Reflecting the directrix over the parabola

@James Wilson – I'm still trying to think of a "proper" way to do this, but the way I plotted the previous image was more or less as you described: I sampled points on the circle, then worked out where they would reflect to through the parabola. As you say, you get a cubic doing this. Points "inside" the parabola lie on multiple normals, so have multiple images under reflection: this corresponds to when the cubic has more than one real root.

If the point on the circle is $P(\cos \theta,\sin \theta)$ , we first need to find $u$ such that $P$ lies on the normal to the parabola through $(u,u^2)$ . The gradient of this normal is $-\frac{1}{2u}$ ; so we need to solve $\frac{u^2-\sin \theta}{u-\cos \theta}=-\frac{1}{2u}$

ie find the roots of $f(u)=u^3+\frac{u}{2}-u \sin\theta - \frac12 \cos\theta=0$

Newton-Raphson works well to find one of these roots, say $u_1$ . Say the other roots are $u_2,u_3$ . Then by Vieta, $u_2+u_3=-u_1$ and $u_2 u_3 = \frac{\cos\theta}{2u_1}$

This gives a quadratic in $u_2,u_3$ . Checking the discriminant shows whether the remaining roots are real.

So now we've got (approximations to) all the real roots; we just need to do the reflections, but this is easy; the image points are $(P_i' \left(2u_i-\cos \theta,2u_i^2-\sin\theta \right)$

---

Of course, this is definitely quick and dirty: it's not an elegant way of finding the roots, but it works well enough to see the shape of the curve, which is what I wanted. I did this in QBasic (!) but there are better languages to use that will already have root-finding libraries (eg Python, Mathematica, MatLab) and be far more accurate.

@James Wilson – Thanks, and likewise - it's definitely fun to explore this idea of reflections through a curve.

Now, I just revealed I was using QBasic there ;-), but depending on the type of animation you're after, gifsmos is great for animating graphs. Essentially it's desmos but you can save gifs from it.

Otherwise there are various free gif makers around that let you upload a deck of frames.

I'm probably not the best placed to advise on animations but some users on here are very good at them - it might be worth either replying to one of those or setting up a discussion on the topic (I'd like to get better at this too!). I seem to remember some very nice animated solutions to the daily problems - that might be a good starting point.