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GMCh3
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GEOMETRIC MODELING: A First Course
Copyright © 19951999 by Aristides A. G.
Requicha
Permission is hereby granted to copy this document for individual student use at USC,
provided that this notice is included in each copy.
3. Representations
Representations were defined in the Introduction as symbol structures that correspond to
(mathematical models of) physical entites. Our focus in this course is on mathematical
models and computer representations that capture the shape of physical objects. This
chapter discusses fundamental properties of representation schemes, and outlines the
known approaches for representing geometric entities.
3.1 Representation Schemes
This section introduces basic notions and properties of representations through some
simple examples. First we need a few definitions. A polygon is a 2D region of the plane
bounded by line segments called
edges
. Adjacent edges cannot be collinear. A
vertex
of a
polygon is a point of the polygon’s boundary where two adjacent edges meet. Two edges
may intersect only at a common vertex. That is, selfintersecting figures are not considered
polygons in this course. A
simple
polygon is a polygon without holes. Figure 3.1.1 shows
examples.
Figure 3.1.1 – A simple polygon with six vertices and six edges (left),
a polygon that is not simple because it has a hole (center),
and a figure that is not a polygon because it selfintersects (right).
A set
X
is
convex
if the line segment
pq
lies entirely within
X
for every pair of points
p
,
q
of
X
. Figure 3.1.2 shows a set that is not convex because it does not contain entirely a line
segment that connects two of the points of the set.
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•
•
p
q
Figure 3.1.2 – A polygon that is not convex
Given any set
X
, one can always construct infinitely many other sets that enclose
X
and are
convex. The smallest such set is called the
convex hull
of
X
. Figure 3.1.3 shows a set of
discrete points in the plane and its convex hull. Note that the convex hull of a set of discrete
points is a convex polygon whose vertices are a subset of the given points.
•
•
•
•
•
•
•
•
Figure 3.1.3 – A set of discrete points and its convex hull
Suppose now that we want to represent in a computer simple polygons. The set of objects
to be represented—in our example, the set of all the simple polygons—is called the
domain
of the representation scheme. We propose a scheme, which we call Scheme 1, defined as
follows.
1.
For each polygon, construct the set of its vertices,
in arbitrary order
.
2.
For each vertex, construct a pair of real numbers with the coordinates of the vertex in
some agreed frame.
3.
Make a list (i.e, a sequence of elements) containing all these pairs of reals.
Therefore the symbol structure used to represent a polygon in Scheme 1 is simply a list of
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