Conference paper Open Access

Analytic Formula for the Difference of the Circumradius and Orthoradius of a Weighted Triangle

Hummel, Michelle H.

Understanding and quantifying the effects of vertex insertion, perturbation, and weight allocation is useful for mesh generation and optimization. For weighted primal-dual meshes, the sensitivity of the orthoradius to mesh variations is especially important. To this end, this paper presents an analytic formula for the difference between the circumradius and orthoradius of a weighted triangle in terms of edge lengths and point weights under certain weight and edge assumptions. Current literature [1] offers a loose upper bound on the this difference, but as far as we know this is the first formula presented in terms of edge lengths and point weights. A formula in these terms is beneficial as these are fundamental quantities which enable a more immediate determination of how the perturbation of a point location or weight affects this difference. We apply this result to the VoroCrust algorithm to obtain the same quality guarantees under looser sampling conditions.

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