Download - The Cardioid
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The CardioidThe CardioidBy Stuart BlankenshipBy Stuart Blankenship
&&Hans ParshallHans Parshall
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What does it mean?What does it mean?
Cardioid, coming Cardioid, coming from the Greek from the Greek roots ‘cardi’ for roots ‘cardi’ for ‘heart’ and ‘-oid’ ‘heart’ and ‘-oid’ for ‘resembling’, for ‘resembling’, roughly translates roughly translates to “heart-shaped.”to “heart-shaped.”
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Who Was Involved?Who Was Involved?
Ole Christensen RømerOle Christensen Rømer 1717thth century Danish century Danish
astronomerastronomer Made the first quantitative Made the first quantitative
measurements of the measurements of the speed of light speed of light
Studied the cardioid to find Studied the cardioid to find the best form of gear teeththe best form of gear teeth
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A Special Case Of…A Special Case Of…
The Limaçon The Limaçon
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A Special Case Of…A Special Case Of…
The EpicycloidThe Epicycloid
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Parametric EquationsParametric EquationsThe parametric equationsThe parametric equationsxx((tt)= )=
22rr(cos(costt – (1/2)*cos – (1/2)*costt))yy((tt)=)=
22rr(cos(costt – (1/2)*cos – (1/2)*costt))create a cardioid similar to create a cardioid similar to
the one shown on the left.the one shown on the left.rr is the radius of each of the is the radius of each of the
circles.circles.
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Other PropertiesOther Properties
Polar EquationPolar Equationrr((θθ)=)=aa(1-cos(1-cosθθ))
where where aa is the radius is the radius of one of the two of one of the two circles in the circles in the previous slideprevious slide
AreaArea
(3/2(3/2)*pi*a)*pi*a
LengthLength
8*8*aa
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Trace of an EpicycloidTrace of an Epicycloid
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Trace of an EpicycloidTrace of an Epicycloid
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Trace of an EpicycloidTrace of an Epicycloid
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The EvoluteThe Evolute
• Cardioid• Evolute• Radius of the Circle of Osculation