A Theory for Locus Ellipticity of Poncelet 3-Periodic Centers

We propose a theory which predicts the ellipticity of a triangle center's locus over a Poncelet 3-periodic family. We show that if the triangle center can be expressed as a fixed affine combination of barycenter, circumcenter, and a third, stationary point over some family, then its locus will be an ellipse. Taking billiard 3-periodics as an example, the third point is the mittenpunkt. We derive conditions under which a locus degenerates to a segment or is a circle. We show a locus turning number is either plus or minus 3 and predict its movement monotonicity with respect to vertices of the 3-periodic family. Finally, we derive a (long) expression for the loci of the incenter and excenters over a generic Poncelet 3-periodic family, showing they are roots of a quartic. We conjecture (i) those loci are convex, and (ii) that they can only be ellipses if the pair is confocal, i.e., within a 1d subspace of the 5d space of ellipse pairs which admit 3-periodics.