CYCLOIDS

April 9Leah Justin

Sections B21 and A17Undergraduate Seminar : Braselton/ Abell

A Parametric Reinvention of the WheelA Super Boring and Very Plain Presentation

BAM! JUST KIDDING! IT IS NOT BORING AT ALL!

TODAY’S OBJECTIVES:1) EAT AND PLAY WITH OUR FOOD

2) INTRODUCE ROULETTES:SPECIFICALLY - CYCLOIDS

3) WALK THROUGH BASIC PROOFS OF AWESOME CYCLOID PROPERTIES

4) SPOIL SOMEONE ELSE’S PRESENTATION ON THE BRACHISTOCHRONE PROBLEM

You should have a candy bag….

Included in your bag:

twizzler pull-and-peeloreo

chewy spreesmint

Don’t eat yet… but if you really can’t help it. Have a spree

• Parametric Equations represent a curve in

terms of one variable using multiple equations

• Equation of a circle:

• x2+y2=r2

• Parametric Representation:

• x = r cos θ• y = r sin θ

What are Parametric Equations?

What is a Roulette?

A roulette is a curve created from a curve rolling along

another curve

Cycloid

The Parametric representation for a

cycloid is: x = a (θ - sin θ)y = a (1 – cos θ)

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.

MORE… YOU ASK???FAMOUS MINDS THAT

WORKED ON THE CYCLOID:• Galileo• Mersenne• Descartes• Torricelli• Fermat• Roberval• Huygens• Bernoulli

• Christopher Wren

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.

Christiaan Huygens

Huygens concluded

an interesting

property about the

cycloid

– tautochrone

property

Astronomer/

Physicist/

Mathematician

Huygens published this in his treatise called Horologium oscillatorium (“The

Pendulum Clock”).

1629-1695

“discovered” the mind of Leibniz

“Cosmotheros”

“Traite de la lumiere”

“De rationiis in ludo aleae”

“Principia Philosophiae”

Martian day is

approximately 24

hoursEarly ideas of the conservation of energy

tautochrone 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.

More Interesting Results:

• 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

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

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

curve

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

expand

integrate and evaluate

remember the cycloid equations

remember cos2 = ½ (1+cos )Θ Θ

substitute, combine like terms, simplify

change bounds of integration. Solve for dx/dΘ

substitute y = a(1-cos ):θ

3πa2 is 3 times the area of rolling circle, πa2

integrate to find area under a curve;

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

remember the cycloid equations

expand, substitute then factor using identity:

cos2 + sinΘ 2 =1Θ

square dx/d and dy/d and addΘ Θ

remember the arc length integral for parametric equations

find derivatives with respect to Θ

half-angle formula

integrate and evaluate

8a is 4 times the diameter (2a) of rolling circle

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

HypocycloidA hypocycloid is the curve traced out by a point on the edge of a circle rolling on the inside of a fixed circle

An astroid is a

hypocycloid of

4 cusps. A cusp is where a cycloid touches the fixed curve the circle rolls on

EpicycloidAn epicycloid is the curve traced out by a point on the edge of a circle rolling on outside of a fixed circle

A cardiod is the curve traced out by a point on the edge of a circle rolling around a circle of the same size.

CardiodAn nephroid is an

epicycloid of 2

cusps.

Brachistochrone Problem

• Which smooth curve connecting two points in a plane would a particle slide down in the shortest amount of time?

• FIRST GUESS?Anyone think of a straight line?

Makes sense, right? The shortest distance between two points?

Brachistochrone Problem

The fastest curve is the cycloid curve!

References• Wikipedia• Wolfram Mathworld• http://scienceblogs.com/startswithabang/upload/2010/05/how_far_to_the_stars/(7-01)Huygens.jpg• http://blog.algorithmicdesign.net/acg/parametric-equations• http://www.dailyhaha.com/_pics/crazy_illusion.jpg• http://www.proofwiki.org/wiki/Area_under_Arc_of_Cycloid