Envelopes created by sphere families in Euclidean 3-space

TL;DR

This paper addresses fundamental problems related to envelopes formed by sphere families in Euclidean 3-space, providing solutions to existence, representation, enumeration, and definitional relationship issues.

Contribution

It offers a comprehensive resolution to four core problems concerning envelopes generated by sphere families in three-dimensional Euclidean space.

Findings

Solved the existence problem for envelopes.

Developed representation methods for these envelopes.

Determined the number of possible envelopes and their relationships.

Abstract

In this paper, on envelopes created by sphere families in Euclidean 3-space, all four basic problems (existence problem, representation problem, problem on the number of envelopes, problem on relationships of definitions) are solved.