Calculation of the Differential Geometry Properties of Implicit Parametric Surfaces Intersection

Abstract

Generally, to calculate the Frenet-Serret apparatus of a curve, it is necessary to have a parameterization of it; but when it is difficult to obtain a parameterization of the curve, as is the case of the curves obtained by the intersection of two implicit parametric surfaces, it is necessary to develop new methods that make it possible to know the geometric properties of said curve. This paper describes a new Mathematica package, Frenet, with the objective of calculating the properties of the differential geometry of a curve obtained by the intersection of two implicit parametric surfaces. The presented package allows us to visualize the Frenet-Serret mobile trihedron, to know the curvature and torsion at a given point of the curve obtained by the intersection of two implicit parametric surfaces. Package performance is discussed using several illustrative examples. Provide the user with an important tool for visualization and teaching.