The following question appears in Section 4.9 of "Computational Geometry: Algorithms and Applications" By Mark de Berg, Marc van Kreveld, Mark Overmars, and Otfried Schwarzkopf:

Show that the problem of finding a center of rotation that allows us to remove $P$ with a single rotation from its mold can be reduced to the problem of finding a point in the common intersection of a set of half-planes.

$P$ here is a simple polygon with a given orientation, cast in a mold - Rotation from mold

Whether $P$ can be removed with a translation can be reduced to whether a common intersection of a set of half-planes is unbounded: Each edge of $P$, continued to a line, defines the half plane touching the outer side of the edge. The intersection of these half-planes is unbounded in a direction $\vec{v}$ if and only if translating $P$ by $\vec{v}$ does not cause any edge to push against the mold.

How can this reduction be modified to solve the case of removing $P$ with a rotation around a point?


1 Answer 1


Consider the marked edge of the following object (light gray) in its mold (dark gray) - edge of object in mold

Where must $p$ be such that an arbitrary points on the edge will not be pushed against the mold? This figure shows the direction a point on the edge will be moving in when rotated around a point $p$ - pivot rotation direction

Notice that the point will be moving perpendicular to the line connecting it with $p$. Now recall that in order not to push on the mold, the points of the edge must move in the direction of the inner half-plane defined by the edge. This means that the possible locations for $p$ lie on a half-plane perpendicular to the edge. Here's the half-plane (in green) corresponding to an inner point of the edge - enter image description here

The half-plane corresponding to the upper vertex of the edge, in this case, is contained in all of the half-planes for inner points, and so a rotation around any point in it moves the entire edge such that it does not push on the mold.

Whether $P$ can be removed with a rotation around $p$, is thus equivalent to whether $p$ is in the common intersection of the half-planes defined by $P$'s edges.

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    $\begingroup$ Welcome to CS.SE! Thank you for the detailed answer and edit to the question! I hope you'll stick around and continue to contribute. $\endgroup$
    – D.W.
    May 1, 2019 at 4:59

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