Abstract
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.
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Notes
The full result admits that the polygon sides be tangent to up to N separate conics in the linear pencil defined by the outer one and a member of the second set (PCT) (del Centina 2016).
Shorthand for inscribed while simultaneously circumscribing.
The affine image which sends the elliptic billiard to a circle also produces an N-periodic family whose invariant cosine product is equal to that conserved by the outer polygons of the confocal pre-image.
A triangle’s orthic has vertices at the feet of the altitudes (Weisstein 2019, Orthic Triangle).
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Acknowledgements
We would like to thank A. Akopyan for pointing out the relationship of some of our results with Akopyan et al. (2020, Thm 6.4).
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Jaud, D., Reznik, D. & Garcia, R. Poncelet plectra: harmonious curves in cosine space. Beitr Algebra Geom 63, 115–131 (2022). https://doi.org/10.1007/s13366-021-00596-x
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DOI: https://doi.org/10.1007/s13366-021-00596-x