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ch3 - GMCh3 3-1 GEOMETRIC MODELING A First Course Copyright...

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GMCh3 2/1/00 3-1 GEOMETRIC MODELING: A First Course Copyright © 1995-1999 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 2-D 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, self-intersecting 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 self-intersects (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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GMCh3 2/1/00 3-2 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 pairs of real numbers: x 1 , y 1 ( 29 x 2 , y 2 ( 29 x n , y n ( 29 ( 29 .
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