Faithful real-time animation of parametrized (semi-) algebraic expressions via cylindrical algebraic decomposition (Q6536548)

The goal of this note is to present a significant improvement implemented in the free software package GeoGebra Discovery. It can be said that GeoGebra Discovery is an experimental version of GeoGebra, which enhances and expands certain automatic reasoning tools from the standard version. For readers unfamiliar with GeoGebra, it is a free dynamic mathematics system with advanced dynamic geometry capabilities, and I encourage you to discover it at \url{https://www.geogebra.org/classic?lang=in}.\N\NWith GeoGebra Discovery, the authors have faced several computational challenges, such as addressing the graphic representation of curves and semi-algebraic sets in the real plane; rigorously preserving the topology of the plotted objects; and accepting as input a) a specific algebraic equation or a (semi-)algebraic formula, or b) a collection of such formulas or equations depending on a parameter, which can be used to animate the plotting of the collection of geometric objects, or c) the result of a locus computation that might (or might not) depend on different parameters.\N\NIn this brief note, the authors present the implementation of the methods described in [\textit{D. S. Arnon} et al., in: Quantifier elimination and cylindrical algebraic decomposition. Proceedings of a symposium, Linz, Austria, October 6--8, 1993. Wien: Springer. 152--165 (1998; Zbl 0900.03050); \textit{C. W. Brown}, Constructing cylindrical algebraic decompositions of the plane quickly. (2002), \url{ttps://www.usna.edu/Users/cs/wcbrown/research/MOTS2002.1.pdf}], using the cylindrical algebraic decomposition tools provided by the software Tarski [\textit{F. Vale-Enriquez} and \textit{C. W. Brown}, Lect. Notes Comput. Sci. 10931, 466--474 (2018; Zbl 1395.68353)], which allows fluid animation of sets of (semi-)algebraic sets of dimension 1 in a desktop application or a web browser. The Tarski library is used to create a cylindrical algebraic decomposition of the input, and its plot2d command, which is processed further in GeoGebra Discovery, to provide the user with a familiar look and feel.