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This is an extension of the question here because the answer seemed to be specific on the shapes involved, and largely rely on some known knowledge of the shapes (e.g. a line can only intersect a circle at most two points)

Suppose I have two shapes: A circle and a square described by

$$x^2+y^2=r^2$$

and

$$\{(x,y):|x−a|≤s,|y−b|≤s\}$$

What function can allow me to count the number of intersections for some given r,a,b,s?

The answer to this question can help generalise the problem to the case where the two shapes $A$ and $B$ may be abstract and does not live in euclidian space which prevent the use of intuition as in the simpler case above

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Secret
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  • Your title seems to acknowledge the importance of "two known shapes", but the body text belittles answers that "largely rely on some known knowledge of the shapes". As the one posing the Question, it is your responsibility to delineate exactly what knowledge of the two shapes is available "to count the number of intersections". – hardmath Nov 01 '15 at 18:09
  • Related: Get location of vector/circle intersection. – hardmath Nov 01 '15 at 18:22
  • You consider the borders, not the surfaces, right ? Maybe you could just look for all the solutions for a circle and a segment. then it you apply to squares or to any polygons. – Fabrice NEYRET Nov 01 '15 at 18:46
  • @hardmath I am looking for something along the lines of: $$$$ Given the only thing we know about the two shapes are their implicit equations (e.g. x^2+y^2=r^2 represents a circle of radius r) or a system of relations, how can one construct some relation $R$ that counts the number of intersections they can have so that one can simply optimise $R$ to minimise or maximise the number of intersections by varying the parameters e.g. r etc.

    Because say if we are working in hyperbolic space, we cannot easily draw a hyperbolic circle or square, thus we lost the intuitive way to solve problems of these

     – Secret Nov 02 '15 at 06:00
  • types. $$$$ That is, i am wondering whether such $R$ can always be constructed (either explicitly or as an algorithm so that it can be applied regardless of the structure of the underlying space (e.g. it might have some unusual inner products, it might be infinite dimensional etc.) – Secret Nov 02 '15 at 06:02

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