Computing Minimum Area Homologies

Abstract

Calculating and categorizing the similarity of curves is a fundamental problem which has generated much recent interest. However, to date there are no implementations of these algorithms for curves on surfaces with provable guarantees on the quality of the measure. In this paper, we present a similarity measure for any two cycles that are homologous, where we calculate the minimum area of any homology (or connected bounding chain) between the two cycles. The minimum area homology exists for broader classes of cycles than previous measures which are based on homotopy. It is also much easier to compute than previously defined measures, yielding an efficient implementation that is based on linear algebra tools. We demonstrate our algorithm on a range of inputs, showing examples which highlight the feasibility of this similarity measure.Calculating and categorizing the similarity of curves is a fundamental problem which has generated much recent interest. However, to date there are no implementations of these algorithms for curves on surfaces with provable guarantees on the quality of the measure. In this paper, we present a similarity measure for any two cycles that are homologous, where we calculate the minimum area of any homology (or connected bounding chain) between the two cycles. The minimum area homology exists for broader classes of cycles than previous measures which are based on homotopy. It is also much easier to compute than previously defined measures, yielding an efficient implementation that is based on linear algebra tools. We demonstrate our algorithm on a range of inputs, showing examples which highlight the feasibility of this similarity measure.