Closed cap condition under the cap construction algorithm

10/01/2022

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by   Mercedes Sandu, et al.

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Every polygon P can be companioned by a cap polygon P̂ such that P and P̂ serve as two parts of the boundary surface of a polyhedron V. Pairs of vertices on P and P̂ are identified successively to become vertices of V. In this paper, we study the cap construction that asserts equal angular defects at these pairings. We exhibit a linear relation that arises from the cap construction algorithm, which in turn demonstrates an abundance of polygons that satisfy the closed cap condition, that is, those that can successfully undergo the cap construction process.