Cycloidy = R - a cos(b t) 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 MathBrachistochrone Problem: Find the curve such that a bead slides without friction between two points in the least possible time. MathWorld link Wikipedia linkIsochrone 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. 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. e-mail: Jim.Swift@nau.edu |