Calculation and Geometric Interpretation of the Curvatures of Implicit Curves and Surfaces with Mathematica

In this paper, based on the formulas proposed by Ronald Goldman, new functions are defined and encoded in the Wolfram Language using the Mathematica symbolic calculation system. With the coded functions, the curvature of implicit plane curves, the curvature and twist of implicit space curves, and the Gaussian and mean curvatures of implicit surfaces can be calculated. All of these functions are included in a Mathematica package. Finally, the package’s functionality is shown by solving several exciting examples. The application of this package allows users to explore and understand the geometry of the analyzed implicit curves and surfaces. This results in a helpful computer tool of great value for educational purposes and to improve the visualization of these concepts.