Graphics Reference
In-Depth Information
CHAPTER 14
Global Geometric
Modeling Topics
Prerequisites:
Basic calculus and vectors, Newton-Raphson method (Section 4.7 in
[AgoM05])
Sections 1.9 and 3.12 in [AgoM05] (for Section 14.5.2)
Chapter 9 in [AgoM05] (for Sections 14.9 and 14.10)
Chapter 10 in [AgoM05] (for Sections 14.5.1)
14.1
Overview
This chapter takes a glimpse at a few properties of curves and surfaces that are global
in nature. Section 14.2 looks at algorithms that compute various distances. Section
14.3 considers the so-far overlooked subject of how to best polygonize curves and sur-
faces. Section 14.4 describes an algorithm for displaying trimmed surfaces. Section
14.5 discusses implicitly defined curves and surfaces. Contour algorithms are
described in Section 14.6. In Section 14.7 we describe some skinning algorithms.
Section 14.8 describes how one can compute arc length and arc-length parameteri-
zation. Finally, in Sections 14.9 and 14.10 we briefly consider offset curves and sur-
faces and, more generally, envelopes. A major application of this subject is in milling
operations using numerically controlled (NC) milling machines. These machines come
in a variety of flavors distinguished mainly by the number of degrees of freedom of
movement that they possess. The more degrees of freedom they have, the more com-
plicated it is to describe their paths.
14.2
Distance Algorithms
This section deals only with distance algorithms between smooth objects with empha-
sis on parameterized objects. Formulas for basic linear objects were already presented
in Chapter 6. Finding the distance between polygonal curves or faceted surfaces easily
reduces to those cases.
