Graphics Reference
In-Depth Information
CHAPTER 15
Local Geometric
Modeling Topics
Prerequisites:
Chapter 4, Sections 8.1-8.5 (parameterization, surface, tangent vectors/
planes, manifolds), Sections 9.1-9.4, 9.8, and 9.9 in [AgoM05]
15.1
Overview
This chapter covers a number of applications that involve local properties of surfaces.
Section 15.2 discusses some curvature-related topics. Sections 15.3-15.4 deal with
geodesics on surfaces. We discuss algorithms for generating both smooth and discrete
geodesics and then describe an application to filament winding and tape laying.
Section 15.5 is on how one can drop curves onto surfaces, something that is impor-
tant in robotic applications. Section 15.6 discusses a few blending topics.
15.2
Curvature
Curvature raises its head at many places in CAD and CAGD. It is important to fairing
curves and surfaces. We have also seen it in the context of finding efficient polygonal
representations for curves and surfaces. For example, if a parametric surface has
many relatively flat spots, then using a polygonization of the surface that is obtained
by evaluating points at a uniform grid in parameter space would be very wasteful. To
minimize the space for the data structure one would want a relatively coarse sub-
division for flat regions and only use a much finer subdivision at the more curved
parts. Clearly, flatness is a function of curvature. This section presents a sample of the
curvature topics that one finds in the literature by summarizing the findings of several
technical papers that have been published on the subject. The interested reader can
find other references in those papers.
