,
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 semi-major axis is thus
|
|
while the length of the semi-minor 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
|
|
.
.
.
,
,
,
.