AI 中文总结
研究圆锥投影，结合自准线和共焦-共准线定理与三个经典事实，通过倒数极性转化参数律，利用固定射线与自准线族交点处法线包络抛物线及直接从原始点和线构造切线，实现圆锥投影相关的一些构造。
AI 中文摘要
设\[ f_R(x)=\frac{x}{1+\norm{x}/R},\qquad R>0, \]为平面到开圆盘$\DR=\{x:\norm{x}<R\}$的径向圆锥投影。先前工作建立了自准线和共焦-共准线定理，据此$f_R$将焦点圆锥弧映射为焦点圆锥弧，同时保留特征焦点和准线。本笔记将这些结果与W.~H.~Besant记录的三个经典事实相结合。首先，关于$\partial\DR$的倒数极性将非线性圆锥参数律转化为圆半径的基本平移$q\mapsto q+R$。其次，固定射线与自准线族交点处的法线包络一条显式抛物线。第三，$f_R(z)$处的切线可直接从原始点$z$和原始线构造，无需对$f_R$求导或求解圆锥曲线。
英文摘要
Let $$ f_R(x)=\frac{x}{1+\norm{x}/R},\qquad R>0, $$ be the radial cone projection of the plane onto the open disk $\DR=\{x:\norm{x}<R\}$. Previous work established the Self-Directrix and Confocal-Codirectrix Theorems, according to which $f_R$ maps focal conic arcs to focal conic arcs while preserving the distinguished focus and directrix. This note combines those results with three classical facts recorded by W.~H.~Besant. First, reciprocal polarity with respect to $\partial\DR$ represents every focal conic under consideration as the reciprocal polar of a unique circle, and in this circle representation the nonlinear action of $f_R$ becomes the elementary radius translation $q\mapsto q+R$. Second, the normals at the intersections of a fixed ray with the self-directrix family envelope an explicit parabola. Third, the tangent at $f_R(z)$ is constructed directly from the original point $z$ and the original line, without differentiating $f_R$ or solving for the conic.
Comments11 pages, 5 figures; exposition changes
