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Affine-Invariant Distances, Envelopes and Symmetry Sets

Publication ,  Journal Article
Giblin, PJ; Sapiro, G
Published in: Geometriae Dedicata
January 1, 1998

Affine invariant symmetry sets of planar curves are introduced and studied in this paper. Two different approaches are investigated. The first one is based on affine invariant distances, and defines the symmetry set as the closure of the locus of points on (at least) two affine normals and affine-equidistant from the corresponding points on the curve. The second approach is based on affine bitangent conics. In this case the symmetry set is defined as the closure of the locus of centers of conics with (at least) 3-point contact with the curve at two or more distinct points on the curve. This is equivalent to conic and curve having, at those points, the same affine tangent, or the same Euclidean tangent and curvature. Although the two analogous definitions for the classical Euclidean symmetry set are equivalent, this is not the case for the affine group. We present a number of properties of both affine symmetry sets, showing their similarities with and differences from the Euclidean case. We conclude the paper with a discussion of possible extensions to higher dimensions and other transformation groups, as well as to invariant Voronoi diagrams.

Duke Scholars

Published In

Geometriae Dedicata

DOI

ISSN

0046-5755

Publication Date

January 1, 1998

Volume

71

Issue

3

Start / End Page

237 / 261

Related Subject Headings

  • General Mathematics
  • 4904 Pure mathematics
  • 0101 Pure Mathematics
 

Citation

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Giblin, P. J., & Sapiro, G. (1998). Affine-Invariant Distances, Envelopes and Symmetry Sets. Geometriae Dedicata, 71(3), 237–261. https://doi.org/10.1023/A:1005099011913
Giblin, P. J., and G. Sapiro. “Affine-Invariant Distances, Envelopes and Symmetry Sets.” Geometriae Dedicata 71, no. 3 (January 1, 1998): 237–61. https://doi.org/10.1023/A:1005099011913.
Giblin PJ, Sapiro G. Affine-Invariant Distances, Envelopes and Symmetry Sets. Geometriae Dedicata. 1998 Jan 1;71(3):237–61.
Giblin, P. J., and G. Sapiro. “Affine-Invariant Distances, Envelopes and Symmetry Sets.” Geometriae Dedicata, vol. 71, no. 3, Jan. 1998, pp. 237–61. Scopus, doi:10.1023/A:1005099011913.
Giblin PJ, Sapiro G. Affine-Invariant Distances, Envelopes and Symmetry Sets. Geometriae Dedicata. 1998 Jan 1;71(3):237–261.
Journal cover image

Published In

Geometriae Dedicata

DOI

ISSN

0046-5755

Publication Date

January 1, 1998

Volume

71

Issue

3

Start / End Page

237 / 261

Related Subject Headings

  • General Mathematics
  • 4904 Pure mathematics
  • 0101 Pure Mathematics