The Cardioid

The CardioidThe CardioidBy Stuart BlankenshipBy Stuart Blankenship

&&Hans ParshallHans Parshall

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.”

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

A Special Case Of…A Special Case Of…

The Limaçon The Limaçon

A Special Case Of…A Special Case Of…

The EpicycloidThe Epicycloid

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.

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

Trace of an EpicycloidTrace of an Epicycloid

Trace of an EpicycloidTrace of an Epicycloid

Trace of an EpicycloidTrace of an Epicycloid

The EvoluteThe Evolute

• Cardioid• Evolute• Radius of the Circle of Osculation