Graphics Reference
In-Depth Information
Y
R
y
t
x
X
Fig. 9.1.
The circle can be drawn by tracing out a series of points on the circumference.
By varying the parameter
t
over the range 0 to 2
π
we trace out the curve of
the circumference. In fact, by selecting a suitable range of
t
we can isolate
any portion of the circle.
9.2 The Ellipse
The equation for an ellipse is
x
2
R
2
maj
y
2
R
2
min
+
= 1
(9.3)
but its parametric form is
x
=
R
maj
cos(
t
)
y
=
R
min
sin(
t
)0
≤
t
≤
2
π
(9.4)
where
R
maj
and
R
min
are the major and minor radii respectively, as shown in
Figure 9.2.
In the previous chapter we saw how a Hermite curve could be developed
using cubic polynomials and tangent slope vectors. Equation (8.27) gave the
x
-and
y
-coordinates for a 2D curve, and there is no reason why it could
not be extended to give the
z
-coordinate for a 3D curve. The tangent slope
vectors would also have to be modified to form the end conditions in three
dimensions.
We will now examine a very useful parametric curve called a Bezier curve,
named after its inventor Pierre Bezier.
