Is there a pure algebraic way to calculate intersection of two disks (extended to spheres, ellipses)?
algebraic way to compute intersection of disks
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geometry
algebraic-geometry
analytic-geometry
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2What exactly do you want to compute? The intersection (if they do intersect non-trivially) is a region of the plane whose boundary consists of part of one circle and part of the other. – 2011-09-02
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2@Robert : I think he is after analytic geometry tag rather than algebraic Geometry tag. – 2011-09-02
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0Perhaps the Question is how to check *if* two disks have a nonempty intersection? This is easy for disks, not so easy for ellipses. – 2011-09-02
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0@hardmath: No i did not mean that. Robert: Intersection of two discs would be a region that is enclosed by two arcs. Can that be categorized with just teh algebbra without any geometry. – 2011-09-02
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0...so what is to be computed, then? Certainly one can shade pixels on a screen depending on whether they're inside the [lens](http://mathworld.wolfram.com/Lens.html) formed by the two circles or not. If you're asking about finding the intersection points of the two circles, that's easy to do. What do you need? – 2011-09-02
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0Here's what I am looking for. I do not want to represent the intersection as a collection of algebraic inequalities. Instead, what I want is a definition that can "generate" points in that region. By that I mean, eg. x<=10; f(x) =x^2, can "generate the region", while ($(x^2 + y^2 \leq 5) \cap (x^2 + y^2 \leq 10), x,y \in \mathbb{R}^n$ cannot (except that you can take the geometry into account and write the intersection in terms of some chord and angles). I don't know much about algebraic geometry so I may totally be wrong in expecting something like this. – 2011-09-02
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0Another way to look at this is: A convex set can be defined by a bunch of supporting hyperplanes. What I am looking at is a single definition for the convex set.. may be in some other space. This might a lot of rambling but still... – 2011-09-02
1 Answers
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I think what you mean is a parametric representation. Suppose the circles $C_1$ and $C_2$ intersect at points $P_1$ and $P_2$. Let $f$ be inversion around $P_1$. This maps the circles to straight lines $L_1 = f(C_1)$ and $L_2 = f(C_2)$, and the intersection of the disks to one of the four regions into which these lines divide the plane. This can be parametrized by $f(P_2) + r (\cos(\theta),\sin(\theta))$, $0 \le r < \infty$, $\theta_1 \le \theta \le \theta_2$. Invert back and you get the intersection of the disks.
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0Thanks. Parametric representation is the right word and indeed this is the right answer (and i am marking it such). However, hidden behind all this, the question I had in my mind was whether there is some algebra/ algebraic geometry which can generalize solutions to problems of this kind. Say intersection of closed well defined shapes in higher dimensions. – 2011-09-02