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Wikipedia (and other sites) essentially define eccentricity (and by extension ellipticity) as a parameter associated with every conic section. It is a measure of how much the conic section deviates from being circular.

From this I'd conclude that eccentricity is not defined for an arbitrary, non conical shape. As an example, claiming to measure the eccentricity or ellipticity of a triangle or square is silly and meaningless. Is that a correct conclusion or not? I am not a mathematician and I'd appreciate some thoughts on this.

Thanks!

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  • $\begingroup$ How would you define it? It is up to you to define something that e.g. characterizes a "thing". That is what eccentricity does, it describes the shape of conic section by a simple number(not uniquely). Calling something eccentricity for a triangle is meaningless until you define it, but it probably would have nothing to do with a conic section. $\endgroup$ Commented May 4, 2018 at 0:53
  • $\begingroup$ I was not looking to define it differently. I wanted to merely confirm that 'eccentricity' as defined for conical shapes is not a parameter you can use to characterize other arbitrary non conical shapes. $\endgroup$ Commented May 4, 2018 at 15:37
  • $\begingroup$ I am referring to a conical section in a plane. I am using a 2D sensor to measure the intensity profile of a laser beam. I have a 3D intensity picture of the beam. The sensor measures the width of the beam in two perpendicular directions, XY. Take the largest width in those two directions and the ratio is calculated as the ellipticity of the beam. The intensity points between which this width is measured is Imax= 100% and 1/e^2 of Imax. I am stuck with this measurement. How could I define a number that would tell me how round the beam is when the beam's 2D slice is not a conical section. $\endgroup$ Commented May 6, 2018 at 20:38

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I'd be slightly careful with “conical shape”: the eccentricity makes sense for conic sections. Those are curves in the plane. There exist various mostly equivalent definitions: what you get when you intersect a (double) cone with a plane, or the union of circles, ellipses, parabolas and hyperbolas, or the set of all quadratic algebraic curves in the plane, and so on. Some corner cases are a bit less clear: conics which factor into a pair of lines, or only consist of a single real point, or no real point at all but only complex solutions. But the key point is: a conic section is not a cone, and is not what most people would picture as looking “conical”.

But apart from this detail, I think people most people will associate eccentricity with conic sections exclusively. I can imagine that some people might have found ways to generalize the concept to higher dimensions, to describe the shape of a quadric using a vector of eccentricities. I could also picture people generalizing the concept to higher degree, although I have no idea what the eccentricity of a cubic curve would be. But I know no actual examples for either of these, and I have a very hard time picturing anyone defining an eccentricity for a triangle in a meaningful way related to that of the ellipse.

That doesn't say there are no other eccentricities out there. In mathematics, names are only a shorthand for some definition, and often names get reused to mean different things in different fields. You have to know which definition is associated in a given context to make sense of it. Since eccentricity literally just refers to the fact that something is outside the center of something else, there could be many situations where this term would get chosen and used by a community agreeing on a definition of the term for their field of study.

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The eccentricity of an arbitrary shape is indeed undefined, but it is certainly neither silly nor meaningless to define one. In particular, shape description as used in pattern matching applications, makes intensive use of geometric parameters that characterize shapes. Among them, for instance, the isoperimetric ratio.

The eccentricity could be another. A way to define it is to pick the eccentricity of the equivalent ellipse (from the inertial point of view). Other measures of elungation can be imagined.

Also read https://en.wikipedia.org/wiki/Eccentricity_(mathematics)#Eccentricity_for_data_shapes.

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