4 papers
Confocal families of plane algebraic curves
Ragni Piene, Boris Shapiro
We study families of plane algebraic curves sharing the same set of foci. We reformulate confocality via a focal map on equiclassical families and analyze its fibers using deformat…
Envelopes and evolutes
Ragni Piene
The study of evolutes of plane curves goes back at least to Huygens, and was continued and extended to space curves by Monge, Darboux, and others. Salmon studied projective curves…
On fundamental forms and osculating bundles
Raquel Mallavibarrena, Ragni Piene
We define higher order fundamental forms and osculating spaces of projective algebraic varieties, using sheaves of principal parts. We show that the th fundamental form can be v…
Adjoints and Canonical Forms of Polypols
Kathlén Kohn, Ragni Piene, Kristian Ranestad +5
Polypols are natural generalizations of polytopes, with boundaries given by nonlinear algebraic hypersurfaces. We describe polypols in the plane and in 3-space that admit a unique…