Banff International Research Station for Mathematical Innovation and Discovery
Abstract
Darboux and isotropic cyclides in RP3 are projections of intersections of certain pairs
of quadrics in RP4. Hence they are particular cases of real Del Pezzo surfaces (of degree 4
or less), that are are known to be rational.
Low-degree rational patches on these cyclides described in a form convenient for shape manipulations
are the most interesting for modeling applications.
Let AP1 be a projective line over two cases of Clifford algebras
A=Cl(R3),Cl(R2,0,1),
generated by euclidean space R3 and pseudo-euclidean space R2,0,1 with signature (++0).
Our approach is to treat AP1 as an ambient space and to consider toric Bezier patches
in the corresponding homogeneous coordinates.
It is proved that such patches of formal degree 2 with standard and non-standard real
structures cover almost all cases of real Darboux and isotropic cyclides.
The corresponding implicitization and parametrization algorithms are studied.
The applications are related to families of circles on surfaces
(including generation of 3-webs of circles) in case of Darboux cyclides and
blending of Pythagorean-normal surfaces in case of isotropic cyclides.Non UBCUnreviewedAuthor affiliation: Vilnius universityFacult