Graphics Reference
In-Depth Information
Table 9.6.
The x, y, z coordinates for different
values of u and v
v
1
2
0
1
1
1
4
,
0
3
0
(0, 0, 0)
2
,
(3,0,0)
0
,
3
2
1
1
2
3
,
3
2
1
2
4
,
1
1
2
,
1
1
2
,
1
1
4
,
1
1
u
1
1
4
,
3
3
1
(0, 0, 3)
2
,
(3,0,3)
Therefore, any point on the surface patch has coordinates
u
2
v
2
)
,z
uv
=3
u
x
uv
=3
v, y
uv
=3(
u
+
v
−
−
Table 9.6 shows the coordinate values for different values of
u
and
v
.Inthis
example, the
y
-coordinates provide the surface curvature, which could be
enhanced by modifying the
y
-coordinates of the control points.
Complex 3D surfaces are readily modelled using Bezier patches. One sim-
ply creates a mesh of patches such that their control points are shared at the
joins. Surface continuity is controlled using the same mechanism for curves.
But where the slopes of trailing and starting control edges apply for curves,
the corresponding slopes of control tiles apply for patches.
9.8 Summary
This chapter has been the most challenging one to write. On the one hand,
the subject is vital to every aspect of computer graphics, but on the other,
the reader is required to wrestle with cubic polynomials and a little calculus.
However, I do hope that I have managed to communicate some essential con-
cepts behind curves and surfaces, and that you will be tempted to implement
some of the mathematics.
