Axiality (geometry)
In the geometry of the Euclidean plane, axiality is a measure of how much axial symmetry 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). A shape that is itself axially symmetric, such as an isosceles triangle, will have an axiality of exactly one, whereas an asymmetric shape, such as a scalene triangle, will have axiality less  than one. Upper and lower bounds. showed that every convex set has axiality at least 2/3. This result improved a previous lower bound of 5/8 by . The best upper bound known is given by a particular convex quadrilateral, found through a computer search, whose axiality is less than 0.816. For triangles and for centrally symmetric convex bodies, the axiality is always somewhat higher: every triangle, and every centrally symmetric convex body, has axiality at least formula_1. In the set of obtuse triangles whose vertices have formula_2-coordinates formula_3, formula_4, and formula_5, the axiality approaches formula_6 in the limit as the  formula_7-coordinates approach zero, showing that the lower bound is as large as possible. 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. 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. Related concepts. 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 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. 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. , 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. In the study of computer vision, proposed to measure the symmetry of a digital image (viewed as a function formula_13 from points in the plane to grayscale intensity values in the interval formula_14) by finding a reflection formula_15 that maximizes the area integral When formula_13 is the indicator function of a given shape, this is the same as the axiality.