论文标题
相似性II相关：poncleet三个周期和brocard porism
Related by Similarity II: Poncelet 3-Periodics in the Homothetic Pair and the Brocard Porism
论文作者
论文摘要
以前，我们在椭圆台球（共聚焦对）中表明了3个周期性的家族是在可变三角形（具有非浓度，固定固定incircle and Crabcircle）的可变相似性变换下的图像。两个家庭都保留了伊拉迪乌斯与圆周的比例，因此也保留了余弦之和。这与一个相似性保持角度的事实一致。在这里，我们研究了两个新的poncleet 3周期家庭也通过可变相似性彼此绑定：（i）第一个属于一对同心，同性恋椭圆形中的第一个，以及（ii）第二个非脑子，称为brocard porism：固定的包围和Brocard inellipse。这个家族的Brocard点处于Inellipse的焦点位置。一个关键的常见不变是brocard角，因此是cotangents的总和。这提出了一个有趣的问题：鉴于一个非集中的poncleet家族（限制或不限制外圆锥为圆），是否总是在同心，轴平衡的椭圆和/或圆锥对中找到类似的doppelgänger？
Previously we showed the family of 3-periodics in the elliptic billiard (confocal pair) is the image under a variable similarity transform of poristic triangles (those with non-concentric, fixed incircle and circumcircle). Both families conserve the ratio of inradius to circumradius and therefore also the sum of cosines. This is consisten with the fact that a similarity preserves angles. Here we study two new Poncelet 3-periodic families also tied to each other via a variable similarity: (i) a first one interscribed in a pair of concentric, homothetic ellipses, and (ii) a second non-concentric one known as the Brocard porism: fixed circumcircle and Brocard inellipse. The Brocard points of this family are stationary at the foci of the inellipse. A key common invariant is the Brocard angle, and therefore the sum of cotangents. This raises an interesting question: given a non-concentric Poncelet family (limited or not to the outer conic being a circle), can a similar doppelgänger always be found interscribed in a concentric, axis-aligned ellipse and/or conic pair?
