• [4][5] Algorithms The axiality of a given convex shape can be approximated arbitrarily closely in sublinear time, given access to the shape by oracles for finding an extreme
  point in a given direction and for finding the intersection of the shape with a line.

• It is also possible to construct a sequence of centrally symmetric parallelograms whose axiality has the same limit, again showing that the lower bound is tight.

• [8] He requires each such measure to be invariant under similarity transformations of the given shape, to take the value one for symmetric shapes, and to take a value between
  zero and one for other shapes.

• In the set of obtuse triangles whose vertices have -coordinates , , and , the axiality approaches in the limit as the -coordinates approach zero, showing that the lower bound
  is as large as possible.

• Lassak (2002), as well as studying axiality, studies a restricted version of axiality in which the goal is to find a halfspace whose intersection with a convex shape has large
  area lies entirely within the reflection of the shape across the halfspace boundary.

• [1] In the study of computer vision, Marola (1989) proposed to measure the symmetry of a digital image (viewed as a function from points in the plane to grayscale intensity
  values in the interval ) by finding a reflection that maximizes the area integral[9] When is the indicator function of a given shape, this is the same as the axiality.

• The set of all possible reflection symmetry lines in the plane is (by projective duality) a two-dimensional space, which they partition into cells within which the pattern
  of crossings of the polygon with its reflection is fixed, causing the axiality to vary smoothly within each cell.

• [2] The best upper bound known is given by a particular convex quadrilateral, found through a computer search, whose axiality is less than 0.816.

 

Works Cited

[‘Lassak, Marek (2002), “Approximation of convex bodies by axially symmetric bodies”, Proceedings of the American Mathematical Society, 130 (10): 3075–3084 (electronic), doi:10.1090/S0002-9939-02-06404-3, MR 1908932. Erratum, doi:10.1090/S0002-9939-03-07225-3.
2. ^
Krakowski, F. (1963), “Bemerkung zu einer Arbeit von W. Nohl”, Elemente der Mathematik, 18: 60–61. As cited by de Valcourt (1966).
3. ^ Choi, Chang-Yul (2006), Finding the largest inscribed axially symmetric polygon for a convex polygon (PDF), Masters
thesis, Department of Electrical Engineering and Computer Science, Korea Advanced Institute of Science and Technology.
4. ^ Nohl, W. (1962), “Die innere axiale Symmetrie zentrischer Eibereiche der euklidischen Ebene”, Elemente der Mathematik, 17:
59–63. As cited by de Valcourt (1966).
5. ^ Buda, Andrzej B.; Mislow, Kurt (1991), “On a measure of axiality for triangular domains”, Elemente der Mathematik, 46 (3): 65–73, MR 1113766.
6. ^ Ahn, Hee-Kap; Brass, Peter; Cheong, Otfried; Na, Hyeon-Suk;
Shin, Chan-Su; Vigneron, Antoine (2006), “Inscribing an axially symmetric polygon and other approximation algorithms for planar convex sets”, Computational Geometry, 33 (3): 152–164, doi:10.1016/j.comgeo.2005.06.001, hdl:10203/314, MR 2188943.
7. ^
Barequet, Gill; Rogol, Vadim (2007), “Maximizing the area of an axially symmetric polygon inscribed in a simple polygon” (PDF), Computers & Graphics, 31 (1): 127–136, doi:10.1016/j.cag.2006.10.006.
8. ^ de Valcourt, B. Abel (1966), “Measures of
axial symmetry for ovals”, Israel Journal of Mathematics, 4 (2): 65–82, doi:10.1007/BF02937452, MR 0203589.
9. ^ Marola, Giovanni (1989), “On the detection of the axes of symmetry of symmetric and almost symmetric planar images”, IEEE Transactions
on Pattern Analysis and Machine Intelligence, 11 (1): 104–108, doi:10.1109/34.23119
Photo credit: https://www.flickr.com/photos/38953258@N08/5675705011/’]