Why Is Computing with Geometry Important?

Up to now, perhaps most of your programs use only integers, except for those you wrote in the numerical methods course which use floating point numbers exclusively. Before you enter computer graphics, you perhaps would not know how to use these numbers (integers and/or reals) for representing geometric objects. But, why is handling geometric objects so important? At a first glance, the answer to this question is simple: it is because this world is full of geometric objects. For example, your car, desk, table, chair, CD-ROM and its player, and TV are all geometric objects. If this is not good enough, just take a look at the field of computer science to see how many branches do need the skill of handling geometric objects.

• Computer Graphics
  This does not require any further explanation. All characters in Toy Story are geometric objects.
• Computer-Aided Design
  Most engineering students use a system for designing something, say a piston engine or a model aircraft. These are, of course, geometric objects.
• Computational Geometry
  What is Computational Geometry? Is it identical to Computing with Geometry? Literally, they should be the same. However, computational geometers study the algorithmic aspect and complexity measure of geometric problems involving simple geometric objects such as points, lines, line segments, polylines and polygons, circles and circular arcs, spheres and polyhedra. Some classical problems are (1) computing the convex hull of a set of points, (2) determining the intersection points of a set of line segments, and (3) triangulating a simple polygon. While the focus is theoretical and algorithmic, the field of computational geometry does deal with geometric objects.
• Visualization
  Richard Hamming once said

  The purpose of computing is insight, not numbers

  Visualization is concerned with exploring data and information graphically - as a means of gaining understanding and insight into the data (R. A. Earnshaw and N. Wiseman, An Introductory Guide to Scientific Visualization, Springer-Verlag, 1992). As long as graphics is involved, geometric objects appear. A good example of visualization is medical scanning. An object is scanned producing many contours on parallel planes. How do we reconstruct the scanned object, including its interior, from the given contours?
• Computer Vision
  In computer vision and image processing, features are extracted from images taken by cameras and used for other purposes. For example, two cameras are mounted on a robot for taken stereo pictures. Features are extracted and feedback to algorithms which control the motion of the robot so that it will not collide with the surrounding environment. What are features? Points, line segments and circles are all features in an image. In many cases, the extracted features are used to reconstructed the scene.