Cycloids: A PARAMETRIC REINVENTION OF THE WHEEL

A Super Cool and Awesome Presentation By: Leah Justin, Undergraduate Seminar 2012 Abell/Braselton

What are Polar Coordinates?

Polar Coordinates are a way to represent points in a plane using an angle (measured from the x-axis ) and distance (r, for radius from the origin)

Parametric Equations?

Parametric Equations represent a curve in terms of one variable using multiple equations

Equation of a circle:

x2+y2=r2

Parametric Representation:

x = rcosθ

y = r sin θ

What is a Roulette? A roulette is a curve created from a curve rolling along another curve

What is a Cycloid?

A cycloid is a roulette; it is a curve traced out by a point on the edge of a circle rolling on a line in a plane. The Parametric representation for a cycloid is:

x = r (θ - sin θ)

y = r (1 – cosθ)

Famous Minds that worked on the Cycloid:

Galileo

Mersenne

Descartes

Torricelli

Fermat

Roberval

Huygens

Bernoulli

Christopher Wren

Christiaan Huygens

Astronomer/ Physicist/ Mathematician

Sometimes called Holland’s Greatest Scientiest, Huygens was from one of the centers of intellect in the 17th century. Holland was filled with printing companies, so many minds came to publish there. In relation to the cycloid, Huygens concluded an interesting property: on an inverted arch of a cycloid, a ball released anywhere on the side of the bowl will reach the bottom in the same time. This is called the tautochrone property. Huygens published this in his treatise called Horologiumoscillatorium (“The Pendulum Clock”).

Properties of the Cycloid:

The area under one arch of a cycloid is 3 times that of the rolling circle

The length of one arch of the cycloid is 4 times the diameter of the rolling circle

A flexible pendulum constrained by cycloid curves swings along a path that is also a cycloid curve

The tangent of a cycloid passes through the top of the rolling circle

Historical Background: Helen of Geometers?

Mathematicians fought over the cycloid just like the Greeks and Trojans fought over Helen of Troy. Both Helen, and the Cycloid are beautiful, however it was tough to get a handle on. The cycloid would become such a topic of dispute, that it earned this reputation as “Helen” in the 1600’s. Galileo named the “cycloid” because of its circle-like qualities.

What is the Brachistochrone Problem?

Which smooth curve connecting two points

In a plane would a particle slide down

in the shortest amount of time?

SPOILER ALERT: THE CYCLOID CURVE

Application: Better Roller Coasters