Iteration of Involutes of Constant Width Curves in the Minkowski Plane

In this paper we study properties of the area evolute (AE) and the center symmetry set (CSS) of a convex planar curve $\gamma$. The main tool is to define a Minkowski plane where $\gamma$ becomes a constant width curve. In this Minkowski plane, the CSS is the evolute of $\gamma$ 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 $\gamma$.Comment: 16 pages, 4 figure

Discrete Integrable Systems and Hodograph Transformations Arising from Motions of Discrete Plane Curves

We consider integrable discretizations of some soliton equations associated with the motions of plane curves: the Wadati-Konno-Ichikawa elastic beam equation, the complex Dym equation, and the short pulse equation. They are related to the modified KdV or the sine-Gordon equations by the hodograph transformations. Based on the observation that the hodograph transformations are regarded as the Euler-Lagrange transformations of the curve motions, we construct the discrete analogues of the hodograph transformations, which yield integrable discretizations of those soliton equations.Comment: 19 page

Involutes of Polygons of Constant Width in Minkowski Planes

Consider a convex polygon P in the plane, and denote by U a homothetical copy of the vector sum of P and (-P). Then the polygon U, as unit ball, induces a norm such that, with respect to this norm, P has constant Minkowskian width. We define notions like Minkowskian curvature, evolutes and involutes for polygons of constant U-width, and we prove that many properties of the smooth case, which is already completely studied, are preserved. The iteration of involutes generates a pair of sequences of polygons of constant width with respect to the Minkowski norm and its dual norm, respectively. We prove that these sequences are converging to symmetric polygons with the same center, which can be regarded as a central point of the polygon P.Comment: 20 pages, 11 figure

Introducing symplectic billiards

In this article we introduce a simple dynamical system called symplectic billiards. As opposed to usual/Birkhoff billiards, where length is the generating function, for symplectic billiards symplectic area is the generating function. We explore basic properties and exhibit several similarities, but also differences of symplectic billiards to Birkhoff billiards.Comment: 41 pages, 16 figure

Multidimensional Inverse and Ill-Posed Problems for Differential Equations

This monograph is devoted to statements of multidimensional inverse problems, in particular to methods of their investigation. Questions of the uniqueness of solution, solvability and stability are studied. Methods to construct a solution are given and, in certain cases, inversion formulas are given as well. Concrete applications of the theory developed here are also given. Where possible, the author has stopped to consider the method of investigation of the problems, thereby sometimes losing generality and quantity of the problems, which can be examined by such a method. The book should be of interest to researchers in the field of applied mathematics, geophysics and mathematical biology

Compressible Flow and Euler's Equations

We consider the classical compressible Euler's Equations in three space dimensions with an arbitrary equation of state, and whose initial data corresponds to a constant state outside a sphere. Under suitable restriction on the size of the initial departure from the constant state, we establish theorems which give a complete description of the maximal development. In particular, the boundary of the domain of the maximal solution contains a singular part where the inverse density of the wave fronts vanishes and the shocks form. We obtain a detailed description of the geometry of this singular boundary and a detailed analysis of the behavior of the solution there.Comment: 505 page

On Holditch's theorem and related kinematics

El teorema de Holditch es un resultado clásico sobre áreas de curvas planas generadas por el movimiento de segmentos. Esta construcción está estrechamente relacionada con otros tipos de curvas como, por ejemplo, curvas paralelas, curvas de anchura constante o curvas de bicicletas. Se compilan las propiedades básicas de este tipo de curvas y se da una revisión histórica sobre el teorema de Holditch y teoremas relacionados de cinemática. Primero, la situación plana de Holditch se define rigurosamente y se consideran ciertos problemas como la existencia de dicha construcción o el modo de evitar movimientos retrógrados en el segmento que se va moviendo. En el enunciado del teorema de Holditch aparece el área de una elipse oculta. En este trabajo se utiliza una aproximación poligonal a dicho teorema para mostrar geométricamente de dónde proviene esta elipse. También es posible dar generalizaciones inmediatas de este teorema y de otros relacionados con la cinemática en otros contextos y situaciones. Así, en la segunda parte, se da una introducción a la geometría no euclidiana y se presenta la extensión de los resultados mencionados a superficies de curvatura constante. Además, se encuentran curvas cerradas ocultas en la superficie de curvatura constante relacionadas con el enunciado generalizado de Holditch. Finalmente, se obtiene una nueva extensión natural del teorema de Holditch para curvas espaciales, lo cual nos lleva al concepto de superficie de Holditch.Holditch's theorem is a classical result on areas of planar curves generated by moving chords. The construction is closely related to other kinds of curves such as parallel curves, constant width curves or bicycle curves. The basic properties of these curves are compiled and a historical review on Holditch's theorem and related theorems in kinematics is given. First, the Holditch planar setting is rigorously defined and problems such as the existence of that construction or the avoidance of retrograde movements of the moving chord are considered. In the statement of Holditch's theorem, the area of a hidden ellipse appears. A polygonal approach to the theorem is used to show geometrically where this ellipse comes from. Moreover, immediate generalizations of Holditch's theorem and related results to other contexts are possible. So, in the second part, an introduction to non-Euclidean geometry is given and the extension of such results to constant curvature surfaces is presented. In addition, hidden closed curves in the constant curvature manifold related to the generalized statement of Holditch's theorem are found. Finally, a new extension of Holditch's theorem to space curves is derived in a natural way leading to the concept of Holditch surface

Minkowski isoperimetric-hodograph curves

International audienceGeneral offset curves are treated in the context of Minkowski geometry, the geometry of the two-dimensional plane, stemming from the consideration of a strictly convex, centrally symmetric given curve as its unit circle. Minkowski geometry permits us to move beyond classical confines and provides us with a framework in which to generalize the notion of Pythagorean-hodograph curves in the case of rational general offsets, namely, Minkowski isoperimetric-hodograph curves. Differential geometric topics in the Minkowski plane, including the notion of normality, Frenet frame, Serret–Frenet equations, involutes and evolutes are introduced. These lead to an elegant process from which an explicit parametric representation of the general offset curves is derived. Using the duality between indicatrix and isoperimetrix and between involutes and evolutes, rational curves with rational general offsets are characterized. The dual Bézier notion is invoked to characterize the control structure of Minkowski isoperimetric-hodograph curves. This characterization empowers the constructive process of freeform curve design involving offsetting techniques