Evolutoids and Pedaloids of Minkowski Plane Curves

2021 ◽  
Author(s):  
Gülşah Aydın Şekerci ◽  
Shyuichi Izumiya
Keyword(s):  
Minkowski Plane ◽  
Plane Curves

Minkowski Symmetry Sets of Plane Curves

2016 ◽  
Vol 60 (2) ◽  
pp. 461-480 ◽  
Author(s):  
Graham Mark Reeve ◽  
Farid Tari
Keyword(s):  
Euclidean Plane ◽  
Medial Axis ◽  
Minkowski Plane ◽  
Smooth Curve ◽  
Connected Components ◽  
Plane Curves ◽  
Symmetry Set ◽  
Blum Medial Axis

AbstractWe study the Minkowski symmetry set of a closed smooth curveγin the Minkowski plane. We answer the following question, which is analogous to one concerning curves in the Euclidean plane that was treated by Giblin and O’Shea (1990): given a pointponγ, does there exist a bi-tangent pseudo-circle that is tangent toγboth atpand at some other pointqonγ? The answer is yes, but as pseudo-circles with non-zero radii have two branches (connected components) it is possible to refine the above question to the following one: given a pointponγ, does there exist a branch of a pseudo-circle that is tangent toγboth atpand at some other pointqonγ? This question is motivated by the earlier quest of Reeve and Tari (2014) to define the Minkowski Blum medial axis, a counterpart of the Blum medial axis of curves in the Euclidean plane.

Self-Circumference in the Minkowski Plane

1989 ◽  
Author(s):  
Mostafa Ghandehari ◽  
Richard Pfiefer
Keyword(s):  
Minkowski Plane

scholarly journals Conjectures on Stably Newton Degenerate Singularities

2021 ◽  
Author(s):  
Jan Stevens
Keyword(s):  
Characteristic Zero ◽  
Arbitrary Number ◽  
Milnor Number ◽  
Plane Curves ◽  
Newton Diagram ◽  
Finite Characteristic ◽  
Vanishing Cycles

AbstractWe discuss a problem of Arnold, whether every function is stably equivalent to one which is non-degenerate for its Newton diagram. We argue that the answer is negative. We describe a method to make functions non-degenerate after stabilisation and give examples of singularities where this method does not work. We conjecture that they are in fact stably degenerate, that is not stably equivalent to non-degenerate functions.We review the various non-degeneracy concepts in the literature. For finite characteristic, we conjecture that there are no wild vanishing cycles for non-degenerate singularities. This implies that the simplest example of singularities with finite Milnor number, $$x^p+x^q$$ x p + x q in characteristic p, is not stably equivalent to a non-degenerate function. We argue that irreducible plane curves with an arbitrary number of Puiseux pairs (in characteristic zero) are stably non-degenerate. As the stabilisation involves many variables, it becomes very difficult to determine the Newton diagram in general, but the form of the equations indicates that the defining functions are non-degenerate.

On an area-preserving inverse curvature flow of convex closed plane curves

2021 ◽  
Vol 280 (8) ◽  
pp. 108931
Author(s):  
Laiyuan Gao ◽  
Shengliang Pan ◽  
Dong-Ho Tsai
Keyword(s):  
Curvature Flow ◽  
Plane Curves ◽  
Area Preserving ◽  
Closed Plane

scholarly journals On 3-syzygy and unexpected plane curves

2021 ◽  
Author(s):  
Grzegorz Malara ◽  
Piotr Pokora ◽  
Halszka Tutaj-Gasińska
Keyword(s):  
Projective Plane ◽  
Plane Curves ◽  
Complex Projective Plane ◽  
Geometric Properties ◽  
Complex Projective

AbstractIn this note we study curves (arrangements) in the complex projective plane which can be considered as generalizations of free curves. We construct families of arrangements which are nearly free and possess interesting geometric properties. More generally, we study 3-syzygy curve arrangements and we present examples that admit unexpected curves.

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