We can think of the circle center as a "focus" of points of its circumference given by distance to center = constant. In case of an ellipse we have sum of distance from points on boundary to the two foci =constant, Can this be generalized ? In fact the reverse problem is of interest i.e. given an arbitrary closed curve, is it possible to determine such criteria( or numerical algorithm to identify such foci and also conditions when it exists). This is a general exploratory question. and all are welcome to comment.
Thank you Ette. Conic sections do have foci but I was thinking of a general approach , for example , given a curve we could do piece wise fitting of successive 3 points with say a set of circles. We obtain a set of foci , perhaps too many. There might be an optimum way to solve this for minimum number of foci.
Cheers
Please see this link
https://projecteuclid.org/download/pdf_1/euclid.chmm/1263315396
The question does not have appropriate answer die to its generality: without telling what is the definition of focus, and then what does it mean that the curve is determined by the (not defined ! ) foci makes the problem practically unsolvable. However, it can produce ideas on what is meant by the Author. I have a fantasy to try the following notions for plane curves:
A focus is any point on the plane, which is a center of a best fitting circle at some point. For the cirle there is exactly one focus, which is identical with the center. The set of all foci of a polygon concides with the set of its vertices (the circles then have radius 0). For smooth functions (without edges) without pieces of straight lines are considered as possessing the best fitting circles whose coordinates are given by well determined formulas (see e.g.
http://www.solitaryroad.com/c361.html
where the set of loci of all centers of curvature are called EVOLUTE. Also formulas are presented there in section Center of curvature. The problem of restoring the curve from its evolute is really challenging. Does someone knows if it is colved? If yes, then where is it published?
Regards, Joachim
@ Joachim Domsta
Thank you for pointing out appropriately the non definition of Foci. I had a a general notion that any curve can be generated with a minimum set of points and defining a rule to define a locus of points.
Cheers
Existence of notes claiming existence of some minimal sets of parameters defining any object is a standard idea in mathematics. However, the existence of some structure determining a class of smooth lines through notions like the foci and/or best fitting simple objects like circles, tangent lines etc. is probably spread in the literature. I didn't work on this kind of problems and therefore we should wait for more reliable comments by other RGaters.
Regards, Joachim
It was a paper of J.C.Maxwell - his first paper - where he he constructed curves that are of the same summ of distances from given finite number of pointes.
You can find it in his collected papers .
Actually, not. There is no common rule to determine a focus of a curve (beyond the second order ones). Of course you can determine infinity of "foci" by different laws. For example, a set of points such that d1^2 + d2^2 - d1*d2 = const. However you ask about inverse problem, so my answer is - will not.
P.S. Naturally, a focus is associated with the whole curve, and if you
transform a piece of circle (with homeomorphism), its focus will be broken. Instead of term "focus" you may take its local analog - center of curvature.
In the complex plane by defining complex Polynomial as
product of (z-ai) it is possible to generate an arbitrary curve.(see pdf enclosed).
It is a trivial case of plotting a complex polynomial , but the inverse problem given a set of points (Image matrix for example) a method for obtaining even approximately the
complex polynomial would be interesting.
Cheers
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