Strange curves
Have you every wondered what the fastest slide would be?
This problem was presented as a maths and physics problem and named the Brachistochrone problem by Galileo. It asks for the most optimum shape of a curve in which “a bead sliding from rest and accelerated by gravity will slip (without friction) from one point to another in the least time” (Wolfram maths). Put simply, he wanted to find the shape of the fastest slide.
A cycloid ended up being the solution to the problem. What they found was the hypotenuse — the shortest distance between two points — was not the best curve. Nor was the path traveling straight down — which maximised acceleration from gravity — with a curve at the turning point. In fact, it was the cycloid which gave the fastest route despite the bead having to travel a longer distance.
https://en.wikipedia.org/wiki/Brachistochrone_curve
Cycloids are created by tracing a point on a circumference of a circle as it travels along a straight line. Imagine the trail a large pencil stuck into the edge of a tire would create as it rolled along.
In addition, a property of the curve means that if you let 3 different beads on three different cycloid curve start from different places, they will all reach the end at the same time.
The equation for a the first hump of a cycloid on a cartesian plane (range 0 to 2a inclusive) is
More commonly used is the parametric equation. It expresses x as a function of time t, and y as a function of time t. Very simply, it takes time t, as a parameter rather than express a direct relationship between x and y.
If you are interested there is more information here. For some visuals, here and a video.
Thinking back to your childhood when you used pencil stencil art (Spirographs) to create doodles — you were unknowingly creating beautiful pieces of complex geometry.
Karina Walker
