Suppose we are given three points, which we will now label with b and use three
subscripts
Think of them as in a triangle:
We've seen we can form the triangle determined by them as
Let's extend this to 6 points, conveniently written in a triangular array:
We can then form triangles from each of the 3 subtriangles
where
Finally, we can form a barycentric combination of these 3 points obtaining:
b0002 = u b1001 + v b0101 + w b0011.
This last point
is said to lie
on the Bezier triangle defined by the control net of original points.
Note that
are called the triangle numbers.
Given a control net
of
points bijk where i+j+k = n and barycentric
coordinates
(which means u+v+w=1), set
bijkr = u b(ijk)+(100)r-1 + v b(ijk)+(010)r-1 + w b(ijk)+(001)r-1
where
and
i+j+k = n-r and
bijk0 = bijk.
Then b000n is the point on the Bezier triangle with coordinates
.
Based on the de Casteljau algorithm we can conclude the following properties
hold for Bezier triangles
- Affine invariance
- Since linear interpolation is affine invariant and
the de Casteljau only uses linear interpolation.
- Invariance under affine parameter transformations
- Convex hull
- The Bezier triangle lies in the convex hull of its
control net.
We can extend the definition of Bernstein polynomials as follows
William Shoaff
1999-02-08