# Collision detection of two circular sectors in mixed polar and Cartesian coordinate

I am trying to solve the following collision detection problem. Suppose we have two circular sectors, each described in their own polar coordinate system with four values $r_1$, $r_2$, $d_1$ and $d_2$, where $r_1$ and $r_2$ ($r_2 \ge r_1$) are angles and $d_1$ and $d_2$ ($d_2 \ge d_1$) are two distances from the polar origin sandwiching the sector. The two sectors are placed on a 2-D Cartesian plane, each described with a coordinate $(x,y)$ (See illustration below).

Illustration:

The problem is to detect whether the two sectors collide. One trivial case is if the sum of the two $d_2$'s is greater than distance of two Cartesian points $\sqrt {(x_1 - x_2)^2 + (y_1 - y_2)^2}$, then the two shapes do not collide. I have yet to figure out how to solve rest of the problem (I thought of using the midline of $r_1$ and $r_2$, but can't exactly figure out how). Any ideas or suggestions?

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I have formatted your question, and am using variables like $x_1$ instead of $x1$ as in your figure, since that is more common. Please feel free to change it back if you want. – Paresh Dec 3 '12 at 6:02