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I. The Problem of Molding Does a given object have a mold from which it can be removed? object not removable mold 1 object removable Assumptions The object is polyhedral. The mold has only one piece. Spherical objects cannot be manufactured using a mold of one piece. The object should be removed by only a single translation. Real screws cannot be removed by just a translation. mold 2

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Castability How to choose the orientation? The object has a horizontal top facet – the only one not in contact with the mold. # possible orientations = # facets. An object is castable if it is removable from its mold for one of the orientations. Because every facet may become horizontal. How to determine that the object is castable? For each potential orientation, determine whether there exists a direction along which the object can be removed from the mold.

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Making Things More Precise polyhedron The mold is a rectangular block with a concavity that exactly matches the polyhedron.

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Necessary Condition for Removal polyhedron (necessary condition) angle between two vectors

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Also a Sufficient Condition Proof ( ) By contradiction. ( ) By contradiction.

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One Trans. vs. Multiple Trans. Polyhedron removable by a sequence of translations. Allowing for multiple translations does not help.

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Representing a Direction

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Geometric Interpretation

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Geometric Formulation The intersection of all such half-planes is the set of points that correspond to a direction in which the polygon can be removed. Given a set of half-planes, compute their common intersection. Castability test: Enumerate all facets as top facet.

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II. Intersection of Half-Planes Their intersection must be a convex set, more specifically, each a convex set! A convex polygonal region. May be unbounded May degenerate into a line, segment, a point, or an empty set.

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Some Possible Cases

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A Divide-and-Conquer Algorithm

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Intersection of Convex Regions #vertices#intersections Modify the algorithm to intersect two convex regions.

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The Recurrence

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Improvement The subroutine for intersection of convex regions is a transplant from that for intersecting two simple polygons. We made use of the convexity in our analysis. Can we do better? But we haven’t taken full advantage of convexity yet … Yes! Assumption (non-degeneracy): The regions to be intersected are 2-dimensional. (The degenerate cases are easier.)

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Representing a Convex Region left boundary right boundary Left and right boundaries as sorted lists of half-planes during traversals from top to bottom. Vertices can be easily computed by intersecting consecutive bounding lines. But they are stored explicitly.

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Plane Sweep Again Assumption: no horizontal edge (easy to dealt with if not true). l_e_C1, r_e_C1, l_e_C2, r_e_C2 At most four edges intersecting the sweep line. Corresponding pointer set to nil if no intersection. r_e_C1r_e_C1 l_e_C1l_e_C1 l_e_C2l_e_C2 r_e_C2 = nil

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No Event Queue Next event point: Highest of the lower endpoints of the four edges that intersect the sweep line. Start at The new edge e is one of the following:

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Left Boundary of Chain 1 This can be determined by checking whether p is between l_e_C2 and r_e_C2. l_e_C2l_e_C2 l_e_C2l_e_C2 2 new edges

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cont’d e intersects r_e_C2. r_e_C2r_e_C2 r_e_C2r_e_C2 Case 1. The new edges start at the intersection. Case 2. The new edges end at the intersection. Do nothing because these edges have been discovered.

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Running Time Recurrence for the total running time:

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