The Cycloid

by Dinoj Surendran

The Cambridge mathematician Godfrey Hardy once said ``There is no room in the world for ugly mathematics'', ie maths is a discipline that studies objects of beauty. Objects of beauty? Surely the likes of Madonna don't appear in the subject? Of course they don't --- fortunately! For a mathematician, an object of beauty is something that can be described simply, but is absolutely - er - beautiful.

Take the circle for instance. It can be described simply but has so many great properties! (why else do you think coins are circular?) Trouble is, most people are so used to seeing circles that they take them for granted. Perhaps that's why the most beautiful object is generally taken to be the cycloid, often called ``The Helen of geometry'', after the legendary Greek beauty.

Cycloid? Not many people have heard of it. This is how it is constructed. Consider a circle rolling along a straight line. Mark a fixed point on it --- the curve created by that fixed point as the circle rolls is called a cycloid. Looking at the diagram below, which represents the position of a circle rolling along a line at four different times, you can see the cycloid traced out by the point A.

Your next question is obviously ``So what?'' Well, turn your cycloid upside-down so it is like a bowl. We now have a brachistochrone . This is the path down which a particle will travel in the shortest time. This problem was first proposed by John Bernouilli in 1696, soon after the birth of calculus. It was solved by John himself, his brother James , Guillaume L'Hopital , Gottfried Leibniz and Isaac Newton.

Newton solved it in a single night after returning home from work! To show that the inverted cycloid is indeed the brachistochrone, they had to minimize t, the time taken for a particle starting from rest to fall along an unknown path y = f(x) from a point (x₁,0) to (x₂,y), where t is given by

(Notice, you calculus minimum-maximum experts, that the variable here is the whole curve - you are not just minimizing t with respect to a number x as in ordinary calculus problems. That's why this problem was so challenging, and why it was the gateway into a whole new subject called the Calculus of Variations.)

Knowledge of a bit of calculus is sufficient to derive the above equation, assuming you ignore frivolous details like friction. Have a go at it! The details will be in the next issue. On to the next property. If you place a particle (eg a marble) at various points on a cycloidical bowl, it will always reach the bottom of the bowl in the same time! This is called the isochronous property, and was proved by the Dutch scientist Christian Huygens . To illustrate, look at the inverted cycloid below. P and Q are two very different points in every respect, but particles placed at both points will reach the bottom at the same time! Countless applications have been made of this.

Now for the part you've all been dreading. Deriving the equation of the cycloid... it's really very easy.

Looking at the diagram above, you can see that I have placed the cycloid on the x-y plane. I have also defined a parameter D (think of it as D for `Dheta'!) which is the angle that the circle has moved through. D ranges from 0 to 2Pi (we measure in radians, for reasons you'll see later). Let the co-ordinates of any `A' point be (x,y), and the radius of the circle be R.

y = R - RcosD

So for any point P(x,y), x = R(D- sinD) and y = R(1 - cosD), where 0 <= D <= 2Pi

The cycloid is only the beginning. We also have several other curves created by rolling a circle on a curve (a line is a curve, remember) and taking the locus of a fixed point on the circle. For instance, the hypocycloid is what you get by rolling a circle on the inside of a larger circle, while an epicycloid is obtained by rolling a circle on the outside of another circle.

For a bit more information, Xah Lee has a nice, but rather over-graphical, page on the cycloid:

Here's a question for you: On a train, travelling from Bulawayo to Harare, what parts are, at a given moment in time, travelling in the opposite direction? Answer is below this big blank space.
























Forget about sneaky answers like the waiter in the buffet car or the conductors - their net velocity would be in the direction of the train anyway - we're talking about mechanical parts of the train. A point on a wheel rim where it touches the rail? No,such a point would move in a cycloid, and is stationary at the moment of touching the rail. But think about a point on the edge of the outer rim which dips below the rail... it describes a curve like this: So in the brief period when it's below the track, it actually moves in the opposite direction to the train.

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