cycloids april 9 leah justin sections b21 and a17 undergraduate seminar : braselton/ abell a...
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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
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