# Cardioid

Cardioid
Field: Geometry
Image Created By: The Math Book

Cardioid

A cardioid is a curve which resembles a heart. Its name is derived from Greek where kardia means heart and eidos means shape, though it is actually shaped more like the outline of the cross section of an apple.

# 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

#### Generating a Cardioid

When $a$ is the radius of the moving circ [...]

#### Generating a Cardioid

When $a$ is the radius of the moving circle, the Cardioid curve is given by:

• Cartesian equation $({x^2} + {y^2} - 2ax)^2 = 4{a^2}({x^2} + {y^2})$.
• Polar equation $r = a (1 - {\cos} {\theta})$
• Parametric equation

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

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

#### Properties of Cardioids

The evolute of a cardioid is equal to itself.

Cardioid is the pedal of a circle with respect to a fixed point on the circle.

Cardioid is the catacaustic of a circle with light source on the circle.

Cardioid is also the envelope of circles with centers on a fixed base circle C and each circle passing through a fixed point P on the base circle C.

Cardioid is the inverse of parabola with respect to its focus.

Cardioid is also the conchoid of a circle of radius r with respect to a fixed point on the circle, and offset 2 r.

# Why It's Interesting

====Cardioid Microphone====

The cardioid microphone...

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