You’re right! The big circle has radius 3/2 times the radius of the small one here, not twice the radius:

Fixed! Thanks.

]]>As you can see, a cardioid can be made by starting with a circle, choosing one point on that circle, and making every other point a circle center. This collection of circles makes a cardioid. I did this same process, but instead of a cicrcle, I started with an equilateral triangle. This created a nephroid, which I put here on ImageShack:

http://imageshack.us/photo/my-images/837/nephroid.jpg/ ]]>

Wow! That’s a great picture and a scarily high degree. I had guessed that epicycloids are ‘generic’ when it comes to the degree of their catacaustics—mainly because if you guess that things are generic, then generically you’re correct. But it seems like I was way off, at least in the degrees you’ve investigated. So that means some potentially interesting mechanism is at work, to make their catacaustics have lower degree than average.

]]>When the rolling circle rolls on the inside of another circle (rather than on the outside), the two rotations cancel rather than add (I managed to puzzle myself for awhile trying to visualize what happens when that cancellation becomes perfect:-)

]]>I think your best bet for a limiting process would be the polynomial lemniscate, see: http://en.wikipedia.org/wiki/Polynomial_lemniscate and http://mathworld.wolfram.com/MandelbrotSetLemniscate.html.

The basic idea is that you take the iterating process behind the mandelbrot set for a finite amount of steps, set it equal to “large value” and solve explicitly. For one, two and three steps the functions are given at Mathworld and the number of terms grows very quickly. I suppose you’d have to let both the large value and the number of steps go to infinity to approximate the actual boundary of M, but there’s no laws against that.

]]>I think the bounds I’ve derived are correct, but still pretty naive; they ignore the correlations between the original polynomial and the various intermediate polynomials derived from it, treating those intermediates as if they were free to be any polynomial of the same degree.

Here’s one more nice example. For the quartic:

the catacaustic with the source at the origin has degree 28.

]]>I hadn’t known the catacaustic of an ellipse could have degree 6, or I wouldn’t have made some of my more optimistic guesses! Here’s a fun site where you can use an applet to move around through the ‘moduli space’ of ellipses and their catacaustics, but only in the case of parallel rays:

• Irina Boyadzhiev, GeoGebra Applet Constructing the Catacaustic of an Ellipse – Parallel Rays.

You’ve got to crank up the number of rays to see anything interesting.

It’s wonderful what computers have done to help explain classical topics in 2d geometry like this! It makes me want to dream up some new questions that’d help revitalize these subjects among professional mathematicians.

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