## FUNDAMENTAL PROBLEMS IN GEOMETRY

Geometry,
especially analytic geometry, offers an effective tool to model graphics
objects. The coordinate systems in analytic geometry facilitate numerical
representations of geometric models. This modeling scheme is fundamental in a
computer graphics system, because geometric entities, properties, and
transformations are directly related to the central problems in computer
graphics.

Complex
numbers and quaternions are highly structured algebraic systems. They have
direct geometric interpretations, which make them valuable tools for solving
certain problems in geometry and graphics.

Linear
algebra is an algebraic subject that further extends the systems of real numbers,
complex numbers, and quaternions to arbitrary dimensional vector spaces and
systematically studies the properties and relationships on vectors. Matrices
provide concrete representations for linear transformations and vectors. They
can be directly implemented in computers to represent geometric transforms
relevant to computer graphics.

Calculus
may be applied to solve problems such as tangent lines and surface normal.
These topics are related to certain modeling and rendering problems in
graphics. Graph theory studies a discrete structure called a graph that has a
wide range of applications in computer science and graphics.