Graphics Reference
In-Depth Information
10
Analytic Geometry
This chapter explores some basic elements of geometry and analytic geome-
try that are frequently encountered in computer graphics. For completeness,
I have included a short review of important elements of Euclidean geome-
try with which you should be familiar. Perhaps the most important topics
that you should try to understand concern the definitions of straight lines in
space, 3D planes, and how points of intersection are computed. Another useful
topic is the role of parameters in describing lines and line segments, and their
intersection.
10.1 Review of Geometry
In the 3rd century
Euclid laid the foundations of geometry that have been
taught in schools for centuries. In the 19th century, mathematicians such as
Bernhard Riemann (1809-1900) and Nicolai Lobachevsky (1793-1856) trans-
formed this traditional Euclidean geometry with ideas such as curved space
and spaces with higher dimensions. Although none of these developments
affect computer graphics, they do place Euclid's theorems in a specific con-
text: a set of axioms that apply to flat surfaces. We have probably all been
taught that parallel lines don't meet, and that the internal angles of a triangle
sum to 180
◦
, but these things are only true in specific situations. As soon as
the surface or space becomes curved, such rules break down. So let's review
some rules and observations that apply to shapes drawn on a flat surface.
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