Robust and accurate computation of geometric distance for Lipschitz continuous implicit curves

Abstract

Computing the geometric distance between a point and an implicitly defined curve is widely used in implicit curve fitting and geometric processing. Iterative numerical methods are often used to compute the distance, which may give quite accurate solutions but are usually time-consuming and non-robust. In this paper, a circle double-and-bisect algorithm is proposed to reliably evaluate the accurate geometric distance of a point to an implicit curve. The method's prerequisite is merely the Lipschitz continuity of the implicit function. Moreover, by using gradient norm's upper bound estimation and minor arcs evolution, the efficiency can be improved apparently. Experimental results show that the evaluated distance can describe the fitting result with more fidelity and that some fitting algorithms (for example, the genetic algorithm) re-implemented with the proposed geometric distance converge much faster. Due to these contributions, the proposed method can be applied in diverse applications such as error evaluation, curve fitting and random point cloud generator.