Is there a standard way of quantifying "closeness to being symmetrical"?

Question

Consider these three polygons:

Object #1 is clearly symmetrical. Object #2 is not, but in some sense it seems "close" to being so (in that it is not very different from object #1). Object #3 appears to be very far from being symmetrical.

Is there a standard way of quantifying these observations, i.e., a measure for "closeness to symmetry"?

Answer 1

Let's pretend we have defined rigorously what we mean by a symmetric shape. Let's also pretend we have some sensible metric metric on the set of shapes in $\mathbb R^2$. Then we can measure how symmetric the shape $A$ is using the non-negative number. . .

$|A|= \inf \{d(A,B): B$ is symmetric$\}$

Symmetric shapes will always satisfy $|A|=0$ because $d(A,A)=0$. Whether there are non symmetric shapes will depend on the measure and the exact notion of symmetry.

So what kind of metrics could we use? Well there's always the Hausdorff metric on $\mathbb R^2$: For any two shapes (subsets) $A$ and $B$ we define the distance. . .

$d(A,B) = \inf \{\varepsilon : $every point of $A$ is within distance $\varepsilon$ of some point of $B$ and vice-versa$\}$

This is a good metric because according to it a 'shape' can be any closed subset of the plane. However the Hausdorff metric might give $(1)$ and $(3)$ as being very far apart because the tip of $(3)$ is very far from the edge of the square would be.

Another metric for shapes is by taking the symmetric difference. . .

$A \Delta B = \{x \in \mathbb R^2 : x$ is an element of exactly one of $A$ or $B$ $\}$

and calculating its area:

$d(A,B) =$ The area of $A \Delta B$

This metric is more likely to give $d((1),(3))< d((1),(2))$. However it only allows for a narrower definition of what a 'shape' is. For example a disc will be distance $0$ from a 'lollipop' shape because the symmetric difference is just the stick, which has measure zero. So we might restrict to, for example, finite unions of polygons. Maybe that's enough for what you want.

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Edit: For either of the two metrics you probably want to be allowed move the shapes around before comparing them. In that case you actually want to consider the distance. . .

$D(A,B) = \inf \{d(A,B+x): x \in \mathbb R^2\}$

where $B+x$ just means $\{b + x : b \in B\}$

Answer 2

The case of reflection symmetry is commonly studied and measured in the fields of human biology and psychology, for the specific case of facial symmetry or fluctuating asymmetry (FA). For example, Berssenbrügge, et. al., 2015 describe a process like this:

The method used to determine a symmetry plane is a crucial factor when creating an objective asymmetry index. Here, the facial symmetry plane is determined on the basis of the shape of the face using the technique described in detail in [Berssenbrügge, et. al., 2014], which is a modified version of the approach proposed by Benz et al. [2002]. The facial point cloud is repeatedly mirrored and registered to the original cloud using the iterative closest point algorithm [Besl and McKay, 1992]. Between each repetition, asymmetrical parts are removed according to a threshold. Finally, from the pair of original and processed clouds, an average symmetry plane is calculated by a least squares fitting. Using this plane, a mirrored copy of the point cloud is then calculated. From these two surfaces, a geometric and a color asymmetry index are determined.

The geometric asymmetry index $AI_{geom}$ is defined as the mean spatial distance $\bar d$ between the facial surface and its mirrored copy, scaled by the size of the face. The face size is estimated by the diagonal $D$ of its bounding box in frontal view. The resulting (typically very small) dimensionless values are multiplied by a factor of 1000, which of course does not influence the statistics:

$AI_{geom} := \frac{\bar d}{D} 1000$.

Other procedures are described in other sources. For example, Grammer and Thornhill, 1994 describe manually selecting a dozen matching morphological points on the left and right sides of the face, drawing a line segment between matching pairs, and computing FA as "sum of all possible nonredundant differences between the midpoints of six horizontal lines between the following pairs of points". Alternatively, they compute a "second measure of facial symmetry, which we call central facial asymmetry (CFA), focuses on the differences between midpoints of adjacent lines, especially in the center of the face."

It would seem that the earlier method of Grammer and Thornhill would be difficult to apply to arbitrary shapes such as polygons that may not have predetermined bilateral matching pairs of points. Berssenbrügge, et. al., therefore may provide the basis of a more general method.

Answer 3

The work of Prof. David Avnir, which works in the meaurement of shape, symmetry and chirality in molecules and related objects (orbitals, electron density,...) might interest you:

http://chem.ch.huji.ac.il/avnir/p_symmetry.html

http://www.csm.huji.ac.il/new/

http://www.ee.ub.edu/index.php?option=com_content&view=article&id=72&Itemid=469