‐valued function of the form:

(2.1)

The functions , and are also called the coordinate functions. If we assume that they are differentiable functions of , then ...