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تسجيل مستخدم جديدConical Existence of Closed Curves on Convex Polyhedra
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Let C be a simple, closed, directed curve on the surface of a convex polyhedron P. We identify several classes of curves C that live on a cone, in the sense that C and a neighborhood to one side may be isometrically embedded on the surface of a cone Lambda, with the apex a of Lambda enclosed inside (the image of) C; we also prove that each point of C is visible to a. In particular, we obtain that these curves have non-self-intersecting developments in the plane. Moreover, the curves we identify that live on cones to both sides support a new type of source unfolding of the entire surface of P to one non-overlapping piece, as reported in a companion paper.
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We extend the notion of a source unfolding of a convex polyhedron P to be based on a closed polygonal curve Q in a particular class rather than based on a point. The class requires that Q lives on a cone to both sides; it includes simple, closed quas
igeodesics. Cutting a particular subset of the cut locus of Q (in P) leads to a non-overlapping unfolding of the polyhedron. This gives a new general method to unfold the surface of any convex polyhedron to a simple, planar polygon.
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