22,115 research outputs found
Darboux cyclides and webs from circles
Motivated by potential applications in architecture, we study Darboux
cyclides. These algebraic surfaces of order a most 4 are a superset of Dupin
cyclides and quadrics, and they carry up to six real families of circles.
Revisiting the classical approach to these surfaces based on the spherical
model of 3D Moebius geometry, we provide computational tools for the
identification of circle families on a given cyclide and for the direct design
of those. In particular, we show that certain triples of circle families may be
arranged as so-called hexagonal webs, and we provide a complete classification
of all possible hexagonal webs of circles on Darboux cyclides.Comment: 34 pages, 20 figure
Constructing reparametrization invariant metrics on spaces of plane curves
Metrics on shape space are used to describe deformations that take one shape
to another, and to determine a distance between them. We study a family of
metrics on the space of curves, that includes several recently proposed
metrics, for which the metrics are characterised by mappings into vector spaces
where geodesics can be easily computed. This family consists of Sobolev-type
Riemannian metrics of order one on the space of
parametrized plane curves and the quotient space of unparametrized curves. For the space of open
parametrized curves we find an explicit formula for the geodesic distance and
show that the sectional curvatures vanish on the space of parametrized and are
non-negative on the space of unparametrized open curves. For the metric, which
is induced by the "R-transform", we provide a numerical algorithm that computes
geodesics between unparameterised, closed curves, making use of a constrained
formulation that is implemented numerically using the RATTLE algorithm. We
illustrate the algorithm with some numerical tests that demonstrate it's
efficiency and robustness.Comment: 27 pages, 4 figures. Extended versio
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