  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.
