Related by Similiarity: Poristic Triangles and 3-Periodics in the Elliptic Billiard
This work connects classical geometry to dynamical systems, providing insights for mathematicians and physicists, but it is incremental as it builds on known families and transforms.
The paper tackles the relationship between Poristic triangles and 3-periodics in the elliptic billiard, showing they are mapped via a similarity transform, which implies that scale-free quantities and invariants in one family hold in the other.
Discovered by William Chapple in 1746, the Poristic family is a set of variable-perimeter triangles with common Incircle and Circumcircle. By definition, the family has constant Inradius-to-Circumradius ratio. Interestingly, this invariance also holds for the family of 3-periodics in the Elliptic Billiard, though here Inradius and Circumradius are variable and perimeters are constant. Indeed, we show one family is mapped onto the other via a varying similarity transform. This implies that any scale-free quantities and invariants observed in one family must hold on the other.