Hah!

One part of the "neat thing" I discovered last week about my Taxonomy paper involves a certain polynomial equation in variables x and y, with a parameter k; the curve defined by that equation, for any specific value of k, helps describe a class of triangles associated with that parameter k. (For a simple example, which is not the one I'm dealing with, you might look at x^2 - 2 k x + y^2 = 0; for any given k, this gives a circle with radius Abs(k) and center (k,0).) What I was interested in is how that curve changes as I change k. (Let's see, if k < 1, it's a fourth-degree oval; if k = 1, it's an ellipse; if 1 < k < 2, it's a fourth-degree oval sitting inside a sort of peanut-shaped curve; if k = 2, it's the union of two circles: etc.)

I was investigating this curve for one specific value of k at a time, and it was a tedious process. I had to find out whether the curve was in two pieces or just one, what the largest and smallest values of x were, what y was in terms of x... it took me several minutes to assemble the graph for each value of k. What I really wanted to see is how the curve changes when I change k from 3/2 to 7/4, or from this value to that nearby value, and the process just wasn't conducive to that kind of close investigation.

I finally figured out how to do it quickly. I've written a single Mathematica routine, plotLocus[k], which does it all at once and shows the graph without me having to check the fiddly details. So far, what it's showing me confirms what I thought was going on, which, of course, is what I was hoping it would do, but in terms of creating figures to go into the paper, suddenly everything is much easier.

I'm pretty happy about that.