September 3, 2012

Rolling Circles and Balls (Part 2)

John Baez

Last time we rolled a circle on another circle the same size, and looked at the curve traced out by a point on the rolling circle:

It's called a cardioid.

But suppose we roll a circle on another circle that's twice as big. Then we get a nephroid:

Puzzle 1. How many times does the small circle rotate as it rolls all the way around the big one here?

By the way, the name 'cardioid' comes from the Greek word for 'heart'. The name 'nephroid' comes from the Greek word for a less noble organ: the kidney! But the Greeks didn't talk about cardioids or nephroids---these names were invented in modern times.

Here are my 7 favorite ways to get a nephroid:

1) The way just described: roll a circle on a circle twice as big, and track the path of a point.

2) Alternatively, take a circle that's 3/2 times as big as another, fit it around that smaller one, roll it around, and let one of its points trace out a curve. Again you get a nephroid!

3) Take a semicircle, point it upwards, shine parallel rays of light straight down at it, and let those rays reflect off it. The envelope of the reflected rays will be half of a nephroid:

This was discovered by Huygens in 1678, in his work on light. He was really big on the study of curves.

As I mentioned last time, a catacaustic is a curve formed as the envelope of rays emanating from a specified point and reflecting off a given curve. We can stretch the rules a bit and let that point be a 'point at infinity'. Then the rays will be parallel. So, we're seeing that the nephroid is a catacaustic of a circle.

Last time we saw the cardioid is also a catacaustic of a circle, but with light emanating from a point on the circle. It's neat that the cardioid and nephroid both show up as catacaustics of the circle. But it's just the beginning of the fun...

4) The nephroid is the catacaustic of the cardioid, if we let the light emanate from the cardioid's cusp!

This was discovered by Jacques Bernoulli in 1692.

5) Let two points move around a circle, starting at the same place, but with one moving 3 times as fast as the other. At each moment connect them with a line. The envelope of these lines is a nephroid!

Last time we saw that if we replace the number 3 by 2 here, we get a cardioid. So, this is yet another way these two curves are related! 6) Draw a circle in blue and draw its diameter as a vertical line. Then draw all the circles that have their center somewhere on that blue circle, and are tangent to that vertical line. You get a nephroid:

The red circle here has the red dot as its center, and it's tangent to a point on the vertical line. Here's a nice animation of the process, made available by the Math Images Project under a GNU Free Documentation License:

7) Finally, here.s how to draw a nephroid starting with a nephroid! Draw all the osculating circles of the nephroid---that is, circles that match the nephroid.s curvature as well as its slope at the points they touch. The centers of these circles give another nephroid:

This trick is an example of an 'evolute'. The evolute of a curve is the set of centers of the osculating circles of that curve. Last time we saw the evolute of a cardioid is another cardioid. Now we're seeing the nephroid shares this property!

Apparently the same is true for all curves formed by rolling one circle on another. These curves are called epicycloids. In a sense, these are the mathematical leftovers of the theory of epicycles in astronomy.

It would be nice if some of the funny relations we've been seeing between the cardioid and the nephroid generalize to relations between the epicycloid with \(k\) cusps and the one with \(k+1\) cusps. But I don't know if that's true.

It would also be nice if the epicycloids with more and more cusps were named after increasingly disgusting organs of the body. But in fact, I don't know any special names for them once we reach \(k = 3\).

Puzzle 2. Use one of the 7 constructions above to get an equation for the nephroid. What is the simplest equation you can find for this curve?

Next time

Most of the pictures above are from Wikicommons, but the picture of the nephroid as a catacaustic of the cardioid is from Xah Lee's wonderful website on plane curves. As usual, you can click on the pictures and get more informatino.

My ultimate goal is to tell you some amazing things about what happens when you roll one ball on another that's exactly 3 times as big. These things have nothing to do with plane curves, actually. But I've been taking many detours, and next time I'll talk about some curves formed by rolling one circle inside another!

Right now, thought I need a break. I need to stop thinking about all these curves. I think I'll get a cup of coffee.

You can see answers to the puzzles on on Azimuth, and make your own comments or ask questions there!

© 2012 John Baez