A cycloid is the path traced out by a point on the rim of a rolling wheel. If we put the cusp of the cycloid at the origin, (x, y) = (0,0), and put the point at the cusp at t = 0, then the parametric equations for the curve are
x = R b t - a sin(b t)
y = R - a cos(b t)
The parameters a and b can be changed with the scrollbar within DPGraph. The radius of the wheel is fixed at R = 1. The distance from the axel to the dot that draws the curve is a, so the curve is not really a cycloid when a is changed from the default value of a = 1. The speed of the wheel is proportional to b.

For your convenience, I saved separate DPGraph files with the initial parameter a = 0, a = 0.5, a = 1 (the true cycloid), and a = 2. (You can get the same effect using the scrollbar with the original graph, but the scrollbar feature doesn't work on the classroom computers.)

Two Famous Problems from the History of Math

Brachistochrone Problem: Find the curve such that a bead slides without friction between two points in the least possible time. MathWorld link   Wikipedia link

Isochrone Problem = Tautochrone Problem: Find the curve such that a bead slides without friction to the lowest point from any other position in the same amount of time. MathWorld link     Wikipedia link

The curve that solves both of these problems is a cycloid with the cusp pointing up. Christiaan Huygens solved the isochrone problem without calculus, using Euclidean geometry. The result was published in his Horologium Oscillatorium (the pendulum clock) in 1673. The result was later proved using the calculus of variations, spawning a new branch of mathematics. Huygens used two cycloids to constrain the motion of the pendulum bob to a cycloid. The graphs here animate Huygens's solution to the isochrone problem, where the parameter A is the amplitude of motion. At the initial amplitude of A = 1 (the maximum amplitude) the dpgraph animation shows the solution to the brachistochrone problem. Click on "Edit" within dpgraph to get additional information about the files, and click on "Scrollbar" to adjust the parameters.

[isochrone]    [isochrone]    [isochrone]
The period of the oscillation is 12 seconds in the dpgraphs, but this can be adjusted with the parameter B. The period of a phyical isochrone is T = sqrt(8 πW/g), where W is the width from cusp to cusp and g is the acceleration of gravity. The width would have to be about 183 feet for the period to be 12 seconds!

These graphs were produced by DPGraph, which is a fast program for viewing 3D objects. NAU students can download the program for free, since the Math/Stat department bought a site license.

Jim Swift's DPGraph page     Jim Swift's home page     Department of Mathematics     NAU