Note 3. The projected curve is an ellipse in standard form.

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

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

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 semi-major axis has length > 1, and the semi-minor axis has length 1. Therefore every similarity class of ellipse is obtained in this way. These classes are parameterized by the length of the semi-major axis.

Where are the foci? We calculate

and so the foci are located at points

and