Computer Graphics and Geometric Modeling

Approaches to Geometric Modeling (Basic Computer Graphics) Part 2

Early Representation Schemes Approaches to geometric modeling have changed over the years. These changes began before computers existed and all one had was pencil and paper. Since the advent of computers, these changes were largely influenced by their power, the essential mathematics behind the changes being basically not new. As computers become more and more […]

Approaches to Geometric Modeling (Basic Computer Graphics) Part 3

Euler Operations Representation schemes based on using Euler operations to build objects are an attempt to have a boundary representation meet at least part of the validity issue head on. The idea is to permit only boundary representations that have a valid Euler characteristic. If we only allow operations that preserve the Euler characteristic or […]

Approaches to Geometric Modeling (Basic Computer Graphics) Part 4

Parametric Representations Many of the representations of solids rest on a representation of their boundaries. That was true even in the case of the csg-rep. Although the primitives were solids, in practice one only had equations or parameterizations for their surfaces, and the interior of the solid was not referenced explicitly. As far as parameterizations […]

Approaches to Geometric Modeling (Basic Computer Graphics) Part 5

Modeling Natural Phenomena Except for the pixel- and voxel-based types, the representation schemes we have discussed so far are not very useful for modeling natural phenomena. Objects such as trees, mountains, grass, or various terrain cannot easily be modeled by linear poly-hedra or smooth surface patches. Using very small pieces in the representation would overwhelm […]

Approaches to Geometric Modeling (Basic Computer Graphics) Part 6

Functions and Algorithms In addition to modeling objects, modelers must also be able to perform a variety of operations on the objects that they have modeled. As indicated in the introduction, modeling involves modeling maps as well as objects. Here is a sample of some queries users may want to make and actions they may […]

Approaches to Geometric Modeling (Basic Computer Graphics) Part 7

Converting Between Representations The need for algorithms that convert from one representation to another exists not only because modelers using different representations may want to exchange geometric data but especially because modelers increasingly seem to maintain multiple representations internally. By in large, the problem seems to have only been dealt with in an ad hoc […]

Approaches to Geometric Modeling (Basic Computer Graphics) Part 8

Round-off Error and Robustness Issues Accuracy and robustness are important issues in numerical computations. For an overview of some common pitfalls see [McCa98]. Because geometric modeling involves a great many numerical algorithms, one wants to minimize the potential of round-off errors and avoid situations where small changes can lead to radically different results. To accomplish […]

Basic Geometric Modeling Tools (Basic Computer Graphics) Part 1

Introduction This topic describes some often-used mathematical tools and formulas in geometric modeling. The author highly recommends the Graphics Gems series of topics to the reader (see the "Miscellaneous" section of the Bibliography). These topics contain many insights into how one can make computations and algorithms more efficient. We begin by discussing bounding objects, such […]

Basic Geometric Modeling Tools (Basic Computer Graphics) Part 2

Surrounding Tests Along with finding intersections, determining whether or not a point belongs to a two-or three-dimensional region is another common task. This section looks at some simple tests to answer the question “Does the point p belong to the linear polyhedron Q?” We call them “surrounding” tests because the question could also be thought […]

Basic Geometric Modeling Tools (Basic Computer Graphics) Part 3

Distance Formulas The next two sections describe a number of formulas that are handy for applications. Formula. Let L be a line defined by a point Q and direction vector v and let P be a point. The point Figure 6.14. Computing the distance from a point to a line. is the unique point of […]

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