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Hofstadter Ellipse


HofstadterEllipse

The Hofstadter ellipses are a family of triangle ellipses introduced by P. Moses in February 2005. The Hofstadter ellipse E(r) for parameter 0<r<1 is defined by the trilinear equation

 alpha^2+beta^2+gamma^2+betagamma(D+1/D)+gammaalpha(E+1/E) 
 +alphabeta(F+1/F)=0,
(1)

where

D=cosA-sinAcot(rA)
(2)
E=cosB-sinBcot(rB)
(3)
F=cosC-sinCcot(rC).
(4)

The ellipses E(r) and E(1-r) are identical. They are plotted above for r=0.1, 0.2, ..., 0.5.

The center of the Hofstadter ellipse E(r) is given by triangle center function

 alpha=4a-a(D+1/D)^2-2b(F+1/F)-2c(E+1/E)+(D+1/D)[b(E+1/E)+c(F+1/F)]
(5)

(P. Moses, pers. comm., Feb. 13, 2005), which does not correspond to any Kimberling center.

The Hofstadter ellipse E(1/2) is an inellipse given by

 alpha^2+beta^2+gamma^2-2(alphabeta+betagamma+gammaalpha)=0
(6)

and passes through Kimberling centers X_i for i=244, 678, 2310, 2632, 2638, and 2643. It has center

 alpha_(37)=b+c,
(7)

corresponding to Kimberling center X_(37), which is the crosspoint of the incenter O and triangle centroid G. It has area

 A=(piabc)/((ab+bc+ca)^(3/2))Delta,
(8)

where Delta is the area of the reference triangle.

HofstadterEllipse0

Taking the limit as r->0 (or r->1) gives the circumellipse E(0) with trilinear equation

 (abetagamma)/A+(bgammaalpha)/B+(calphabeta)/C=0.
(9)

This has center with trilinear center function

 alpha=a((b^2)/B+(c^2)/C-(a^2)/A),
(10)

and its fourth intersection with the circumcircle (other than the vertices A, B, C) given by triangle center function

 alpha=a/(A(B-C)).
(11)

See also

Hofstadter Point, Hofstadter Triangle

Explore with Wolfram|Alpha

References

Kimberling, C. "Encyclopedia of Triangle Centers: X(359)=Hofstadter One Point." http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X359.

Referenced on Wolfram|Alpha

Hofstadter Ellipse

Cite this as:

Weisstein, Eric W. "Hofstadter Ellipse." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HofstadterEllipse.html

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