Affine transformations of circle and sphere

TL;DR

This paper investigates the average deformation of circles and spheres under linear transformations, providing explicit mean values in 2D and estimates in 3D, enhancing understanding of geometric distortions.

Contribution

It computes the mean deformation coefficient for 2D circles and estimates it for 3D spheres under linear transformations, a novel analysis in geometric deformation.

Findings

Mean deformation coefficient in 2D is explicitly calculated.

An estimation for the mean deformation coefficient in 3D is provided.

The work advances understanding of geometric distortions under linear operators.

Abstract

A non-degenerate two-dimensional linear operator $\varphi$ transforms the unit circle into ellipse. Let $p$ be the length of its bigger axis and $q$ -- the length of smaller. We can define the deformation coefficient $k(\varphi)$ as $q/p$. Analogously, if $\varphi$ is a non-degenerate three-dimensional operator, then it transforms the unit sphere into ellipsoid. If $p>q>r$ are lengths of its axes, then deformation coefficient $k(\varphi)$ will be defined as $r/p$. In this work we compute the mean value of deformation coefficient in two-dimensional case and give an estimation of mean value in three-dimensional case.