Hyperplane families creating envelopes

TL;DR

This paper introduces a geometric mechanism involving intersections of perpendicular bisectors and normal lines to solve fundamental problems related to envelopes of hyperplane families, providing a unified approach.

Contribution

It presents a novel application of a simple geometric mechanism to address core problems in the theory of hyperplane envelopes, unifying multiple issues.

Findings

Provides solutions to existence, representation, equivalence, and uniqueness problems.

Demonstrates the mechanism's effectiveness across various hyperplane envelope scenarios.

Offers a new geometric perspective for analyzing hyperplane families.

Abstract

A simple geometric mechanism: "the locus of intersections of perpendicular bisectors and normal lines", often arises in many guises in Nonlinear Sciences. In this paper, a new application of this simple geometric mechanism is given. Namely, we show that this mechanism gives answers to all four basic problems on envelopes created by hyperplane families (existence problem, representation problem, equivalence problem of definitions, uniqueness problem) at once.