Even Dimensional Improper Affine Spheres

There are exactly two different types of bi-dimensional improper affine spheres: the non-convex ones can be modeled by the center-chord transform of a pair of planar curves while the convex ones can be modeled by a holomorphic map. In this paper, we show that both constructions can be generalized to arbitrary even dimensions: the former class corresponds to the center-chord transform of a pair of Lagrangian submanifolds while the latter is related to special K\"ahler manifolds. Furthermore, we show that the improper affine spheres obtained in this way are solutions of certain exterior differential systems. Finally, we also discuss the problem of realization of simple stable Legendrian singularities as singularities of these improper affine spheres.Comment: 26 page

Discretization of asymptotic line parametrizations using hyperboloid patches

Two-dimensional affine A-nets in 3-space are quadrilateral meshes that discretize surfaces parametrized along asymptotic lines. The characterizing property of A-nets is planarity of vertex stars, so for generic A-nets the elementary quadrilaterals are skew. We classify the simply connected affine A-nets that can be extended to continuously differentiable surfaces by gluing hyperboloid surface patches into the skew quadrilaterals. The resulting surfaces are called "hyperbolic nets" and are a novel piecewise smooth discretization of surfaces parametrized along asymptotic lines. It turns out that a simply connected affine A-net has to satisfy one combinatorial and one geometric condition to be extendable - all vertices have to be of even degree and all quadrilateral strips have to be "equi-twisted". Furthermore, if an A-net can be extended to a hyperbolic net, then there exists a 1-parameter family of such C^1-surfaces. It is briefly explained how the generation of hyperbolic nets can be implemented on a computer. The article uses the projective model of Pluecker geometry to describe A-nets and hyperboloids.Comment: 27 pages, 17 figure

A Compact Representation of Drawing Movements with Sequences of Parabolic Primitives

Some studies suggest that complex arm movements in humans and monkeys may optimize several objective functions, while others claim that arm movements satisfy geometric constraints and are composed of elementary components. However, the ability to unify different constraints has remained an open question. The criterion for a maximally smooth (minimizing jerk) motion is satisfied for parabolic trajectories having constant equi-affine speed, which thus comply with the geometric constraint known as the two-thirds power law. Here we empirically test the hypothesis that parabolic segments provide a compact representation of spontaneous drawing movements. Monkey scribblings performed during a period of practice were recorded. Practiced hand paths could be approximated well by relatively long parabolic segments. Following practice, the orientations and spatial locations of the fitted parabolic segments could be drawn from only 2–4 clusters, and there was less discrepancy between the fitted parabolic segments and the executed paths. This enabled us to show that well-practiced spontaneous scribbling movements can be represented as sequences (“words”) of a small number of elementary parabolic primitives (“letters”). A movement primitive can be defined as a movement entity that cannot be intentionally stopped before its completion. We found that in a well-trained monkey a movement was usually decelerated after receiving a reward, but it stopped only after the completion of a sequence composed of several parabolic segments. Piece-wise parabolic segments can be generated by applying affine geometric transformations to a single parabolic template. Thus, complex movements might be constructed by applying sequences of suitable geometric transformations to a few templates. Our findings therefore suggest that the motor system aims at achieving more parsimonious internal representations through practice, that parabolas serve as geometric primitives and that non-Euclidean variables are employed in internal movement representations (due to the special role of parabolas in equi-affine geometry)

Alpha-expansions: a class of frame decompositions

AbstractThis article analyzes a scheme for frame decompositions that is called α-expansion. In this scheme, the choice of a parameter α adequate to a given frame is a central point. We develop a theory that helps choosing the parameter α and also suggests algorithms for obtaining the α-expansions. The method is applied to frame expansions coding. In this context we give conditions under which, for high rate coding, α-expansions are better, in a rate × distortion sense, than schemes that find the frame coefficients first and quantize them in a second step