Cardioid

History

Roemer first conceived the cardioid in 1674 while investigating the best form for gear teeth.

The name cardioid comes from the Greek root cardi, which means heart.

The cardioid is an epycycloid and a special case of a limacon of Pascal, a family of curves studied and named after Pascal.

 car·di·oid

Pronunciation: 'kär-dE-"oid
Function: noun
Date: 1753
A heart-shaped curve that is traced by a point on the circumference of a circle rolling completely around an equal fixed circle.

Merriam-Webster OnLine.

 Parametric Equations

x = r cos 2t + 2r cos t

y = r sin 2t + 2r sin t

0 <= t <= 2 pi

 Deriving the Equations:

The circles c1 and c2 have the radius r and are centered at the origin and point D, respectively.

The point D is a distance of 2r from the origin, at the angle BAC. The coordinates of D are:

x_D = 2r cos BAC

y_D = 2r sin BAC

The point F is on the circle c2, which has a radius r and is centered at the point D. The angle is the angle BAC plus the angle EDF. Therefore the coordinates of point F are:

x_F = x_D + r cos (BAC + EDF)

y_F = y_D + r sin (BAC + EDF)

Circle c2 rolls around the fixed circle c1 without slipping. Therefore we know that the arc length from B to C must be equal to the arc length from E to F. If the arc lengths are equal then we know that the angles BAC and EDF must also be equal. Therefore the coordinates of point F are:

x_F = x_D + r cos (2 BAC )

y_F = y_D + r sin (2 BAC )

If we define t as the angle BAC, then the cardioid as traced by the point F is defined by the parametric equations:

x = 2r cos t + r cos 2t

y = 2r sin t + r sin 2t