$q-$spherical surfaces in Euclidean space

TL;DR

This paper introduces $q$-spherical surfaces in Euclidean space, focusing on their algebraic properties, singular points, and visualization, expanding understanding of these geometric objects.

Contribution

It defines $q$-spherical surfaces, analyzes their singularities, and provides algebraic equations and visualizations, which are novel contributions to geometric surface theory.

Findings

Identified surfaces with one and two $n$-fold points.

Derived algebraic equations for these surfaces.

Visualized the surfaces using Mathematica.

Abstract

In this paper we define $q$-spherical surfaces as the surfaces that contain the absolute conic of the Euclidean space as a $q-$fold curve. Particular attention is paid to the surfaces with singular points of the highest order. Two classes of such surfaces, with one and two $n-$fold points, are discussed in detail. We study their properties, give their algebraic equations and visualize them with the program {\it Mathematica}.