In this paper we study properties of the area evolute (AE) and the center symmetry set (CSS) of a convex planar curve γ. The main tool is to define a Minkowski plane where γ becomes a constant width curve. In this Minkowski plane, the CSS is the evolute of γ and the AE is an involute of the CSS. We prove that the AE is contained in the region bounded by the CSS and has smaller signed area.The iteration of involutes generate a pair of sequences of constant width curves with respect to the Minkowski metric and its dual, respectively. We prove that these sequences are converging to symmetric curves with the same center, which can be regarded as a central point of the curve γ.Mathematics Subject Classification (2010). 53A15, 53A40.
Abstract. The area distance to a convex plane curve is an important concept in computer vision. In this paper we describe a strong link between area distances and improper affine spheres. Based on this link, we propose an extremely fast algorithm to compute the inner area distance. Moreover, the concepts of the theory of affine spheres lead to a new definition of an area distance on the outer part of a convex plane curve. On the other hand, area distances provide a good geometrical understanding of improper affine spheres.
There are exactly two different types of bi-dimensional improper affine spheres: the non-convex ones can be modeled by the centerchord 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 Kähler 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.Mathematics Subject Classification (2010). 53A15, 53D12.
Asymptotic net is an important concept in discrete differential geometry. In this paper, we show that we can associate affine discrete geometric concepts to an arbitrary non-degenerate asymptotic net. These concepts include discrete affine area, mean curvature, normal and co-normal vector fields and cubic form, and they are related by structural and compatibility equations. We consider also the particular cases of affine minimal surfaces and affine spheres.The expansion of computer graphics and applications in mathematical physics have recently given a great impulse to discrete differential geometry. In this discrete context, surfaces with indefinite metric are generally modelled as asymptotic nets. ([2],[1]). In particular, there are several types of discrete affine surfaces with indefinite metric modelled as asymptotic nets: Affine spheres ([3]), improper affine spheres ([7], [6]) and minimal surfaces ([5]). In this work we define a general class of discrete affine surfaces that includes these types as particular cases.Beginning with an arbitrary asymptotic net, with the mild hypothesis of non-degeneracy, we define a discrete affine invariant structure on it. The discrete geometric concepts defined are affine metric, normal vector field, co-normal vector field, cubic form and mean curvature, and they are related by equations comparable with the correspondent smooth equations. The asymptotic net is called minimal when its mean curvature is zero, while the asymptotic net is an called an affine sphere when certain discrete derivatives described in section 4 vanishes. Asymptotic discretizations of the one-sheet hyperboloid are basic examples of discrete surfaces that satisfy our definition of affine spheres.The main characteristic of our definition of the discrete affine geometric concepts is that they are related by discrete equations that closely resembles the smooth equations of affine differential geometry of surfaces. The simplicity of these equations are somewhat surprising, since the construction is very general. Structural equations describe the discrete immersion in terms of the affine metric, cubic form and mean curvature, and these concepts must satisfy compatibility equations. When the discrete affine metric, cubic form and mean curvature satisfy the compatibility equations, one can define an asymptotic net, unique up to eqüi-affine transformations of R 3 , that satisfies the structural equations.The paper is organized as follows: Section 2 reviews the basic facts of smooth affine differential geometry with asymptotic parameters. Section 3 contains the main results: It shows how you can define the affine metric, normal and co-normal vector fields, cubic form and mean curvature from a given asymptotic net. It also relate this work with the discrete affine minimal surfaces of [5] and give an example of a "non-minimal" asymptotic net. Section 4 describes the structural equations and propose a new definition of affine spheres that is more general than the one proposed in [3]. Section 5 describes the compati...
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