Cardioid

Herzkurve

Caustique

Cardiod animation

Kardioide

Kardioide-2

Cardioid is a distinctive, heart-shaped curve that is the path traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It is a type of epicycloid specifically with one cusp and is a special case of the sinusoidal spiral. The cardioid is significant in various fields such as mathematics, physics, and engineering, particularly in acoustics and antenna theory.

Etymology

The term cardioid was first used by de Castillon in a letter to Montucla in 1741, deriving from the Greek word kardia for heart, due to its heart-like shape.

Equations

The cardioid can be described by several mathematical equations in different coordinate systems. In the Cartesian coordinate system, its equation can be given as:

\[(x^2 + y^2 + 2ax)^2 = 4a^2(x^2 + y^2)\]

where \(a\) is the radius of the generating circle.

In polar coordinates, the cardioid's equation is simpler:

\[r = 2a(1 + \cos \theta)\]

This form clearly shows the cardioid's symmetry around the horizontal axis.

Properties

The cardioid has several interesting properties:

• It is a dimpled curve, with a single cusp at the origin when the generating circle starts rolling.
• It has an area equal to \(6\pi a^2\) and a perimeter of \(8a\), where \(a\) is the radius of the generating circle.
• The cardioid can also be generated by plotting the points of reflection of a fixed point on a circle (the caustic) through rays emanating from another circle.

Applications

1. 1. 1. Acoustics

In acoustics, cardioid microphones are valued for their ability to pick up sound primarily from one direction, minimizing background noise. This directional pattern helps in isolating the sound source from unwanted ambient sound.

1. 1. 1. Antenna Theory

In antenna theory, cardioid patterns are used in directional antennas to focus the transmission power in one direction, improving signal strength and reducing interference from other directions.

1. 1. 1. Optics

The cardioid shape is also found in optics, where it describes the caustic pattern formed by light rays reflecting or refracting through a curved surface.

See Also

• Epicycloid
• Cycloid
• Lemniscate
• Sinusoidal spiral