Approximate Implicitization Using Monoid Curves and Surfaces

A least-squares solution for h and d is arrived at by replacing q with H in the steps of our approximate
implicitization method.
3.3 Error estimation
3.3.1 Estimation of the distance from a point to a monoid curve
Given a point P 1 (s 1 ;t 1 ;u 1 ), we nd point P 0 , other than T 0 , at which the monoid curve intersects the line going
through T 0 and P 1 (see Figure 7). In order to estimate the smallest distance between P 1 and the curve, we
consider the distance from P 1 to P 0 . It is obvious that the former is bounded by the latter.
T 0
T 2
P 0
P 1
T 1
I
Figure 7: Distance between P 1 and P 0 .
Assume that the barycentric coordinates of P 0 are (s 0 ;t 0 ;u 0 ) and that line T 0 P 1 intersects line T 1 T 2 at
I(1 , ;;0). Hence
= t 1 =(1 , u 1 );
s 0 =(1, u 0 )(1 , ); t 0 =(1, u 0 ); u 0 = u 0 ;
s 1 =(1, u 1 )(1 , ); t 1 =(1, u 1 ); u 1 = u 1 :
Note that point P 0 is on the monoid curve. From the parametric representation (10),
(21)
u 0 = f b (1 , ;)=(f b (1 , ;) , f i (1 , ;)): (22)
The distance between P 1 and P 0 is :
kP 1 P 0 k =
k(s 1 , s 0 )T 0 T 1 +(t 1 , t 0 )T 0 T 2 = ju 1 , u 0 jk(1 , )T 0 T 1 + T 0 T 2 k
f
= ju 1 , b (1,;)
f b (1,;),f i(1,;)
jk(1 , )T 0 T 1 + T 0 T 2 k:
(23)
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