ch6-1 - GMCh6-1 6-1 GEOMETRIC MODELING A First Course...

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GMCh6-1 3/29/99 6-1 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. 6. Fundamental Algorithms Fundamental algorithms are the building blocks we use to construct computational solutions to application problems. This chapter covers some of the fundamental algorithms that underly many geometric modeling computations. We begin with a short introduction that emphasizes the connections between algorithms and the representations on which they operate. 6.1 Algorithms and Representations We consider a very simple example, which is a lower-dimensional version of the familiar problem of computing the image of a 3-D object by orthographically projecting it onto a 2- D screen. Our objective is to design an algorithm for computing the orthographic projections of convex polygons on a 1-D screen. 1-D images are not visually very exciting (to put it mildly), but they are simple, and suffice to illustrate some important concepts. A convex polygon may be defined as the convex hull of a set on non-collinear points—see Chapter 3. The orthographic projection of a set S on a line L may be defined as follows. First we choose a coordinate system such that L coincides with the x axis. Then, for each point p = ( x , y ) of S , we construct the projected point q = ( x , 0). The set of all such q is the projection we want. With these definitions, we can state our problem in standard mathematical terms: Given: A convex polygon S Find: The orthographic projection of S on a line L This is a well-defined mathematical problem, but it is not a well-posed computational problem, because we have not specified how the polygon is to be “given”, and what is the format of the result. In other words, we have not specified how the input and output are represented . It is also interesting to note that the definitions of convex polygon and projection are mathematically correct but not computationally effective, in the sense that they cannot be directly embodied in algorithms. A convex hull is the smallest convex set that encloses the given points, and the projection is obtained by zeroing the y coordinate of every point of the polygon. Both of these are infinite processes that cannot be implemented directly in computers. We cannot compute all the enclosing convex sets to choose the smallest one, nor can we project and infinite set of points on a line. This does not mean we cannot compute projections of convex polygons. Although a polygon contains an infinite number of points, it can be represented by a finite number of vertices.
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GMCh6-1 3/29/99 6-2 Let us specify the input and output representations as follows. The convex polygon is represented in Scheme 2 of Chapter 3, i.e. , by a list of its vertex coordinates; the projection is represented by the x coordinates of its endpoints. This output representation is unambiguous because the orthographic projection of a convex polygon on a line is a line
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ch6-1 - GMCh6-1 6-1 GEOMETRIC MODELING A First Course...

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