We consider triangular orbits in elliptic billiards and the loci of their triangular centers [1], specifically \(X(1)\) to \(X(100)\) as defined in [2], for a billiard with \(a/b=1.5\). In the pictures below, the billiard is shown in black, loci of the orbit’s center in blue, and loci of the orbit’s extriangle shown in green, referred below as \(X'(n)\).

Notes on the Loci:

\(X'(6)\), the excentral Symmedian, is congruent with \(X(9)\), the Mittenpunkt, and both loci are a point.
\(X(9)\), the Mittenpunk, fixed at the origin.
\(X(11)\), the Feuerbach point, tracing the caustic.
\(X(14)\), 2nd Isogonic, closely tracking the billiard.
\(X(30)\), Euler Infinity Point, is the intersection of the Euler line with the line at infinity. Is there any interesting info here?
\(X(37)\), the crosspoint of incenter and centroid, seem to generate identical perpendicular loci, with major semiaxis = \(1/2\), i.e, one third of the original.
\(X'(41)\) generates a nearly circular locus.
\(X'(46)\) horizontal axis is exactly 1.
\(X(49)\) has cusps.
\(X(50)\) has lines to infinity at intriguing directions.
\(X'(56)\) is an almost perfect circle, std. dev. of radius within 0.75% of its average.
\(X(59)\) and \(X'(59)\) are marvellously self-intersecting.
\(X(67)\) tracks the billiard.
\(X(73)\) is diamond-shaped, though smooth.
\(X(74)\) has north-south cusps.
\(X(76)\) Brocard point ellipse is almost flat along \(y\).
\(X(77)\) nicely self-intersecting.
\(X(87)\) nicely self-intersecting.
\(X(88)\) locus = billiard.
\(X(89)\) axes almost vanish, Mittenpunk-style. Why?
\(X(92)\) pillow-shaped, with north and south inward concavities.
\(X(93)\) has interesting escape directions.
\(X(94)\) has a high-order curvature function.
\(X(94)\) has a very small, hourglass locus.
\(X(99)\), the Steiner point, almost perfectly tracks the billiard. What is the reason for this close tracking?
\(X(100)\), the anticomplement of the Feuerbach point, has the billiard as locus.

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Triangular Orbits in Elliptic Billards: Loci of Points X(1)~X(100)

Dan Reznik

May, 2019

Notes on the Loci:

References