## Cycloidx, 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)
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
## 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 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 e-mail: Jim.Swift@nau.edu |