Poncelet Plectra: Harmonious Properties of Cosine Space

Poncelet N-periodics in a confocal pair (elliptic billiard) conserve the same sum of cosines as their affine image with fixed incircle. For N=3, the vector of cosines in either family sweep the same planar curve: an equilateral cubic resembling a plectrum (guitar pick). We also show that the family of excentral triangles to the confocal family conserves the same product of cosines as its affine image with fixed circumcircle. In cosine space cosine triples in either family sweep the same spherical curve. The associated planar curve in log cosine space is also plectrum-shaped, though rounder than the one swept by its parent confocal family.