Note 10. Coordinates adapted to the projected ellipse

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

 

.