Approximate Implicitization Using Monoid Curves and Surfaces

Written in triangular Bernstein-Bezier form, it looks like this:
with
0
B
@
s
t
u
v
1
C
A =
P
P
i+j+k=n
i+j+k=n
,
w ijk Q n
ijk ijk i j k
w ijk
, n
ijk
the weights w ijk = c ijk0 , ic i,1;j;k;1 , jc i;j,1;k;1 , kc i;j;k,1;1
and the control points Q ijk =
0
B
@
i j k (40)
,ic i,1;j;k;1
1
C
A =w ijk:
c i;j;k;0
,jci;j,1;k;1
,kci;j;k,1;1
Similarly, the coecients c ijkl satisfy the \same-sign" condition if the weights w ijk are all non-negative orall
non-positive, and the weights at the three corners are nonzero.
4.2 Approximate implicitization and inversion map
We now consider approximate implicitization for a rational Bezier surface of degree n k with control points
P ij and weights ! ij :
p(; ) =
nP kP
i=0 j=0
nP
i=0 j=0
! ij P ij B n i ()Bk j ()
kP
! ij B n i ()Bk j () ; ; 2 [0; 1]: (41)
In general, the implicit degree of p(; ) is 2mk. An approximate monoid surface q() = 0 for this implicit
equation can be found using the following procedure.
1. Determine the reference tetrahedron for the monoid surface.
A method for nding the optimum tetrahedron that ensures maximum numerical stability and the best
accuracy remains to be discovered. As of now, we are able to propose only an intuitive way of constructing
the reference tetrahedron, a direct extension of the approach we used to determine the reference triangle
for a curve. Namely, for a given rational parametric Bezier patch, we compute an oriented bounding box
and nd the \concave" side of the patch. The multiple point is chosen on that side. The other three
vertices are determined on the base plane of the bounding box such that the tetrahedron contains the box.
Since the construction is similar to that for curves, we omit further details.
2. Convert the parametric surface p(; ) into barycentric form.
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