Bezier surface in triangular patches
de Castlejau Algorithm of triangular Bezier surfaces
Degree raising in triangular patches
Derivative of a triangular Bezier surface
Subdivision of a triangular Bezier surface
Bezier surfaces in triangular patches
The idea behind the Bezier surface in triangular patches
is to polarize the variables u and v separately. This way if F is a
bipolynomial surface of degree , we get a (p+q) multiaffine map which
is symmetric separately in its first p arguments and in its last q arguments,
but not symmetric in all its arguments. We get what is traditionally
called a tensor product Bezier surface. The advantage of this method
is that it allows the use of many algorithms applying in the case of
curves. Note that in this case, since the surface F is really a map
However, since is isomorphic to , we can view F as a polynomial surface
We can also polarize F in a different way: treating the coordinates
of a point (u,v) in , and to polarize the polynomials in both variables
simultaneously. This way, if m is such that F is a polynomial surface
of total degree m, we get an mmultiaffine map which is symmetric in
all of its m arguments. In some sense, this method is the immediate
generalization of Bezier curves. Indeed, since the surface F is indeed
a map The resulting surface in this case is traditionally called a triangular
Bezier surface (or simply Bezier triangles).
Bernstein basis: Following the example, we are going to use the generalized
Bernstein polynomials as a basis to define We call X(u) the parametric
representation of a triangular Bezier surface of degree n. The coefficients
in (**) are called Bezier points. They form the Bezier net or Bezier
polyhedron associated with the surface. The coefficients can be vectorvalued
(in or ), or they can be real numbers ,which we then refer to as Bezier
ordinates. In the latter case, the triangular Bezier surface is actually
a function defined on the base triangle. As in the case of Bezier curves,
we associate Bezier ordinates with points I/n on the base triangle according
to the following scheme: with Bezier ordinates we associate the parameter
values
de Castlejau algorithm of triangular Bezier surfaces
The de Castlejau Recurrence Relation The de Castlejau algorithm can
also be extended to triangular Bezier surfaces. The classical de Castlejau
algorithm is based on the following decomposition of Bernstein polynomials:
Now since i+j+k=n, we have and it follows analogous to (*) that or equivalently
In view of the basic idea of the algorithm that the desired surface
point X(u) can be found stepwise by computing the Bernstein polynomials
of lower degree, we can write where are triangular Bezier patches of
degree r, and are generalized Bernstein polynomials of degree nr. On
the other hand, from (**) we have Comparing coefficients leads to the
de Castlejau recursion relation
Geometric Interpretation In the recurrence relation, we set for r=0,
and the step r=n1 produces The recurrence formula asserts that geometrically,
the new coefficient is computed as an average of the three coefficients
, i=1,2,3, using the barycentric coordinates (u,v,w) with respect to
the upright triangle
Degree Raising for triangular Bezier surfaces
Degree Raising Rule Suppose are the Bezier points of a triangular Bezier
surface of degree n, and are the Bezier points of the same surface after
transforming to a basis of degree n+1. Then with whenever any of the
i,j,k is smaller than 0 or greater than n. Equivalently, we can write
where To see why this is true please click here. Remark: 1. This says
that can be obtained by computing the values at (i/(n+1)), (i/(n+1)),
(i/(n+1)) of the linear function interpolating the values 2. On the
boundary of the net, the degree raising formula reduces to one for curves.
The figure illustrates the construction of new Bezier points by degree
raising from n=2 to n=3.
Derivative of triangular Bezier surfaces
First Order Directional Derivatives: It will be convenient to work
with the operator notation with and Since u+v+w=1, the first directional
derivative is given by where we have used w=1uv. Alternately, we could
have eliminated v=1uw, in which case the first directional derivative
becomes Similarly, we also have The figure gives a geometric interpretation
of the directional derivatives along the boundary curve correspoding
to constant u. At the corner (0,0,1) of the triangular patch, the directional
derivatives are i.e. the tangent plane at a vertex is determined by
the Bezier point at the corner and its two neighbors.
Subdivision of a triangular Bezier surface 
As for tensorproduct surfaces, the de Castlejau algorithm can also
be used to subdivide a triangular Bezier surface into three spline patches
touching at a given point (u0, v0, w0). The figure shows the boundaries
of the three new nets in the case n=3. The Bezier points of the three
new surface patches are again boundary points of the de Castlejau scheme
(which we now organize as a tetrahedron). Patch I has Bezier points
Patch II has Bezier points Patch III has Bezier points where r+j+k=n.
By combining several steps of the de Castlejau algorithm,we can subdivide
a traingular Bezier patch into an arbitrary number of subtriangles of
degree n. The sequence of piecewise linear surfaces interpolating the
Bezier nets converges to the Bezier surface.
