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Herbert Edelsbrunner develops algorithms for computational geometry and computational topology. His research account describes a progression from data structures and geometry to topological methods, guided by applications.^[ref] He coauthored Computational Topology: An Introduction by Herbert Edelsbrunner and John L. Harer with John L. Harer.^[ref]

A central computational challenge is to replace a continuous shape by finite combinatorial data while preserving the invariant being studied. For example, a triangulated surface can be described by vertices, edges and faces, allowing homology to be computed from boundary relations. The representation supplies an algorithmic route to a geometric question; the quality of the input model remains part of interpreting the answer.

John L. Harer works at the intersection of topology, geometry and computational data analysis. His Duke research description emphasizes geometric, combinatorial and computational methods for analyzing shape and data.^[ref] With Herbert Edelsbrunner, he coauthored Computational Topology: An Introduction by Herbert Edelsbrunner and John L. Harer, connecting topological invariants to computational methods.^[ref]

An elementary illustration is a triangular outline: it has one connected component and one unfilled loop. Filling its interior keeps the connected component but removes the hole. Homology distinguishes these cases, showing why a description of shape can capture something different from coordinates or pairwise distances. The example illustrates the subject rather than attributing a particular theorem or discovery to the book's authors.