Recalling again that the gradient of at is
, 
we may represent the tangent plane as the graph of a function of x and y:

If we intersect the graph with the standard cone
, 
we see with a little algebra that
(N3.1) 
. 
This is the projection of in the plane. Its image under the affine function is .
What sort of curve is defined by (N3.1) ? For our purposes, we need only look at the quadratic form that defines it in the plane,
. 
The coefficients of the terms are both positive, and the discriminant

is certainly negative. So the curve in the plane is an ellipse.
Now, let us give a parametric representation of so that we can see that all similarity classes of ellipses in the plane are represented by the of this construction. We will see in particular that and do not in general belong to the same similarity class, but that they do have a simple relationship.
The projected ellipse consists of the points of the form
(N3.2) 
. 
If we let

and
, 
with , then the parametric equation for is
where , , and .
This is obviously an ellipse with center and with major axis directed along the line through the origin generated by . The semimajor axis has length > 1, and the semiminor axis has length 1. Therefore every similarity class of ellipse is obtained in this way. These classes are parameterized by the length of the semimajor axis.
Where are the foci? We calculate
, 
and so the foci are located at points

and
. 