Approximate Implicitization Using Monoid Curves and Surfaces

2 Approximate implicitization method
Dokken developed a general method to approximately implicitize g-manifolds represented by parametric equations
[11]. Here, we outline his approach.
Let p(s) be a parametric || say, Bezier || curve or surface. We would like to nd an approximate
low-degree implicit equation q(x) =0forp(s). In general, the distance between the parametric curve or surface
and its implicit approximation depends not only on the value of q p(s), but also on the gradient of q(x).
However, in practice, the value of jq p(s)j alone can serve as a measure of the goodness of an approximation,
provided the coecients of q are normalized. This reduces the implicitization problem to nding a degree m
algebraic hypersurface q(x) =0with normalized coecients such that jq p(s)j, where is a prescribed
tolerance.
Let S be an arbitrary simplex containing p(s), and q(x) = P I b I m!
I! xI be represented in barycentric coordinates
x over S. Then the composition of q(x) = 0 and p(s) can be expressed as
q p(s) =(Db) T (s); (1)
where D is a constant matrix, called composite matrix, b is a vector consisting of the coecients of q, and (s)
is a vector whose components are basis functions of the composition.
If the components of (s) form a unit partition for s 2 , the following properties hold:
Property 1. max
s2
jq p(s)jkDbk 2
Property 2.
min
kbk 2=1
kDbk 2 = 1 , where 1 0 is the smallest singular value of D.
Thus we can obtain q(x) by setting b to the eigenvector of D D that corresponds to the eigenvalue 2 1 . This
will ensure q(x) satises jq p(s)j when 1