Symmetry sets

Bruce, J. W.; Giblin, Peter; Gibson, C. G.

doi:10.1017/s0308210500026263

Cited by 82 publications

(95 citation statements)

References 10 publications

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“…As we mentioned, two tangent segments to an oval γ are equal if and only if there is a circle touching γ at these tangency points. The locus of centers of such bitangent circles is called the symmetry set of γ [2]. Thus our problem is closely related to the geometry of symmetry sets.…”

Section: Constructionmentioning

confidence: 99%

The (Un)equal Tangents Problem

Tabachnikov

2012

The American Mathematical Monthly

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The problemGiven a point A outside of a closed strictly convex plane curve γ, there are two tangent segments from A to γ, the left and the right ones, looking from point A.Problem: Does there exist a curve γ such that one can walk around it so that, at all moments, the right tangent segment is smaller than the left one?In other words, does there exist a pair of simple closed curves, γ and Γ, the former strictly convex, the latter containing the former in its interior, such that for every point A of Γ the right tangent segment to γ is smaller than the left one?Over the years, I have polled numerous colleagues, mostly as a dinner table topic. Most of them thought that the answer was negative, and quite a few tried to provide a proof, but each attempt had a flaw. I invite the reader to think about this question too before reading any further.Up until recently, I have believed that for any oval 1 γ and every closed curve Γ going around γ, there existed a point A ∈ Γ from which the tangent segments to γ were equal. In fact, I conjectured in [7] that there existed at least four such points. Figure 1 illustrates the situation for an ellipse. The extensions of the axes partition the plane into four quadrants marked + and − according to the sign of the difference between the left and the right tangent segments. The axes themselves are the locus of points from which the tangent segments * Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA; e-mail: tabachni@math.psu.edu 1 a smooth strictly convex closed curve.

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Section: Constructionmentioning

confidence: 99%

The (Un)equal Tangents Problem

Tabachnikov

2012

The American Mathematical Monthly

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“…If the restriction in the definition of the MA regarding maximal circles is omitted, the Symmetry Set is obtained: The Symmetry Set (SS) is defined as the closure of the set of centers of circles that are tangent to the shape at least two points [9,11,16,15]. Obviously, the MA and the R-skeleton are a subset of the SS [15].…”

Section: Symmetry Setsmentioning

confidence: 99%

“…The MA is a sub-set of the Symmetry Set (SS) [11]. The SS exhibits nice mathematical properties, but is more difficult to compute than the MA.…”

Section: Introductionmentioning

confidence: 99%

Alternative 2D Shape Representations using the Symmetry Set

et al. 2006

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Abstract. Among the many attempts made to represent families of 2D shapes in a simpler way, the Medial Axis (MA) takes a prominent place. Its graphical representation is intuitively appealing and can be computed efficiently. Small perturbations of the shape can have large impact on the MA and are regarded as instabilities, although these changes are mathematically known from the investigations on a super set, the Symmetry Set (SS). This set has mainly been in a mathematical research stage, partially due to computational aspects, and partially due to its unattractive representation in the plane.In this paper novel methods are introduced to overcome both aspects. As a result, it is possible to represent the SS as a string is presented. The advantage of such a structure is that it allows fast and simple query algorithms for comparisons.Second, alternative ways to visualize the SS are presented. They use the distances from the shape to the set as extra dimension as well as the so-called pre-Symmetry Set and antiSymmetry Set. Information revealed by these representations can be used to calculate the linear string representation structure.Example shapes from a data base are shown and their data structures derived.

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“…Thus, the computation of skeletons and symmetry sets of planar shapes is a subject that received a great deal of attention from the mathematical (see Bruce et al, 1985;Bruce and Giblin, 1992 and references therein), computational geometry (Preparata and Shamos, 1990), biological vision (Kovács and Julesz, 1994;Lee et al, 1995;Leyton, 1992), and computer vision communities (see for example Ogniewicz, 1993;Serra, 1982 and references therein) since the original work by Blum (1967Blum ( , 1973.…”

Section: Introductionmentioning

confidence: 99%

Area-Based Medial Axis of Planar Curves

Niethammer

Betelú

Sapiro

et al. 2004

International Journal of Computer Vision

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A new definition of affine invariant medial axis of planar closed curves is introduced. A point belongs to the affine medial axis if and only if it is equidistant from at least two points of the curve, with the distance being a minimum and given by the areas between the curve and its corresponding chords. The medial axis is robust, eliminating the need for curve denoising. In a dynamical interpretation of this affine medial axis, the medial axis points are the affine shock positions of the affine erosion of the curve. We propose a simple method to compute the medial axis and give examples. We also demonstrate how to use this method to detect affine skew symmetry in real images.

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Symmetry sets

Cited by 82 publications

References 10 publications

The (Un)equal Tangents Problem

The (Un)equal Tangents Problem

Alternative 2D Shape Representations using the Symmetry Set

Area-Based Medial Axis of Planar Curves

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