Conformal dual of a quadruplet of points

TL;DR

This paper introduces a conformal duality for quadruplets of points in three-dimensional space, demonstrating that duality is an involution and relating cross ratios through complex conjugation.

Contribution

It defines a conformal dual for quadruplets in $S^3$ and proves properties like involution and the relationship of cross ratios, advancing geometric understanding.

Findings

Dual of a dual quadruplet is the original quadruplet.

Cross ratio of dual quadruplet is the complex conjugate of the original.

Establishes conformal invariance properties of quadruplets.

Abstract

Let $\{P_1, P_2, P_3, P_4\}$ be a quadruplet of points in $S^3$ . We define a ``dual'' quadruplet of it in a conformal geometric way. We show that the dual of a dual quadruplet coincides with the original one. We also show that the cross ratio of the dual quadruplet is equal to the complex conjugate of that of the original one.