Triangular Orbits in Elliptic Billards: Loci of Points X(1)~X(100)
Dan Reznik
May, 2019
An elliptic billiard (black), its \(N=3\) caustic (dashed black), and a particular \(N=3\) orbit (blue) are shown below, as well as the following triangles derived from the orbit:
- medial: medians of each side
- orthic: feet of altitudes
- contact: incircle touchpoints with sides
- excentral: excenters
- extouch: excircle touchpoints with sides
- anticompl: anticomplementary triangle
- feuerbach: 9-point-circle touchpoints w excircles
The pictures below show the loci of \(X_1\) to \(X_{100}\) calculated for the orbit and the above derived triangles, \(a/b=1.5\).
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.
- \(X(100)\), the anticomplement of the the Feuerbach point of the intouch triangle sweeps the caustic.
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