The Plasticity of Some Fittable Surfaces on a Given Quadruple of Points in the Three-Dimensional Euclidean Space

We construct a two-dimensional sphere in the three-dimensional Euclidean space which intersects a circular cylinder in three given points and the corresponding weighted Fermat–Torricelli point for a geodesic triangle such that these three points and the corresponding weighted Fermat–Torricelli point remain the same on the sphere for a different triad of weights which correspond to the vertices on the surface of the sphere. We derive a circular cone which passes from the same points that a circular cylinder passes. By applying the inverse weighted Fermat–Torricelli problem for different weights, we obtain the plasticity equations which provide the new weights of the weighted Fermat–Torricelli point for fixed geodesic triangles on the surface of a fittable sphere and a fittable circular cone with respect to the given quadruple of points on a circular cylinder, which inherits the curvature of the corresponding fittable surfaces.