# Is the projection of an ellipse still an ellipse? [duplicate]

I’ve been trying to convince myself of the veracity of this for a while and I can’t seem to do it.

Given two planes in R3, will an ellipse on one plane necessarily project to an ellipse on the other?

If the planes are parallel, it’s trivially true. Similarly, if the ellipse is a circle, I know it projects to an ellipse. But if they’re not either of those cases? Assuming an orthographic projection, it seems to me the projection could end up some skewed, egg-like shape where the ellipse axes’ intersections point has been shifted away from the center of the ellipse.

Can someone please shed some light on this?

A parallel projection from a plane to another is an affine transformation (it turns a parallelogram into another parallelogram).

If you plug the affinely transformed coordinates in the equation of the ellipse, you still get a quadratic equation. And as the point set is bounded, it must be another ellipse.

Even a perspective projection can preserve an ellipse. Indeed, analytically it is expressed by an homographic transformation, which is of the form

$$x'=\frac{ax+by+c}{gx+hy+i},y'=\frac{dx+ey+f}{gx+hy+i}$$

and when you plug this into a quadratic equation, you still get a quadratic equation. Anyway, due to the possibly cancelling denominators, you can get an hyperbola as well.