Approximate Implicitization Using Monoid Curves and Surfaces

multiple point T 0 is chosen such that it lies on the \concave" side of the curve and on the other axis of
symmetry a 2 . Finally, the other vertices of the reference triangle, T 1 and T 2 , are found as the intersections
between the bounding box's base DC and the lines joining the corners A and B with T 0 .
T 0
T 2
|AB|
A
P 0
P 3
P 2
P 4
P1
T 1
C
D
B
a 1
a 2
Figure 6: Reference triangle determination.
2. Convert the parametric curve p(v) into barycentric form.
Let T 0 T 1 T 2 be the reference triangle with T 0 as its multiple point. Then arbitrary point P can be converted
into barycentric form
P =
x
y
= sT 1 + tT 2 + uT 0 = T 0 +[T 1 , T 0 ;T 2 , s
T 0 ] ; (13)
t
or 8 <
:
s
t
= [T 1 , T 0 ;T 2 , T 0 ] ,1 (P , T 0 );
u = 1 , s , t:
For a rational Bezier curve, the conversion can be completed simply by applying transformation (14) to
each control point P i . The converted curve isnow put in the form
p(v) =
0
@ s t
u
1 P n
A =
i=0
nP
i=0
! i
P i B n i (v)
(14)
! i B n i (v) ; v 2 [0; 1]: (15)
12