kardioida

A kardioida, commonly known as a cardioid, is a heart-shaped plane curve that appears in various mathematical and applied contexts. It is often described as the locus traced by a point on the circumference of a circle of radius a rolling externally around an identical circle, or equivalently as an epicycloid with the ratio radius of fixed to rolling circle equal to 1:1.

In polar form the cardioid can be written as r = a(1 ± cos θ), where a is a positive constant that sets the size. In Cartesian coordinates it has the implicit equation (x^2 + y^2 − a x)^2 = a^2(x^2 + y^2). The curve has a single cusp at the origin, is symmetric about the x-axis, and is a special case of the limaçon family (a limaçon with a cusp). The standard cardioid has arc length 8a and area (3/2)πa^2.

The cardioid can also arise as a caustic of a circle under parallel light or a point light on the circle’s circumference, highlighting its optical significance. Its geometric simplicity and distinctive shape make it a common figure in design and computer graphics, as well as in acoustic and antenna patterns where cardioid shapes describe directional properties.

Overall, the kardioida is a classical and widely studied plane curve with several equivalent constructions, a precise polar and Cartesian description, and characteristic features such as a cusp, symmetry, and a well-defined area and length.