Yesterday’s parametric curve was a warm up for another I wanted to draw. I’ve played with it a little bit, in the same spirit as yesterday’s, but haven’t gotten an answer yet. It looks sufficiently more complicated, so I thought I’d just go ahead and post my work so far up here, and see if anybody else was interested. Here’s a picture, explanation follows:

The curve I want is the one in the upper left. I’ve decided to base this on the graph of cosine again, so I am thinking

The two other graphs in the picture, on the right, are approximations to (on top) and (on the bottom).

While scares me a bit, I’ve got an idea for finding a formula for . Near multiples of , the curve looks vaguely like a sine curve with small amplitude and period (so, ). Near odd multiples of , the curve looks more like the curve from yesterday, . So we want to sort of mix these curves, more of the first near -s, and more of the second near -s.

A decent way to mix two functions, is to use what I’ll call a mixing function (probably other people have a name for it, and probably it’s something I should know, but whatever). The only requirement I have for a function to be a mixing function is that it takes values between 0 and 1 (and continuity is preferred). I’ll mix the two functions as . Notice that when is near 1, the mixed function is near , and when is closer to 0, the mixed function is near .

It’s not too hard to determine that to mix my two sine curves like I want, my will be based on , namely . You can make the mixing ‘quicker’ (more time near each curve, and quick transitions to the other function) via for small positive values of (I’m thinking for some positive odd integer ). To get some idea how this looks, here’s a graph mixing with , using (the basic , fooplot doesn’t seem to want to take odd roots of negative values):

I think that by modifying the various parameters (), one could get a pretty decent function for the I was looking for. Like I said, though, looks a bit harder. Perhaps you’ve got a suggestion? I’m tempted to believe you could base it off of … It might almost be a mixing of with (or various other amplitudes on both).

I feel like maybe I should mention why I was thinking these curves, and why I want to draw that parametric curve. When I was playing with my quadric polynomials a while ago, I wanted to draw animations, but I had two parameters floating around. So I thought that if I made a parametric curve that occupied ‘most’ of some square region in the plane, then I could use just one parameter, and make a nice animation. Probably I’m not explaining that too well, but it also isn’t entirely worth it. My goal was to draw a curve that occupied ‘lots’ of the region in the plane, and the curve I came up with is the one I’ve been trying to draw in this post.