CYCLOIDS A Parametric Reinvention of the Wheel A

CYCLOIDS A Parametric Reinvention of the Wheel A Super Boring and Very Plain Presentation April 9 Leah Justin Sections B 21 and A 17 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

What are Parametric Equations? • Parametric Equations represent a curve in terms of one variable using multiple equations • Equation of a circle: • x 2+y 2=r 2 • Parametric Representation: • x = r cos θ • y = r sin θ

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

A cycloid Cycloid is a roule tte; it is a out by a p curve trac oint on th ed e edge of on a line i a circle ro n a plane. lling The Parametric representation for a cycloid is: x = a (θ - sin θ) y = a (1 – cos θ)

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 r/ e m ono Astr cist/ n i Phys ematicia h Mat “disc o mind vered” th e of Le ibniz Martian day is 4 approximately 2 hours Early i de conse as of the rv energ ation of y ros” ere” e h t i ” mo “Cos de la lum do aleae lu te “Trai tioniis in phiae” a so “De r ipia Philo c “Prin 1629 -169 5 luded c n o c s n e g Huy ing an interest t the u o b a y t r e prop cycloid rone – tautoch property Huyg ens p ublish in his ed th treat i s e call is Horo ed logiu oscill m atori u m (“T Pend ulum h Clock e ”).

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 cos 2 Θ = ½ (1+cos Θ) integrate and evaluate 3πa 2 is 3 times the area of rolling circle, πa 2

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: cos 2 Θ + sin 2 Θ =1 half-angle formula integrate and evaluate 8 a is 4 times the diameter (2 a) 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 a s i id of o r t s id a o l An ocyc hyp sps. A cusp u is whe c 4 re a cycloid t fixed c ouches the ur rolls o ve the circle n

Epicycloid An epicycloid is the curve traced out by a point on the edge of a circle rolling on outside of a fixed circle Cardiod is an d i o r ph An ne id of 2 lo epicyc. cusps 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