$ \newcommand{\sech}{\mathop{\rm sech}\nolimits} \newcommand{\csch}{\mathop{\rm csch}\nolimits} \newcommand{\R}{\mathbb{R}} \newcommand{\basis}{\underline{\mathbf{e}}} $


Object type: Plane curve


A cardioid is a curve traced out by a point (the 'drawing point') on the perimeter of a circle as the circle is rolling without slipping along the perimeter of another, stationary, circle of the same radius. Below, the common radius is 1, the stationary circle is centred at the origin, and the point of contact between the drawing point and the stationary circle is chosen to be $(1,0)$.

A cardioid

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The specific cardioid shown above is the image $\mathbf{r}(\left[0,2\pi\right[)$ where $$\mathbf{r}(t) = \basis\begin{pmatrix}2\cos t-\cos{2t}\\2\sin t-\sin{2t}\end {pmatrix}.$$