Why Is Computing with Geometry Important?
Up to now, perhaps most of your programs use only integers,
except for those you wrote in a 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
Engineers use computer-aided design systems for
designing something (e.g., 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.
Richard Hamming once said
The purpose of computing is insight, not numbers
Visualization explores 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 sent to
algorithms that 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