Graphics Reference
In-Depth Information
9
Curves and Patches
In this chapter we investigate the foundations of curves and surface patches.
This is a very large and complex subject, and it is impossible for us to delve
too deeply. However, we can explore many of the ideas that are essential to
understanding the mathematics behind 2D and 3D curves and how they are
developed to produce surface patches. Once you have understood these ideas
you will be able to read more advanced texts and develop a wider knowledge.
In the previous chapter we saw how polynomials were used as interpolants
and blending functions. We will now see how these can form the basis of
parametric curves and patches. To begin with, let's start with the humble
circle.
9.1 The Circle
The circle has a very simple equation:
x
2
+
y
2
=
R
2
(9.1)
where
R
is the radius. Although this equation has its uses, it is not very
convenient for drawing the curve. What we really want are two functions that
determine the coordinates of any point on the circumference in terms of some
parameter. Figure 9.1 shows a scenario where the
x
-and
y
-coordinates are
given by
x
=
R
cos(
t
)
y
=
R
sin(
t
)0
≤
t
≤
2
π
(9.2)
