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Creators/Authors contains: "Christopher J. Bishop"

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  1. (, Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA))
    For any simple polygon P we compute the optimal upper and lower angle bounds for triangulating P with Steiner points, and show that these bounds can be attained (except in one special case). The sharp angle bounds for an N-gon are computable in time O(N), even though the number of triangles needed to attain these bounds has no bound in terms of N alone. In general, the sharp upper and lower bounds cannot both be attained by a single triangulation, although this does happen in some cases. For example, we show that any polygon with minimal interior angle θ has a triangulation with all angles in the interval I = [θ, 90°–min(36°, θ)/2], and for θ ≤ 36° both bounds are best possible. Surprisingly, we prove the optimal angle bounds for polygonal triangulations are the same as for triangular dissections. The proof of this verifies, in a stronger form, a 1984 conjecture of Gerver. 
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