## A Trickier Curve

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

$\begin{array}{rcl}x(t) &=& t+o_x(t)\\ y(t) &=& \cos(t)+o_y(t) \end{array}$

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

While $o_y(t)$ scares me a bit, I’ve got an idea for finding a formula for $o_x(t)$. Near multiples of $\pi$, the curve looks vaguely like a sine curve with small amplitude and period $\pi/6$ (so, $c_1\cdot \sin(12t)$). Near odd multiples of $\pi/2$, the curve looks more like the curve from yesterday, $c_2\cdot \sin(2t)$. So we want to sort of mix these curves, more of the first near $\pi$-s, and more of the second near $\pi/2$-s.

A decent way to mix two functions, $f_1(t),f_2(t)$ is to use what I’ll call a mixing function $\mu(t)$ (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 $f(t)=\mu(t)f_1(t)+(1-\mu(t))f_2(t)$. Notice that when $\mu(t)$ is near 1, the mixed function is near $f_1(t)$, and when $\mu(t)$ is closer to 0, the mixed function is near $f_2(t)$.

It’s not too hard to determine that to mix my two sine curves like I want, my $\mu$ will be based on $\cos(2t)$, namely $\mu(t)=(1/2)(1+\cos(2t))$. You can make the mixing ‘quicker’ (more time near each curve, and quick transitions to the other function) via $\mu_p(t)=(1/2)(1+\cos^{p}(2t))$ for small positive values of $p$ (I’m thinking $p=1/n$ for some positive odd integer $n$). To get some idea how this looks, here’s a graph mixing $(.2)\sin(12t)$ with $(1)\sin(2t)$, using $\mu_1$ (the basic $\mu$, fooplot doesn’t seem to want to take odd roots of negative values):

I think that by modifying the various parameters ($c_1,c_2,p$), one could get a pretty decent function for the $o_x(t)$ I was looking for. Like I said, though, $o_y(t)$ looks a bit harder. Perhaps you’ve got a suggestion? I’m tempted to believe you could base it off of $-\cos(t)$… It might almost be a mixing of $-\cos(t)$ with $(1/2)\cos(t)$ (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 $-1\leq y\leq 1$ in the plane, and the curve I came up with is the one I’ve been trying to draw in this post.