Graphics Reference
In-Depth Information
Y
P
y
Z
x
X
z
Fig. 5.6.
A right-handed axial system showing the coordinates of a point P.
5.2.3 Euler's Rule
In 1619, Descartes discovered quite a nice relationship between vertices, edges
and the faces of a 3D polygonal object:
faces
+
vertices
=
edges
+ 2
(5.4)
As a simple test, consider a cube; it has 12 edges, 6 faces and 8 vertices, which
satisfies this equation. This rule can be applied to a geometric database to dis-
cover whether it contains any spurious features. Unfortunately for Descartes,
for some unknown reason, the rule is named after Euler!
5.3 Summary
The Cartesian plane and its associated coordinates are the basis for all math-
ematics used for computer graphics. We will see in following chapters how
shapes can be manipulated using simple functions, and how the plane can be
extended into a 3D Cartesian space that becomes the domain for creating ob-
jects, curves, surfaces, and a virtual environment where they can be animated
and visualized.
