Isoperimetric equalities for rosettes

In this paper, we study the isoperimetric-type equalities for rosettes, i.e. regular closed planar curves with nonvanishing curvature. We find the exact relations between the length and the oriented area of rosettes based on the oriented areas of the Wigner caustic, the Constant Width Measure Set and the Spherical Measure Set. We also study and find new results about the geometry of affine equidistants of rosettes and of the union of rosettes.

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Non-planar curves of constant width

Czechoslovak Mathematical Journal ◽

10.21136/cmj.1967.100801 ◽

1967 ◽

Vol 17 (4) ◽

pp. 540-549

Author(s):

Zbyněk Nádeník

Keyword(s):

Constant Width ◽

Planar Curves ◽

Curves Of Constant Width

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The geometry of the Wigner caustic and a decomposition of a curve into parallel arcs

Analysis and Mathematical Physics ◽

10.1007/s13324-021-00617-x ◽

2021 ◽

Vol 12 (1) ◽

Author(s):

Wojciech Domitrz ◽

Michał Zwierzyński

Keyword(s):

Rotation Number ◽

Optical Lattice ◽

Quantum Particle ◽

Singular Points ◽

Planar Curves ◽

Lattice Potential ◽

Global Properties ◽

Important Object ◽

Cusp Singularities ◽

Regular Closed

AbstractIn this paper we study global properties of the Wigner caustic of parameterized closed planar curves. We find new results on its geometry and singular points. In particular, we consider the Wigner caustic of rosettes, i.e. regular closed parameterized curves with non-vanishing curvature. We present a decomposition of a curve into parallel arcs to describe smooth branches of the Wigner caustic. By this construction we can find the number of smooth branches, the rotation number, the number of inflexion points and the parity of the number of cusp singularities of each branch. We also study the global properties of the Wigner caustic on shell (the branch of the Wigner caustic connecting two inflexion points of a curve). We apply our results to whorls—the important object to study the dynamics of a quantum particle in the optical lattice potential.

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Affine Subspaces of Curvature Functions from Closed Planar Curves

Results in Mathematics ◽

10.1007/s00025-021-01378-6 ◽

2021 ◽

Vol 76 (2) ◽

Author(s):

Leonardo Alese

Keyword(s):

Arc Length ◽

Planar Curves ◽

Affine Subspaces ◽

Planar Curve ◽

Real Functions ◽

Discrete Counterpart ◽

Equivalent Conditions

AbstractGiven a pair of real functions (k, f), we study the conditions they must satisfy for $$k+\lambda f$$ k + λ f to be the curvature in the arc-length of a closed planar curve for all real $$\lambda $$ λ . Several equivalent conditions are pointed out, certain periodic behaviours are shown as essential and a family of such pairs is explicitely constructed. The discrete counterpart of the problem is also studied.

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Clothoid fitting and geometric Hermite subdivision

Advances in Computational Mathematics ◽

10.1007/s10444-021-09876-5 ◽

2021 ◽

Vol 47 (4) ◽

Author(s):

Ulrich Reif ◽

Andreas Weinmann

Keyword(s):

Building Block ◽

Numerical Results ◽

Normal Vector ◽

Subdivision Schemes ◽

Planar Curves ◽

Nonlinear Subdivision ◽

New Strategy ◽

Vector Information ◽

Hermite Subdivision

AbstractWe consider geometric Hermite subdivision for planar curves, i.e., iteratively refining an input polygon with additional tangent or normal vector information sitting in the vertices. The building block for the (nonlinear) subdivision schemes we propose is based on clothoidal averaging, i.e., averaging w.r.t. locally interpolating clothoids, which are curves of linear curvature. To this end, we derive a new strategy to approximate Hermite interpolating clothoids. We employ the proposed approach to define the geometric Hermite analogues of the well-known Lane-Riesenfeld and four-point schemes. We present numerical results produced by the proposed schemes and discuss their features.

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New exact relations for steady irrotational two-dimensional gravity and capillary surface waves

Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rsta.2017.0220 ◽

2017 ◽

Vol 376 (2111) ◽

pp. 20170220 ◽

Cited By ~ 2

Author(s):

Didier Clamond

Keyword(s):

Free Surface ◽

Gravity Waves ◽

Water Waves ◽

Two Dimensional ◽

Physical Plane ◽

Theme Issue ◽

Nonlinear Water Waves ◽

Dimensional Gravity ◽

Irrotational Motion ◽

Exact Relations

Steady two-dimensional surface capillary–gravity waves in irrotational motion are considered on constant depth. By exploiting the holomorphic properties in the physical plane and introducing some transformations of the boundary conditions at the free surface, new exact relations and equations for the free surface only are derived. In particular, a physical plane counterpart of the Babenko equation is obtained. This article is part of the theme issue ‘Nonlinear water waves’.

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