[contravariant]

Yeah I think you're right, they're pretending the projection is a linear transformation (in cartesian coordinates) and using it to transform the Gaussian.

Or viewed alternatively they're approximating the projection by assuming all of the Gaussian is at a fixed depth, which I suppose works if it is far enough away.

A projective transformation of a Gaussian seems somewhat annoying, though I assume someone will have done it before. Seems like it should be possible to do it with projective coordinates but the final projection to cartesian coordinates is tricky.

For what it's worth, projecting a contour is also wrong, the whole density changes which also affects the contours.