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Étude des différences de corps convexes plans

Yves Martinez-Maure (1999)

Annales Polonici Mathematici

We characterize the linear space ℋ of differences of support functions of convex bodies of 𝔼² and we consider every h ∈ ℋ as the support function of a generalized hedgehog (a rectifiable closed curve having exactly one oriented support line in each direction). The mixed area (for plane convex bodies identified with their support functions) has a symmetric bilinear extension to ℋ which can be interpreted as a mixed area for generalized hedgehogs. We study generalized hedgehogs and we extend the...

Gielisova transformace logaritmické spirály

Luděk Spíchal (2020)

Pokroky matematiky, fyziky a astronomie

Logaritmická spirála byla od okamžiku svého objevu studována z mnoha různých pohledů. Prvotní fascinace matematiků, z nichž někteří věnovali logaritmické spirále značnou část svého tvůrčího potenciálu, se postupně přenesla do dalších oblastí nejen přírodních věd a promítá se tak např. do fyziky, biologie, ale také různých inženýrských disciplín či architektury. Článek ukazuje, že logaritmická spirála popisovaná jako hladká křivka s exponenciálně rostoucím poloměrem může být transformována do řady...

How to unify the total/local-length-constraints of the gradient flow for the bending energy of plane curves

Yuki Miyamoto, Takeyuki Nagasawa, Fumito Suto (2009)

Kybernetika

The gradient flow of bending energy for plane curve is studied. The flow is considered under two kinds of constraints; one is under the area and total-length constraints; the other is under the area and local-length constraints. The fundamental results (the local existence and uniqueness) were obtained independently by Kurihara and the second author for the former one; by Okabe for the later one. For the former one the global existence was shown for any smooth initial curves, but the asymptotic...

Indice d'un hérisson: étude et applications.

Yves Martínez-Maure (2000)

Publicacions Matemàtiques

Hedgehogs are a natural generalization of convex bodies of class C+2. After recalling some basic facts concerning this generalization, we use the notion of index to study differential and integral geometries of hedgehogs.As applications, we prove a particular case of the Tennis Ball Theorem and a property of normals to a plane convex body of constant width.

Integral formula for secantoptics and its application

Witold Mozgawa, Magdalena Skrzypiec (2012)

Annales UMCS, Mathematica

Some properties of secantoptics of ovals defined by Skrzypiec in 2008 were proved by Mozgawa and Skrzypiec in 2009. In this paper we generalize to this case results obtained by Cieślak, Miernowski and Mozgawa in 1996 and derive an integral formula for an annulus bounded by a given oval and its secantoptic. We describe the change of the area bounded by a secantoptic and find the differential equation for this function. We finish with some examples illustrating the above results.

Integral formulas related to wave fronts

Sergeĭ Anisov (1999)

Banach Center Publications

In the first section of the paper we study some properties of oriented volumes of wave fronts propagating in spaces of constant curvature. In the second section, we generalize to an arbitrary isometric action of a Lie group on a Riemannian manifold the following principle: an extra pression inside of a ball does not move it.

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