The Hofstadter ellipses are a family of triangle ellipses introduced by P. Moses in February 2005. The Hofstadter ellipse 
for parameter 
is defined by the trilinear equation
 |
(1)
|
where
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
The association with Douglas R. Hofstadter comes through the Hofstadter point family. For the same parameter 
, the Hofstadter
point has trilinear coordinates 
, where 
and 
and 
are defined cyclically (Kimberling 1994). The coefficients
in the ellipse equation satisfy 
, 
, and 
.
The ellipses 
and 
are identical. They are plotted above for 
, 0.2, ..., 0.5.
The center of the Hofstadter ellipse 
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)]](/images/equations/HofstadterEllipse/NumberedEquation2.svg) |
(5)
|
(P. Moses, pers. comm., Feb. 13, 2005), which does not correspond to any Kimberling center.
The Hofstadter ellipse 
is an inellipse given by
 |
(6)
|
and passes through Kimberling centers 
for 
, 678, 2310, 2632, 2638, and
2643. It has center
 |
(7)
|
corresponding to Kimberling center 
, which is the crosspoint
of the incenter 
and triangle centroid 
. It has area
 |
(8)
|
where 
is the area of the reference triangle.
Taking the limit as 
(or 
) gives the circumellipse 
with trilinear equation
 |
(9)
|
This has center with trilinear center function
 |
(10)
|
and its fourth intersection with the circumcircle (other than the vertices 
, 
, 
) given by triangle center function
 |
(11)
|
