In geometry, a cardioid (from Greek καρδιά (kardiá) 'heart') is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It can also be defined as an epicycloid having a single cusp. It is also a type of sinusoidal spiral, and an inverse curve of the parabola with the focus as the center of inversion.^[1] A cardioid can also be defined as the set of points of reflections of a fixed point on a circle through all tangents to the circle.^[2]

The name was coined by Giovanni Salvemini in 1741^[3] but the cardioid had been the subject of study decades beforehand.^[4] Although named for its heart-like form, it is shaped more like the outline of the cross-section of a round apple without the stalk.

A cardioid microphone exhibits an acoustic pickup pattern that, when graphed in two dimensions, resembles a cardioid (any 2d plane containing the 3d straight line of the microphone body). In three dimensions, the cardioid is shaped like an apple centred around the microphone which is the "stalk" of the apple.

Equations

Let ${\displaystyle a}$ be the common radius of the two generating circles with midpoints ${\displaystyle (-a,0),(a,0)}$, ${\displaystyle \varphi }$ the rolling angle and the origin the starting point (see picture). One gets the

• parametric representation:
  $${\displaystyle {\begin{aligned}x(\varphi )&=2a(1-\cos \varphi )\cdot \cos \varphi \ ,\\y(\varphi )&=2a(1-\cos \varphi )\cdot \sin \varphi \ ,\qquad 0\leq \varphi <2\pi \end{aligned}}}$$

  
  and herefrom the representation in
• polar coordinates:
  $${\displaystyle r(\varphi )=2a(1-\cos \varphi ).}$$

  
• Introducing the substitutions ${\displaystyle \cos \varphi =x/r}$ and ${\textstyle r={\sqrt {x^{2}+y^{2}}}}$ one gets after removing the square root the implicit representation in Cartesian coordinates:
  $${\displaystyle \left(x^{2}+y^{2}\right)^{2}+4ax\left(x^{2}+y^{2}\right)-4a^{2}y^{2}=0.}$$

  

Proof for the parametric representation

A proof can be established using complex numbers and their common description as the complex plane. The rolling movement of the black circle on the blue one can be split into two rotations. In the complex plane a rotation around point ${\displaystyle 0}$ (the origin) by an angle ${\displaystyle \varphi }$ can be performed by multiplying a point ${\displaystyle z}$ (complex number) by ${\displaystyle e^{i\varphi }}$. Hence

the rotation ${\displaystyle \Phi _{+}}$ around point ${\displaystyle a}$ is${\displaystyle :z\mapsto a+(z-a)e^{i\varphi }}$,

the rotation ${\displaystyle \Phi _{-}}$ around point ${\displaystyle -a}$ is: ${\displaystyle z\mapsto -a+(z+a)e^{i\varphi }}$.

A point ${\displaystyle p(\varphi )}$ of the cardioid is generated by rotating the origin around point ${\displaystyle a}$ and subsequently rotating around ${\displaystyle -a}$ by the same angle ${\displaystyle \varphi }$:

$${\displaystyle p(\varphi )=\Phi _{-}(\Phi _{+}(0))=\Phi _{-}\left(a-ae^{i\varphi }\right)=-a+\left(a-ae^{i\varphi }+a\right)e^{i\varphi }=a\;\left(-e^{i2\varphi }+2e^{i\varphi }-1\right).}$$