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POLine(s) Equidistant From Two Convex Polygons In 2D
 

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  1. POLine(s) Equidistant From Two Convex Polygons In 2D
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    <p>Given two convex polygons in 2D space, how would you go about constructing the line segment(s ) which, at any point on the lines, is equidistant from the closest point of either convex polygon?</p> <p>I'm looking towards an implementation of Voronoi diagrams for convex polygons instead of points, but I'm unsure how to even begin calculating the line for just two polygons. So I figured I'd take this one step at a time and start here.</p> <p><strong>Edit</strong> To try to make the question a little clearer, I want to bisect the plane (or a subset thereof). </p> <p>Suppose we have polygon A on the left and polygon B on the right. There will be some line of bisection that divides the plane into points on the left and points on the right. Every point on the line is equally distance from either polygon. Every point left of the line is closer to polygon A than to polygon B. Every point right of the line is closest to polygon B.</p> <p>Here's an image generated by a Matlab script I wrote that brute-forces an approximation:</p> <p><img src="https://i.stack.imgur.com/ZgV1U.jpg" alt="approximate bisecting line"></p> <p>The problem, I believe, is not as simple as examining the space in "between" the two polygons, since the line must extend beyond the area directly between the two shapes. And ideally I'd like to find a solution that generalizes to more than two shapes, which, to me, seems to complicate the problem a great deal more. Here's a (obviously very rough) approximation of how that might look:</p> <p><img src="https://i.stack.imgur.com/9DM1q.jpg" alt="more complicated example"></p>
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