Symmetry sets

1985 ◽  
Vol 101 (1-2) ◽  
pp. 163-186 ◽  
Author(s):  
J. W. Bruce ◽  
P. J. Giblin ◽  
C. G. Gibson
Keyword(s):  
Smooth Manifold ◽  
Normal Forms ◽  
Shape Recognition ◽  
Uniqueness Result ◽  
Potential Functions ◽  
Plane Curves ◽  
Symmetry Set ◽  
Space Curves ◽  
The Family ◽  
Set Of Points

SynopsisFor a smooth manifold M ⊆ ℝn, the symmetry set S(M) is defined to be the closure of the set of points u∈ℝn which are centres of spheres tangent to M at two or more distinct points. (The idea has its origin in the theory of shape recognition.) The connexion with singularities is that S(M) can be described alternatively as the levels bifurcation set of the family of distance-squared functions on M. In this paper a multi-germ version of the standard uniqueness result for versal unfoldings of potential functions is used to obtain a complete list of local normal forms (up to diffeomorphism) for the symmetry sets of generic plane curves, generic space curves, and generic surfaces in 3-space. For these cases the authors verify that M can be recovered as the envelope of a family of spheres centred at smooth points of S(M).

Geometry of projection-generic space curves

2009 ◽  
Vol 147 (1) ◽  
pp. 115-142
Author(s):  
C. T. C. WALL
Keyword(s):  
Local Structure ◽  
Double Point ◽  
Normal Forms ◽  
Blow Up ◽  
Complex Case ◽  
Plane Curves ◽  
Space Curves ◽  
Open Set ◽  
The Family ◽  
Flat Family

AbstractIn earlier work I defined a class of curves, forming a dense open set in the space of maps from S1 to P3, such that the family of projections of a curve in this class is stable under perturbations of C: we call the curves in the class projection-generic. The definition makes sense also in the complex case. The partition of projective space according to the singularities of the corresponding projection of C is a stratification. Its local structure outside C is the same as that of the versal unfoldings of the singularities presented.To study points on C we introduce the blow-up BC of P3 along C, and a family of plane curves, parametrised by z ∈ BC; we saw in the earlier work that this is a flat family.Here we show that near most z ∈ BC, the family gives a family of parametrised germs which versally unfolds the singularities occurring. Otherwise we find that the double point number δ of Γz drops by 1 for z ∉ EC. We establish a theory of versality for unfoldings of A or D singularities such that δ drops by at most 1, and show that in the remaining cases, we have an unfolding which is versal in this sense.This implies normal forms for the stratification of BC; further work allows us to derive local normal forms for strata of the stratification of P3.

scholarly journals Centre Symmetry Sets of Families of Plane Curves

2015 ◽  
Vol 48 (2) ◽  
Author(s):  
Peter Giblin ◽  
Graham Reeve
Keyword(s):  
Normal Forms ◽  
Direct Calculation ◽  
Parameter Family ◽  
Plane Curves ◽  
The Family ◽  
Study Centre

AbstractWe study centre symmetry sets and equidistants for a I-parameter family of plane curves where, for a special member of the family, there exist two inflexions with parallel tangents. Some results can be obtained by reducing a generating family to normal forms, but others require direct calculation from the generating family.

Symmetry sets of piecewise-circular curves

1993 ◽  
Vol 123 (6) ◽  
pp. 1135-1149
Author(s):  
P. J. Giblin ◽  
T. F. Banchoff
Keyword(s):  
Shape Recognition ◽  
Tangent Line ◽  
Plane Curves ◽  
Smooth Curves ◽  
Symmetry Set ◽  
Geometrical Properties ◽  
Mathematical Properties ◽  
Circular Arcs ◽  
Circular Curves ◽  
Straight Lines

SynopsisPiecewise-circular (PC) curves are made up of circular arcs and segments of straight lines, joined so that the (undirected) tangent line turns continuously. PC curves have arisen in various applications where they are used to approximate smooth curves. In a previous paper, the authors introduced some of their geometrical properties. In this paper they investigate the ‘symmetry sets’ of PC curves and one-parameter families of such curves. The symmetry set has also arisen in applications (this time to shape recognition) and its mathematical properties for smooth curves have been investigated by Bruce, Giblin and Gibson. It turns out that the symmetry sets of general one-parameter families of plane curves are mirrored remarkably faithfully by the symmetry sets arising from the much simpler class of PC curves.

scholarly journals General-affine invariants of plane curves and space curves

2019 ◽  
Vol 70 (1) ◽  
pp. 67-104
Author(s):  
Shimpei Kobayashi ◽  
Takeshi Sasaki
Keyword(s):  
Plane Curves ◽  
Space Curves ◽  
Affine Invariants ◽  
General Affine

Towards a Dynamic Font Respecting the Arabic Calligraphy

2011 ◽  
pp. 359-388
Author(s):  
Abdelouahad Bayar ◽  
Khalid Sami
Keyword(s):  
Mathematical Model ◽  
Optimization Techniques ◽  
Plane Curves ◽  
Time Processing ◽  
Cpu Time ◽  
Dynamic Character ◽  
Context Dependent ◽  
Set Of Points ◽  
Arabic Calligraphy

To justify texts, Arabic calligraphers use to stretch some letters with small flowing curves; the kashida instead of inserting blanks among words. Of course, such stretchings are context dependent. An adequate tool to support such writing may be based on a continuous mathematical model. The model has to take into account the motion of the qalam. The characters may be represented as outlines. Among the curves composing the characters outlines, some intersections are to be determined dynamically. In the Naskh style, the qalam‘s head behaves as a rigid rectangle in motion with a constant inclination. To determine the curves delimiting the set of points to shade when writing, we have to find out a mathematical way to compare plane curves. Moreover, as the PostScript procedure to produce a dynamic character, should be repeated whenever the letter is to draw, the development of a font supporting a continuous stretching model, allowing stretchable letters with no overlapping outlines, without optimization would be of a high cost in CPU time. In this chapter, some stretching models are given and discussed. A method to compare curves is presented. It allows the determination of the character encoding with eventually overlapping outlines. Then a way to approximate the curves intersection coefficients is given. This is enough to remove overlapping outlines. Some evaluations in time processing to confirm the adopted optimization techniques are also exposed.

Inflectional Convex Space Curves

1984 ◽  
Vol 36 (3) ◽  
pp. 537-549 ◽  
Author(s):  
Tibor Bisztriczky
Keyword(s):  
Convex Hull ◽  
Convex Space ◽  
Intersection Point ◽  
List Type ◽  
Multiple Points ◽  
Space Curves ◽  
Euclidean Three Space ◽  
Set Of Points ◽  
Three Space ◽  
Regular Closed

Let Φ be a regular closed C2 curve on a sphere S in Euclidean three-space. Let H(S)[H(Φ) ] denote the convex hull of S[Φ]. For any point p ∈ H(S), let O(p) be the set of points of Φ whose osculating plane at each of these points passes through p.1. THEOREM ([8]). If Φ has no multiple points and p ∈ H(Φ), then |0(p) | ≧ 3[4] when p is [is not] a vertex of Φ.2. THEOREM ( [9]). a) If the only self intersection point of Φ is a doublepoint and p ∈ H(Φ) is not a vertex of Φ, then |O(p)| ≧ 2.b) Let Φ possess exactly n vertices. Then(1)|O(p)| ≦ nforp ∈ H(S) and(2)if the osculating plane at each vertex of Φ meets Φ at exactly one point, |O(p)| = n if and only if p ∈ H(Φ) is not vertex.

scholarly journals Parameterizations of 1-Bridge Torus Knots

2003 ◽  
Vol 12 (04) ◽  
pp. 463-491 ◽  
Author(s):  
Doo Ho Choi ◽  
Ki Hyoung Ko
Keyword(s):  
Normal Form ◽  
Fundamental Group ◽  
Explicit Formula ◽  
Normal Forms ◽  
Double Cover ◽  
Torus Knots ◽  
Number Of Components ◽  
Torus Knot ◽  
Branched Cover ◽  
The Family

A 1-bridge torus knot in a 3-manifold of genus ≤ 1 is a knot drawn on a Heegaard torus with one bridge. We give two types of normal forms to parameterize the family of 1-bridge torus knots that are similar to the Schubert's normal form and the Conway's normal form for 2-bridge knots. For a given Schubert's normal form we give algorithms to determine the number of components and to compute the fundamental group of the complement when the normal form determines a knot. We also give a description of the double branched cover of an ambient 3-manifold branched along a 1-bridge torus knot by using its Conway's normal form and obtain an explicit formula for the first homology of the double cover.

scholarly journals Equilibrium states for Mañé diffeomorphisms

2018 ◽  
Vol 39 (9) ◽  
pp. 2433-2455 ◽  
Author(s):  
VAUGHN CLIMENHAGA ◽  
TODD FISHER ◽  
DANIEL J. THOMPSON
Keyword(s):  
Large Deviations ◽  
Existence And Uniqueness ◽  
Thermodynamic Formalism ◽  
Potential Functions ◽  
Equilibrium States ◽  
Unique Equilibrium ◽  
The Family ◽  
Natural Classes ◽  
Uniqueness Of Equilibrium

We study thermodynamic formalism for the family of robustly transitive diffeomorphisms introduced by Mañé, establishing existence and uniqueness of equilibrium states for natural classes of potential functions. In particular, we characterize the Sinaĭ–Ruelle–Bowen measures for these diffeomorphisms as unique equilibrium states for a suitable geometric potential. We also obtain large deviations and multifractal results for the unique equilibrium states produced by the main theorem.

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