Poncelet Plectra: Harmonious Curves in Cosine Space
This work addresses theoretical geometry problems related to Poncelet polygons and cosine conservation, likely incremental for mathematicians and geometric researchers.
The paper investigates the geometric properties of Poncelet N-gons in confocal pairs, showing that for N=3, cosine triples sweep a planar equilateral cubic curve resembling a plectrum, and that excentral triangles conserve the same product of cosines as their affine images inscribed in a circle, with cosine triples sweeping a spherical curve that becomes planar when using log-cosines.
It has been shown that the family of Poncelet N-gons in the confocal pair (elliptic billiard) conserves the sum of cosines of its internal angles. Curiously, this quantity is equal to the sum of cosines conserved by its affine image where the caustic is a circle. We show that furthermore, (i) when N=3, the cosine triples of both families sweep the same planar curve: an equilateral cubic resembling a plectrum (guitar pick). We also show that (ii) the family of triangles excentral to the confocal family conserves the same product of cosines as the one conserved by its affine image inscribed in a circle; and that (iii) cosine triples of both families sweep the same spherical curve. When the triple of log-cosines is considered, this curve becomes a planar, plectrum-shaped curve, rounder than the one swept by its parent confocal family.