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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0$A$nother 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