Conveniently, we can define variance ellipsoids in such a way that
projections of variance ellipsoids of a random variable correspond to the
variance ellipsoids of the projection of the random variable. In particular,
projections of the variance spheroid of 
onto 
correspond to the variance ellipsoids of projections of y
. We have used the cylinder to depict a nonorthogonal projection of the
sphere onto the plane of 
.
In Variance Ellipsoid Projections, we show that the variance ellipsoid of Py is equivalent to the image of the variance ellipsoid of y under the linear transformation P. This is a key element of the geometric interpretation: that we can visualize the variance ellipsoid of a projection as the projection of the variance ellipsoid.
The cylinder has the same radius as the sphere and the intersection of the cylinder with the plane corresponds to the variance ellipsoid of the projector of y along the axis of the cylinder.
One can see that the orthogonal projection of the spheroid produces a
variance ellipsoid in 
that is contained within all other
possible projections. Smaller variance spheroids correspond to smaller
variance matrices. The conclusion of the Gauss-Markov Theorem corresponds to
the minimal projection of the sphere onto the plane.
