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
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 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)
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
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
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
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
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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