Undergraduate Seminar : Braselton/ Abell CYCLOIDS A Parametric Reinvention of the Wheel A Super Boring and Very Plain Presentation April 9 Leah Justin Sections B21 and A17 Undergraduate Seminar : Braselton/ Abell

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-peel oreo chewy sprees mint Don’t eat yet… but if you really can’t help it. Have a spree

x2+y2=r2 x = r cos θ y = r sin θ What are 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 = r cos θ y = r sin θ What are Parametric Equations?

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

The Parametric representation for a cycloid is: 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 = a (θ - sin θ) y = a (1 – cos θ)

MORE… YOU ASK??? 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. FAMOUS MINDS THAT WORKED ON THE CYCLOID: Galileo Mersenne Descartes Torricelli Fermat Roberval Huygens Bernoulli Christopher Wren

Christiaan Huygens Astronomer/ Physicist/ Mathematician 1629-1695 Huygens concluded an interesting property about the cycloid tautochrone property “discovered” the mind of Leibniz Martian day is approximately 24 hours Early ideas of the conservation of energy Huygens published this in his treatise called Horologium oscillatorium (“The Pendulum Clock”). “Cosmotheros” “Traite de la lumiere” “De rationiis in ludo aleae” “Principia Philosophiae”

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 remember the cycloid equations integrate to find area under a curve; substitute y = a(1-cosθ): change bounds of integration. Solve for dx/dΘ substitute, combine like terms, simplify expand remember cos2 Θ = ½ (1+cos Θ) integrate and evaluate 3πa2 is 3 times the area of rolling circle, πa2

The length of one arch of the cycloid is 4 times the diameter of the rolling circle remember the cycloid equations find derivatives with respect to Θ remember the arc length integral for parametric equations square dx/dΘ and dy/dΘ and add expand, substitute then factor using identity: cos2 Θ + sin2 Θ =1 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

Hypocycloid A 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

Epicycloid An epicycloid is the curve traced out by a point on the edge of a circle rolling on outside of a fixed circle An nephroid is an epicycloid of 2 cusps. Cardiod A cardiod is the curve traced out by a point on the edge of a circle rolling around a circle of the same size.

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