Cardioid

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Cardioid
Field: Geometry
Image Created By: The Math Book

Cardioid

A cardioid is a curve which resembles a heart.

Basic Description

In geometry, a cardioid is the curve traced by a point on the circumference of a circle that rolls around the circumference of another equal circle.

A More Mathematical Explanation

The curve is given by:

• Cartesian equation $({x^2} + {y^2} - 2ax)^2 = 4{a^2}({x^2} + {y^2})$, where '"`UNIQ--math-00 [...]

The curve is given by:

• Cartesian equation $({x^2} + {y^2} - 2ax)^2 = 4{a^2}({x^2} + {y^2})$, where $a$ is the radius of the moving circle.
• Polar equation $r = a (1 - {\cos} {\theta})$
• Parametric equation

$x = a {\cos} t (1 - {\cos t})$

$y = a {\sin} t (1 - {\cos t})$

Properties

• It has a cusp at the origin.
• There are exactly three tangents to the cardioid with any given gradient
• The tangents at the ends of any chord through the cusp point are at right angles
• The length of any chord through the cusp point is 2$a$

Generating a Cardioid

Draw a circle $C$, and pick a fixed point $A$ on it. Then, draw a set of circles centered on the circumference of $C$ and passing through $A$. The envelop of the chords of these circles is a cardioid, as in the main image. If the fixed point A is not on the circle, then the figure becomes a limacon

The Cardioid in Real Life

An instance where one could see a cardioid is when looking into a cup of coffee. The caustic seen at the bottom of a cup of coffee could be a cardioid, depending on the angle of light relative to the bottom of the cup.

Also, all unidirectional microphones are cardioid-shaped.

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