The points of the form
, 
define the Apollonian curve.
To project this ellipse, consider the unit vectors in the plane of , defined as
(N10.1) 


From the definitions of , it follows that the projections of the vectors of the form
into the plane of have the form
(N10.2) 
. 

The center of the ellipse is the projection of :
. 
The endpoints on the major axis of the ellipse are
. 
The length of the semimajor axis is thus
, 
while the length of the semiminor axis is 1.
Recalling the standard form for an ellipse for which ,
, 
we see that the foci of this ellipse must be at
, 
because the distance from the center to the focus is easily seen from these observations to be
. 