Page 93 - Geometric Modeling and Algebraic Geometry

5 Canal Surfaces Defined by Quadratic Families of Spheres 91
We have deg Γ(C)=8 and one double hyperboloid

{x =0, b − a 2 z − 2 b rz + a + b 2 r + a − a y =0}.
4
2 2
2
2
2
2
2
[1] 2
The case H corresponds to the hyperbola C(t)=(0, 0, 2at/(1 − t ),a(1 +
+−
t) /(1 − t )), and deg Γ(C)=6. Here the two families of lines are lying on the
2
2
hyperboloid {z = r, x + y − z = a /2} which is not double. This is caused by
2
2
2
2
the following fact: natural projections from C to all of these lines are 1–1 but not 2–1
as in all previous cases, since one asymptote of C is isotropic.
5.7 Conclusions
Quadratic canal surfaces are natural generalizations of Dupin cyclides with the po-
tential applications in geometric modeling, since
• they have a relatively simple rational parametrization of bi-degree (3, 2) or (4, 2);
• their B´ ezier representation is invariant with respect to Laguerre transformations
(in particular, offsets have B´ ezier representations of the same bi-degree);
• their implicit degree is 6 or 8;
• they are tangent with families of circular cones and cylinders (also along non-
circular curves).
In Fig. 5.9 we see three blendings between natural quadrics . The first two use (4, 2)-
patches of a quadratic canal surface of type E ++ for blending a cone and a cylinder.
The third uses the biangle patch of bi-degree (6, 2) of the same surface as fixed radius
rolling ball blend of two cylinders with a common inscribed sphere.

Fig. 5.9. Blendings with patches of canal surfaces of type E ++.

References

1. Degen, W., Cyclides, in Handbook of Computer Aided Geometric Design, 2002, p.575–
601