Rational function as a locus??
Help me out here. I think I can define a kind of rational function as a locus of points, but my locus definition is still too long and confusing.
It all started with these doodles I like to make showing two points (A and B) and parallel lines going through those points. At the endpoints (A' and B') of any perpendicular to the parallel lines I'd connect the points A and B' and then B and A' so that they cross. Then I'd track the arc traced by the intersection of AB' and BA'. In this drawing to the left you can see the case where the lines from A and B are parallel, so no intersection... that is the the same point the function becomes undefined (I think) when you are working with it using algebra. In any case, this is great fun if you are in a long boring meeting. Try it some time.
At first I didn't know if it was a curve. Ha! I thought it'd be a line! But it's never *looked* like it was linear...
Then, I tried using a little algebra and found the parametric equation:
Xt=tm/(n+2t)
Yt=t(t+n)/(n+2t)
n/m is the slope from A to B (not reduced... though, I don't know if it matters.)
After solving for t in both parts of the parametric equation I found a general rational function. That part was quite a mess and I still need to check my answer. But, whatever it is, it's not a poly-function.
Bigger and better constructions held up this idea as some lovely rational curves emerged:
OK. So, now the question is, how can I define these curves as a locus. I want it to be shorter and sweeter than what I've written above. Are there and fun geometric applications for the rational function? (I mean a situation that mirrors the requirements I've described for these curves.)
Also, where can I learn more about this topic?
Thanks to anyone who can help!