Three Dollar Quill

I haven't yet buckled down and experimented with the formulas I've been deriving, but I have been thinking about them, and some of those thoughts have been promising.

One thing I did was try to figure out whether the formulas for ellipse coefficients can convey any useful information in the event that the "full" version collapses due to linear dependence. The answer so far is a resounding maybe. If the source ellipse passes through the origin, then I believe the quadric collapses to a linear equation, which is promising, since the projection of an ellipse passing through the origin is a line.

But what I want to know is what kind of quadrics appear if the source ellipse doesn't pass through the origin. Suppose we set x to 0, confining the ellipse to the yz plane, and therefore confining the projection to the y axis. Let's put the center at y = 0, and make the axes the y and z unit vectors, with the center at some z greater than 1. Like 2. This causes the A coefficient to take on a non-zero value, and the C and F coefficients to take on zero values. Regardless of what is happening with the other coefficients, this is not the equation of an ellipse. So what is it? Well, all other coefficients in this restricted case are simply 0. So, what we have is a strange equation of a line. This line includes the desired segment, but also a bunch of other stuff.

My intuition could be leading me astray, but it's currently having me ask "so, can we say something about the rates of change of the coefficients?" In other words, if we subtract out the non-zero coefficients, does what remains have some interesting infinitesimal behavior? If we slightly perturb the x value of one of the axes, then some of the coefficients take on values close to but distinct from zero, sufficient to qualify as an ellipse. The coefficient A is unaffected by this process, which means that its rate of change is zero. The rates of change therefore form another non-ellipse, but one with quantitative and perhaps qualitative differences. My intuition is really hopeful that this will produce an equation that can be factored into a pair of parallel lines that are perpendicular, or at least non-parallel, to the original line, but I'm not at all confident in this idea.

By the way, numerical stability was one of the things I studied for my degree; I just kind of forgot what I learned.

One thing that it just occurred to me to wonder is if the numerical instability can be "dodged" by plugging in the formulas for the implicit form coefficients directly into the formulas for the canonical forms. The hope I have with this is that the "segment" form would fall out directly in the case of linear dependence, in the form of a minor axis length of zero. This is something that I can try without too much trouble, but it's late enough that I don't want to try right now.

Right now, I need to get to bed.

Good night.