481:, 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. He shows that such an intersection can always be found to have area at least 1/8 that of the whole shape.
285:) 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. They thus reduce the problem to a numerical computation within each cell, which they do not solve explicitly. The partition of the plane into cells has
474:
of the given shape, to take the value one for symmetric shapes, and to take a value between zero and one for other shapes. Other symmetry measures with these properties include the ratio of the area of the shape to its smallest enclosing symmetric superset, and the analogous ratios of perimeters.
28:
a shape has. It is defined as the ratio of areas of the largest axially symmetric subset of the shape to the whole shape. Equivalently it is the largest fraction of the area of the shape that can be covered by a mirror reflection of the shape (with any orientation).
274:
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.
357:
cells for convex polygons; it can be constructed in an amount of time which is larger than these bounds by a logarithmic factor. Barequet and Rogol claim that in practice the area maximization problem within a single cell can be solved in
700:
900:
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",
117:
559:
240:
181:
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consider the problem of computing the axiality exactly, for both convex and non-convex polygons. The set of all possible reflection symmetry lines in the plane is (by
579:
385:
723:
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260:
201:
157:
137:
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262:-coordinates approach zero, showing that the lower bound is as large as possible. It is also possible to construct a sequence of centrally symmetric
470:
lists 11 different measures of axial symmetry, of which the one described here is number three. He requires each such measure to be invariant under
74:
convex bodies, the axiality is always somewhat higher: every triangle, and every centrally symmetric convex body, has axiality at least
1058:
903:
829:, Masters thesis, Department of Electrical Engineering and Computer Science, Korea Advanced Institute of Science and Technology
984:
1021:
Marola, Giovanni (1989), "On the detection of the axes of symmetry of symmetric and almost symmetric planar images",
77:
471:
523:
206:
821:
726:
282:
162:
33:
1053:
841:
Nohl, W. (1962), "Die innere axiale
Symmetrie zentrischer Eibereiche der euklidischen Ebene",
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25:
949:
695:{\displaystyle {\frac {\int \int w(p)w(\sigma (p))dx\,dy}{\int \int w(p)^{2}dx\,dy}}.}
1047:
493:
263:
60:
749:
Lassak, Marek (2002), "Approximation of convex bodies by axially symmetric bodies",
950:"Maximizing the area of an axially symmetric polygon inscribed in a simple polygon"
783:
765:
916:
55:
has axiality at least 2/3. This result improved a previous lower bound of 5/8 by
866:
266:
whose axiality has the same limit, again showing that the lower bound is tight.
36:, will have an axiality of exactly one, whereas an asymmetric shape, such as a
968:
52:
823:
Finding the largest inscribed axially symmetric polygon for a convex polygon
517:
67:
998:
63:, found through a computer search, whose axiality is less than 0.816.
1034:
982:
de
Valcourt, B. Abel (1966), "Measures of axial symmetry for ovals",
925:
870:
796:
Krakowski, F. (1963), "Bemerkung zu einer Arbeit von W. Nohl",
1023:
IEEE Transactions on
Pattern Analysis and Machine Intelligence
59:. The best upper bound known is given by a particular convex
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119:. In the set of obtuse triangles whose vertices have
80:
32:
A shape that is itself axially symmetric, such as an
729:
of a given shape, this is the same as the axiality.
387:time, giving (non-rigorous) overall time bounds of
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451:
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111:
871:"On a measure of axiality for triangular domains"
752:Proceedings of the American Mathematical Society
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112:{\displaystyle 2({\sqrt {2}}-1)\approx 0.828}
8:
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492:proposed to measure the symmetry of a
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48:
948:Barequet, Gill; Rogol, Vadim (2007),
7:
40:, will have axiality less than one.
14:
581:that maximizes the area integral
520:intensity values in the interval
554:{\displaystyle 0\leq w(p)\leq 1}
235:{\displaystyle 2({\sqrt {2}}-1)}
321:cells in the general case, and
759:(10): 3075–3084 (electronic),
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985:Israel Journal of Mathematics
784:10.1090/S0002-9939-03-07225-3
766:10.1090/S0002-9939-02-06404-3
917:10.1016/j.comgeo.2005.06.001
516:from points in the plane to
279:Barequet & Rogol (2007)
176:{\displaystyle {\sqrt {2}}}
1075:
561:) by finding a reflection
472:similarity transformations
203:, the axiality approaches
969:10.1016/j.cag.2006.10.006
24:is a measure of how much
1059:Euclidean plane geometry
957:Computers & Graphics
820:Choi, Chang-Yul (2006),
452:{\displaystyle O(n^{5})}
423:for the convex case and
416:{\displaystyle O(n^{4})}
350:{\displaystyle O(n^{3})}
314:{\displaystyle O(n^{4})}
875:Elemente der Mathematik
843:Elemente der Mathematik
798:Elemente der Mathematik
574:{\displaystyle \sigma }
16:In the geometry of the
904:Computational Geometry
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496:(viewed as a function
459:for the general case.
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44:Upper and lower bounds
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380:{\displaystyle O(n)}
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242:in the limit as the
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72:centrally symmetric
999:10.1007/BF02937452
865:Buda, Andrzej B.;
853:de Valcourt (1966)
808:de Valcourt (1966)
727:indicator function
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468:de Valcourt (1966)
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283:projective duality
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51:showed that every
34:isosceles triangle
718:{\displaystyle w}
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509:{\displaystyle w}
255:{\displaystyle y}
221:
196:{\displaystyle 1}
171:
152:{\displaystyle 0}
132:{\displaystyle x}
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484:In the study of
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38:scalene triangle
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18:Euclidean plane
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1029:(1): 104–108,
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963:(1): 127–136,
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911:(3): 152–164,
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851:. As cited by
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806:. As cited by
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264:parallelograms
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26:axial symmetry
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2:
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494:digital image
491:
490:Marola (1989)
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479:Lassak (2002)
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139:-coordinates
126:
106:
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69:
64:
62:
61:quadrilateral
58:
54:
50:
49:Lassak (2002)
43:
41:
39:
35:
30:
27:
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1026:
1022:
1016:
992:(2): 65–82,
989:
983:
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960:
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895:
881:(3): 65–73,
878:
874:
867:Mislow, Kurt
860:
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797:
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756:
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704:
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778:. Erratum,
1048:Categories
733:References
270:Algorithms
53:convex set
926:10203/314
652:∫
649:∫
619:σ
598:∫
595:∫
569:σ
546:≤
531:≤
518:grayscale
224:−
104:≈
95:−
68:triangles
1054:Symmetry
869:(1991),
70:and for
22:axiality
1008:0203589
935:2188943
887:1113766
849:: 59–63
804:: 60–61
775:1908932
725:is the
1006:
933:
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183:, and
953:(PDF)
827:(PDF)
705:When
107:0.828
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780:doi
761:doi
757:130
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