Comments on: Rolling Circles and Balls (Part 2) http://johncarlosbaez.wordpress.com/2012/09/03/rolling-balls-and-circles-part-2/ Sat, 18 May 2013 10:57:29 +0000 hourly 1 http://wordpress.com/ By: Rolling Circles and Balls (Part 4) « Azimuth http://johncarlosbaez.wordpress.com/2012/09/03/rolling-balls-and-circles-part-2/#comment-23532 Wed, 02 Jan 2013 00:38:12 +0000 http://johncarlosbaez.wordpress.com/?p=11899#comment-23532 In Part 2 we rolled a circle on a circle that’s twice as big [...]

]]> By: John Baez http://johncarlosbaez.wordpress.com/2012/09/03/rolling-balls-and-circles-part-2/#comment-19771 Tue, 11 Sep 2012 11:29:25 +0000 http://johncarlosbaez.wordpress.com/?p=11899#comment-19771 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.

]]> By: Rolling Circles and Balls (Part 3) « Azimuth http://johncarlosbaez.wordpress.com/2012/09/03/rolling-balls-and-circles-part-2/#comment-19761 Tue, 11 Sep 2012 09:18:10 +0000 http://johncarlosbaez.wordpress.com/?p=11899#comment-19761 In Part 1 and Part 2 we looked at the delightful curves you get by rolling one circle on another. Now let’s see what happens when you roll one circle inside another!

]]> By: arch1 http://johncarlosbaez.wordpress.com/2012/09/03/rolling-balls-and-circles-part-2/#comment-19743 Mon, 10 Sep 2012 21:05:35 +0000 http://johncarlosbaez.wordpress.com/?p=11899#comment-19743 It’s likely redundant w/ comments already made, but I find this stuff easier to think about if I add in a non-black (say, blue) radius from the center of the rolling cirle to the point of tangency. This more graphically distinguishes rotating reference frame from what’s happening relative to that frame: In general, the black radius crosses the blue one n times as the blue radius executes a single rotation, thus the black radius executes n+1 rotations.

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:-)

]]> By: P. Bloom (@pbloemesquire) http://johncarlosbaez.wordpress.com/2012/09/03/rolling-balls-and-circles-part-2/#comment-19736 Mon, 10 Sep 2012 19:04:33 +0000 http://johncarlosbaez.wordpress.com/?p=11899#comment-19736 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.

Let’s try those images again:

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.

]]> By: Greg Egan http://johncarlosbaez.wordpress.com/2012/09/03/rolling-balls-and-circles-part-2/#comment-19713 Mon, 10 Sep 2012 08:10:49 +0000 http://johncarlosbaez.wordpress.com/?p=11899#comment-19713 That applet’s great! And there are links from that page to other versions for all the conics, for both parallel and radial rays.

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:

x^4 + y^4 =1

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

]]> By: John Baez http://johncarlosbaez.wordpress.com/2012/09/03/rolling-balls-and-circles-part-2/#comment-19709 Mon, 10 Sep 2012 05:03:10 +0000 http://johncarlosbaez.wordpress.com/?p=11899#comment-19709 All your comments are very interesting to me, Greg, but also rather intimidating…. especially because I feel some algebraic geometers out there must have been studying this question for at least a century, and I’m reluctant to dive in and compete with them without even the help of Mathematica.

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.

]]> By: Greg Egan http://johncarlosbaez.wordpress.com/2012/09/03/rolling-balls-and-circles-part-2/#comment-19682 Sun, 09 Sep 2012 09:40:38 +0000 http://johncarlosbaez.wordpress.com/?p=11899#comment-19682 For the record, all my dimensions of polynomial spaces above are too large by a factor of 2. The actual dimension of the space of bivariate polynomials of degree at most d is (d+1)(d+2)/2.

Unfortunately, this has no effect on the results! Hopefully the next error I find will actually improve the bounds …

]]> By: Greg Egan http://johncarlosbaez.wordpress.com/2012/09/03/rolling-balls-and-circles-part-2/#comment-19681 Sun, 09 Sep 2012 07:53:48 +0000 http://johncarlosbaez.wordpress.com/?p=11899#comment-19681 Though the bounds I’ve derived above are much too high, my earlier guess that a curve of degree d would have a catacaustic of degree 2d is much too low. A generic catacaustic for an ellipse has degree 6!

You can see pictures at MathWorld. That article doesn’t give any formulas for the generic case, but the picture clearly shows two cusps (like a nephroid, which is degree 6), and working through the algebra it’s possible to show that the degree really is 6.

]]> By: Greg Egan http://johncarlosbaez.wordpress.com/2012/09/03/rolling-balls-and-circles-part-2/#comment-19677 Sun, 09 Sep 2012 05:53:00 +0000 http://johncarlosbaez.wordpress.com/?p=11899#comment-19677 I can get lower-degree expressions for X and Y by expanding out the definitions of A(x,y) and B(x,y). The numerator polynomials are then:

2 \left(-2 x \frac{\partial P}{\partial x} \frac{\partial P}{\partial y} \left(r^2 \frac{\partial ^2P}{\partial x\, \partial y}+y \frac{\partial P}{\partial x}\right)+\left(\frac{\partial P}{\partial y}\right)^2 \left(r^2 x \frac{\partial ^2P}{\partial x^2}-y^2 \frac{\partial P}{\partial x}\right)+x \left(\frac{\partial P}{\partial x}\right)^2 \left(r^2 \frac{\partial ^2P}{\partial y^2}-x \frac{\partial P}{\partial x}\right)\right)

and

2 \left(-\frac{\partial P}{\partial x} \frac{\partial P}{\partial y} \left(2 r^2 y \frac{\partial ^2P}{\partial x\, \partial y}+x^2 \frac{\partial P}{\partial x}\right)+y \left(\frac{\partial P}{\partial y}\right)^2 \left(r^2 \frac{\partial ^2P}{\partial x^2}-2 x \frac{\partial P}{\partial x}\right)+r^2 y \left(\frac{\partial P}{\partial x}\right)^2 \frac{\partial ^2P}{\partial y^2}-y^2 \left(\frac{\partial P}{\partial y}\right)^3\right)

and the denominator is:

-\frac{\partial P}{\partial x} \frac{\partial P}{\partial y} \left(4 r^2 \frac{\partial ^2P}{\partial x\, \partial y}+y \frac{\partial P}{\partial x}\right)+\left(\frac{\partial P}{\partial y}\right)^2 \left(2 r^2 \frac{\partial ^2P}{\partial x^2}-x \frac{\partial P}{\partial x}\right)+\left(\frac{\partial P}{\partial x}\right)^2 \left(2 r^2 \frac{\partial ^2P}{\partial y^2}-x \frac{\partial P}{\partial x}\right)-y \left(\frac{\partial P}{\partial y}\right)^3

The degrees of the numerators are now bound by 3n-1, and the denominator by 3n-2, so we now have s=d(3n-1). But that still only bounds the degree of the catacaustic of a quadratic to be at most 18.

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