Graphics Reference
In-Depth Information
12.12.3
Triangular Bézier Surfaces
In the last section we laid the groundwork for triangular blossoms. This section will
show how one works with them. Using them for surface parameterizations means that
such parameterizations are based on triangular domains. Triangles are in a sense more
natural for surfaces because a triangular grid of points on a surface gives a truly linear
approximation of a surface, whereas a rectangular grid may very well have rectangles
whose vertices do not lie in a plane so that they would not be planar.
First of all, a triangular control net { b i,j,k } is usually displayed in a triangular form
(even though the points themselves can be arbitrary points of R 3 ). For example, see
the right side of Figure 12.19 for the case when d = 3. (That representation is not uni-
versal however because some authors show a rotated version of this triangle.) If we
let r = (1,0), s = (0,1), and t = (0,0) in Theorem 12.12.2.5, then according to that
theorem the points b i,j,k define a surface p(u,v). Again looking at Figure 12.19 note
that when we represent all points in the triangle rst by means of barycentric coordi-
nates there is a natural correspondence between the point b i,j,k and the point in the
triangle rst with barycentric coordinates (i/d,j/d,k/d). Note also how the ith to the right
slanting column of the b i,j,k array all have the same index i. Such a diagram general-
izes in the obvious way to triangular b i,j,k arrays for other values of d. With this rep-
resentation it is now easy to explain how the de Casteljau algorithm works in this
situation. Consider the case d = 3 again. See Figure 12.20. To simplify the notation
we have dropped the superscripts, so that the points b i,j, m can be recognized by the
fact that their indices add up to d - m. The start points b i,j, 0 = b i,j,k are the points
marked with solid circles along the outside of the region along with the one center
point b 1,1,1 . The points b i,j,k
1
are the points marked as circles. For example,
1
b
=
a
b
+
b
b
+
c
b
,, .
102
,,
012
,,
003
002
,,
2
The points b i,j,k
are the points marked with a cross. For example,
2
b
=
a
b
+
b
b
+
c
b
,, .
101
,,
011
,,
00 2
001
,,
Figure 12.19.
The standard triangular net versus an arbitrary one for d = 3.
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