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An excerpt from the 2023 Department Newsletter research article...
In 1983, Edelsbrunner, Kirkpatrick and Seidel introduced the concept of the alpha shape, which is a piecewise-linear region surrounding a collection of points in the plane $\mathbb{R}^{2}$, including the 2-dimensional convex hull as a special case [3]. Edelsbrunner and Mucke extended the alpha shape to three dimensions, a construction which was later used to create beautiful models of the shapes of certain biomolecules.
I will share a new algorithm developed with J. Carlsson for computing a closely related combinatorial structure known as the alpha complex in higher dimension. The reason we began looking at the alpha complex has do with the discovery of a different type of shape associated to Gaussian mixtures, which is naturally approximated by weighted versions of the alpha complexes.
For the full article, as well as example illustrations, check out our 2023 Department Newsletter! This article starts on page 6.