Pathological and Test Cases For Reeb Analysis

After two decades in computational topology, it is clearly a computationally challenging area. Not only do we have the usual algorithmic and programming difficulties with establishing correctness, we also have a class of problems that are mathematically complex and notationally fragile. Effective development and deployment therefore requires an additional step - construction or selection of suitable test cases. Since we cannot test all possible inputs, our selection of test cases expresses our understanding of the task and of the problems involved. Moreover, the scale of the data sets we work with is such that, no matter how unlikely the behaviour mathematically, it is nearly guaranteed to occur at scale in every run. The test cases we choose are therefore tightly coupled with mathematically pathological cases, and need to be developed using the skills expressed most obviously in the constructing mathematical counterexamples. This paper is therefore a first attempt at reporting, classifying and analysing test cases previously used in computational topology, and the expression of a philosophy of how to test topological code.