Page 170 - Geometric Modeling and Algebraic Geometry

172 S. Chau et al.
Boundary points.

Such points correspond to intersections of the boundary parameter lines of one sur-
face with the other one. In order to compute them, we apply the algorithm for inter-
secting parameter lines with a biquadratic patch to the 2·4 boundary parameter lines
of the two surfaces.

Turning points.

We consider the turning points of x(u, v) in respect to u. Let y r and y s denote the
partial derivatives of y(r, s). Several possibilities for computing the turning points
exist.
1. The two surfaces x(u 0 ,v) and y(r, s) intersect, x(u 0 ,v)= y(r, s), and the
tangent vector of the parameter line lies in the tangent plane of the second patch,
x u · (y r × y s )=0. (9.22)

These conditions lead to a system of four polynomial equations for four un-
knowns, which has to be solved for u.
2. By using the previous geometric result, we may eliminate the variable v,asfol-
lows. First, the plane spanned by the parameter line has to contain the point
y(r, s),
π(u 0 )(y(r, s)) = 0, (9.23)

which gives an equation of degree (6, 2, 2) in (u 0 ,r,s). Second, the cylinder has
to contain the point,
ζ(u 0 )(y(r, s)) = 0, (9.24)
which leads to an equation of degree (16, 4, 4). Finally, the tangent vector of the
parameter line has to be contained in the tangent plane of the second patch. Since
the tangent of the parameter line is parallel to the cross product of the gradient of
the plane and the gradient of the cylinder, the third condition gives an equation
of degree (18, 5, 5),

det [y r , y s , ∇π(u 0 )(y(r, s)) ×∇ζ(u 0 )(y(r, s))] = 0. (9.25)
For solving either of these two systems of polynomial equations, we use again a
B´ ezier–clipping–type algorithm [14, 25, 28].

9.6 Self–intersections of biquadratic surface patches

In order to detect the self–intersection curves of any of the two patches, the methods
for surface–surface intersections have to be modified. The computation of the self–
intersection locus by using approximate implicitization is not discussed here, since
it was already treated in [31]. Instead we focus on the other two techniques.