I Heart Cardioids

a year ago

A cardioid is a heart-shaped curve formed by rolling a circle around another circle of the same radius.
The cardioid was possibly discovered by Philippe de la Hire in 1708, who computed its length, and named by Johann Castillon in 1741.
Cardioids appear in unexpected places: in the reflection of light in a coffee cup, forming a caustic; in the main heart-shaped region of the Mandelbrot set; and in the sensitivity graph of cardioid microphones used in audio engineering.
The cardioid can be constructed as the envelope of a family of lines or circles, with specific geometric constructions detailed in the text.
A practical example involves drawing lines between points on a circle to form a cardioid, with printable templates provided for creating a cardioid flip book.
The envelope of a family of curves is defined using calculus, where each curve in the family is tangent to the envelope, and the cardioid serves as an example of such an envelope.
The coffee cup caustic is more accurately a nephroid when light rays are parallel, a related curve formed by different geometric constructions.

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