Rectangles conformally inscribed in lines

TL;DR

This paper explores the geometric properties of parallelograms inscribed in four lines and introduces a model to analyze the flow of such inscribed rectangles in three-dimensional space.

Contribution

It provides a detailed geometric description of conformally inscribed parallelograms and introduces a compact model for studying the flow of inscribed rectangles.

Findings

Describes the geometry of the space of conformally inscribed parallelograms.

Introduces a compact model for the rectangle inscription problem.

Analyzes the flow of inscribed rectangles in a new geometric framework.

Abstract

A parallelogram is conformally inscribed in four lines in the plane if it is inscribed in a scaled copy of the configuration of four lines. We describe the geometry of the three-dimensional Euclidean space whose points are the parallelograms conformally inscribed in sequence in these four lines. In doing so, we describe the flow of inscribed rectangles by introducing a compact model of the rectangle inscription problem.