标题： 圆和球的仿射变换 显示英文标题

标题： Affine transformations of circle and sphere

摘要： 一个非退化的二维线性算子 $\varphi$ 将单位圆变换为椭圆。 设 $p$ 为其长轴长度， $q$ 为其短轴长度。 我们可以定义形变系数 $k(\varphi)$ 为 $q/p$。 类似地，若 $\varphi$ 是一个非退化的三维算子，则它将单位球变换为椭球。 若$p>q>r$是其轴的长度，则变形系数$k(\varphi)$将定义为$r/p$。 本文中我们计算了二维情况下变形系数的均值，并给出了三维情况下均值的一个估计。 显示英文摘要

摘要： 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.