We provide a theory as to why the locus of a triangle center over Poncelet 3-periodics in an ellipse pair is an ellipse or not. For the confocal pair (elliptic billiard), we show that if the  center can be expressed as a fixed affine combination of barycenter, circumcenter, and mittenpunkt (which is stationary over the confocal family) , then its locus will be an ellipse. We also provide conditions under which a particular locus will be a circle or a segment. We also analyze locus  turning number and monotonicity with respect to vertices of the 3-periodic family. Finally we write out expressions for the convex quartic locus of the incenter for a generic Poncelet family, conjecturing it can only be an ellipse if the pair is confocal  .
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 .
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provide a theory as to why the locus
propose a theory which predicts the ellipticity
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over Poncelet 3-periodics in an ellipse pair is an ellipse or not. For the confocal pair (elliptic billiard), we
's locus over a Poncelet 3-periodic family. We
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triangle
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mittenpunkt (which is stationary over the confocal family)
a third, stationary point over some family
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We also provide
Taking billiard 3-periodics as an example, the third point is the mittenpunkt. We derive
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particular locus will be a circle or a segment. We also analyze
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degenerates to a segment or is a circle. We show a locus
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and
is either plus or minus 3 and predict its movement
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we write out expressions for the convex quartic locus
, we derive a (long) expression for the loci
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for
and excenters over
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family, conjecturing it
3-periodic family, showing they are roots of a quartic. We conjecture (i) those loci are convex, and (ii) that they
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an ellipse
ellipses
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, i.e., within a 1d subspace of the 5d space of ellipse pairs which admit 3-periodics