Certified numerical algorithm for isolating the singularities of the plane projection of generic smooth space curves

• Published in: Journal of computational and applied mathematics Vol. 394; p. 113553
• Main Authors: Krait, George, Lazard, Sylvain, Moroz, Guillaume, Pouget, Marc
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.10.2021; Elsevier
• Subjects: Certified numerical algorithms; Computational Geometry; Computer Science; Generic singularities; Interval analysis; Singular curve topology; Transversality; Interval analysis; Certified numerical algorithms; Transversality; Generic singularities; Singular curve topology; Certified Numerical Algorithms; Generic Singularities; Singular Curve Topology; Interval Analysis
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2021.113553

Isolating the singularities of a plane curve is the first step towards computing its topology. For this, numerical methods are efficient but not certified in general. We are interested in developing certified numerical algorithms for isolating the singularities. In order to do so, we restrict our attention to the special case of plane curves that are projections of smooth curves in higher dimensions. This type of curve appears naturally in robotics applications and scientific visualization. In this setting, we show that the singularities can be encoded by a regular square system whose solutions can be isolated with certified numerical methods. Our analysis is conditioned by assumptions that we prove to be generic using transversality theory. We also provide a semi-algorithm to check their validity. Finally, we present experiments, some of which are not reachable by other methods, and discuss the efficiency of our method.