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
. |