Cycloidal Excursions

1. Cycloidal Excursions by Alexander Tir and John de St. Germain

2. What is the path followed by a point on a wheel as it rolls along a straight line on level ground?

3. Answer: It’s a cycloid curve!

4. The cycloid has been a subject of study because of its unique properties. Two of these properties are: • The length of one arc is exactly 4 times the diameter of the generating circle. • The area of under the cycloid is 3 times the area of the generating circle. This is astonishing because these ratios are both very simple, yet the curve itself is completely non-circular.

5. If two balls are dropped from any point in a cycloidal bowl they will both arrive at the bottom of the bowl at exactly the same instant. Therefore, the cycloid is the tautochrone, or curve of equal descent. If a ball were to travel along any path from Point A to a lower Point B under the influence of gravity alone, which path would the ball move from Point A to B in the least possible time? The quickest path is a cycloid curve; therefore, the curve is the brachistochrone, or curve of quickest descent.

6. ThePendulum Suppose that we invert the cycloid curve and tie a string to the beginning and ending points of two cycloid curves equal to the length of exactly twice the diameter of the generating circle. In other words, the string is half the length of one arc. If we unwind the string, keeping it taut, it traces out another cycloid curve. This is called an involute. A simple pendulum swings with a period approximately independent of the amplitude. To adjust the rate of a pendulum, you would adjust the length. Is there any kind of pendulum whose period is exactly the same for all amplitudes? The answer is yes; if the length of the string is the same as the string used to generate an involute of the cycloid curve, then the time stays constant for any small or wide swing.

7. Here are some more curves related to the cycloid.

8. The prefix hypo- implies that the generating circle is rolling on the inside of another circle. Here is an example of a hypocycloid: More curves can be generated with circles of smaller diameter.

9. The prefix epi- implies that the generating circle is rolling on the outside of another circle. Here is an example of a epicycloid: More curves can also be generated with circles of smaller diameter.

10. The prefix tro- implies that the generating circle creates a loop or curve rather a cusp. Here are examples of trochoids: The curve is created by any point extended from the center of the generating circle.

11. The Reuleaux Triangle The basis of a cycloid is a circle, a curve of constant width; however, a circle is not the only one. When equal arcs are drawn from three vertices of an equilateral triangle and each arc with radius equal to the side of the triangle, a shape called the Reuleaux triangle is formed. Cylindrical pipes are often used as rollers to move crates or other heavy objects. A roller in the shape of a Reuleaux triangle can be used just as well. Since its cross-section is of constant width, the crate would roll along absolutely smoothly without any friction or up and down motion!

12. The End