My brain was not functioning at all well today, but I still ground through almost all of the remaining exercises on parametric curves. It was fairly miserable, but I am proud to have done it. I did skip one exercise that I simply couldn’t get a handle on. I may or may not return to it another day.
I’m now ready to start the review exercises for the chapter on functions. After I have finished those, and possibly some of the challenge problems that follow them, I will be able to start my calculus review in earnest with the chapter on limits and derivatives. Hooray!
The most interesting problem I worked on today concerned the alternative construction of an ellipse that is illustrated in the diagram below.
It turns out that the figure traced by point $P$ as $\theta$ varies from $0$ to $2\pi$ is the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. (The red right triangle always has legs parallel and perpendicular to the x-axis and changes proportions as $\theta$ varies. It does not tilt in order to maintain its proportions.) That this is so can be shown by using parametric equations to define the curve traced by $P$, then converting them to the Cartesian equation for the ellipse. I thought this was pretty neat, and I wonder if anything interesting would come out of examining the relationship between this construction and the usual one for an ellipse.
(I may demonstrate how to show the figure is an ellipse at some time in the future. Let me know if you are interested.)
How cool. Once more thing you’ve taught me. I guess the result isn’t shocking once you’ve seen the picture. The curve has to go through $(\pm a, 0)$ and $(0,\pm b)$, and it probably has to be some sort of quadratic thing. An ellipse would be the simplest choice. But it’s not a result I knew.
You worked before on properties of ellipses (showing that sound emitted at one focus bounced to the other, I think?). My instinct is that this new description won’t make it easier to prove that, but I could be wrong.
Yes, if you know that the point traces a familiar curve, it’s not difficult to guess which one it is.
I conjectured that the angles of incidence and reflection of the sound/light/whatever are equal when measured relative to the line tangent to the ellipse at the point of reflection. I showed the similar fact for parabolas, but I haven’t gotten to ellipses yet.