The talented Mr. Inversive Triangle in the elliptic billiard

Reznik, Dan; García, Ronaldo; Helman, Mark

doi:10.1007/s40879-021-00489-2

Cited by 5 publications

(5 citation statements)

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“…In [21,22,34,37,38] loci of triangle centers are studied over various 1d triangle families, Poncelet or otherwise. Works [29,30] follow in their footsteps and identify new curious properties and invariants of Poncelet families. Proofs that loci of certain triangle centers in over the confocal family are ellipses appear in [9,10,13,32].…”

Section: Related Workmentioning

confidence: 99%

“…Right: As one sweeps the triangle family, the incenter changes position: X1, X 1 , X 1 , sweeping a locus. Video 1, Video 2 Over the same family, the mittenpunkt X9 (purple) is stationary [30], the symmedian point X6 (red) sweeps a quartic [13], the Feuerbach point X11 (green) is a curve identical to the caustic (inner ellipse), and X59 sweeps a self-intersected curve (blue), studied in [29]. app triangle centers in [19], over one dozen Poncelet families, including the ones of Figure 3.…”

Section: The Locus Rendering Appmentioning

confidence: 99%

“…7: Left: elliptic loci of the incenter X1 (red), barycenter X2 (green), circumcenter X3, and orthocenter X4 over Poncelet triangles between two confocal ellipses. Right:Over the same family, the mittenpunkt X9 (purple) is stationary[30], the symmedian point X6 (red) sweeps a quartic[13], the Feuerbach point X11 (green) is a curve identical to the caustic (inner ellipse), and X59 sweeps a self-intersected curve (blue), studied in[29]. app…”

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The Wrought Iron Beauty of Poncelet Loci

Reznik¹

2022

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We've built a web-based tool for the real-time interaction with loci of Poncelet triangle families. Our initial goals were to facilitate exploratory detection of geometric properties of such families. During frequent walks in my neighborhood, it appeared to me Poncelet loci shared a palette of motifs with those found in wrought iron gates at the entrance of many a residential building. As a result, I started to look at Poncelet loci aesthetically, a kind of generative art. Features were gradually added to the tool with the sole purpose of beautifying the output. Hundreds of interesting loci were subsequently collected into an online "gallery", with some further enhanced by a graphic designer. We will tour some of these byproducts here. An interesting question is if Poncelet loci could serve as the basis for future metalwork and/or architectural designs.

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Section: Related Workmentioning

confidence: 99%

Section: The Locus Rendering Appmentioning

confidence: 99%

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The Wrought Iron Beauty of Poncelet Loci

Reznik¹

2022

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“…In [13] triangle centers whose locus over 3-periodics was the billiard boundary were called "swans". As it was shown, both X 88 and X 100 are swans [13]. Curiously, the former's (resp.…”

Section: Proposition 8 the Locus Of X †mentioning

confidence: 99%

The Talented Mr. Inversive Triangle in the Elliptic Billiard

Reznik,

Garcia,

Helman

2020

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Inverting the vertices of elliptic billiard N-periodics with respect to a circle centered on one focus yields a new "focus-inversive" family inscribed in Pascal's Limaçon. The following are some of its surprising invariants: (i) perimeter, (ii) sum of cosines, and (iii) sum of distances from inversion center (the focus) to vertices. We prove these for the N = 3 case, showing that this family (a) has a stationary Gergonne point, (b) is a 3-periodic family of a second, rigidly moving elliptic billiard, and (c) the loci of incenter, barycenter, circumcenter, orthocenter, nine-point center, and a great many other triangle centers are circles.

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“…This is a continuation of our investigation of Euclidean phenomena of Poncelet families [11,12,14,20]. Recall Poncelet's porism: specially-chosen pairs of conics C , C admit a one-parameter family of polygons inscribed in C while simultaneously circumscribed about C [5,7,8].…”

Section: Introductionmentioning

confidence: 96%