Approximate Implicitization Using Monoid Curves and Surfaces

form:
with
0
@ s t
u
1 P n
A =
i=0
nP
i=0
w i Q i B n i ()
the weights w i = c n,i;i;0 , ic n,i;i,1;1 , (n , i)c n,1,i;i;1
and the control points Q i =
w i B n i () (10)
0
@ ,(n , 1
i)c n,1,i;i;1
,ic n,i;i,1;1
c n,i;i;0
A =wi :
If the weights are all non-negative or all non-positive and w 0 w n 6=0,wesay that the coecients c i;j;k satisfy
the \same-sign" condition. Under this condition, the monoid curve has many useful properties, one of which is
convex hull property.
Finally, it follows from the above derivation that the inversion map of the monoid curve is a linear fractional
transformation
3.2 Approximate implicitization and inversion map
= t=(1 , u): (11)
Given a planar degree n rational Bezier curve
p(v) =
nP
i=0
nP
i=0
! i P i B n i (v)
! i B n i (v) ; v 2 [0; 1]; (12)
where P i are the control points, ! i the weights and Bi n(t) =, n
i
(1 , t) n,i t i ,wewant to nd a monoid curve
q() = 0 to approximate the parametric curve. The process of approximate implicitization involves the following
steps.
1. Determine the reference triangle for the monoid curve.
To dene an algebraic curve, an arbitrary triangle can be used as the reference triangle. Dokken showed
that the numerical stability of the approximation process benets from choosing a bounding triangle as the
reference triangle. However, for a monoid curve one vertex of the reference triangle must be the multiple
point ifwe express the curve as in (6). When a monoid curve is used to approximate a parametric curve
segment, the location of the multiple point has a signicant impact on the accuracy of the approximation.
As an example, Figure 5 demonstrates the relationship between the maximum approximation error and
the location of the multiple point for a cubic Bezier curve.
10