Kardioid

Kardioid refers to the cardioid, a heart-shaped plane curve that is a special case of a limaçon of Pascal and is widely studied in geometry and applied fields. The term cardioid comes from the Greek kardia, meaning heart. In some languages and contexts, the spelling Kardioid is used as a variant of cardioid.

Mathematically, the cardioid can be described in polar coordinates by r = a(1 − cos θ), where a is a positive constant that sets the size. A commonly cited form is r = 2a(1 − cos θ), which simply scales the curve. In Cartesian coordinates, the cardioid satisfies (x^2 + y^2 − ax)^2 = a^2(x^2 + y^2). The curve is symmetric about its x-axis and has a cusp at the origin, where the curve changes direction abruptly. It is the envelope of circles with diameter on the x-axis and is also the result of a circle of radius a rolling around another circle of radius a, traced by a point on the rolling circle. Equivalently, it is the epicycloid generated by a point on a circle of radius a rolling around a fixed circle of the same radius.

Properties and relationships: the cardioid is a special instance of the Limaçon of Pascal with equal radii, and it can be generated as an epicycloid. The area enclosed by r = a(1 − cos θ) is (3/2)πa^2. The curve is often used as a model in both pure mathematics and applied contexts due to its simple form and distinctive shape.

Applications: the cardioid shape appears in optics and acoustics, notably in microphone polar patterns, where the term cardioid (and the variant Kardioid) denotes a heart-shaped directional sensitivity. It also informs antenna design and other directional systems.