structural:reformatting, information:filtering, structural:scholar_edition
Equation of the tangent of the cardioid with polar representation r = 2(1 + cos 𝜑)

From the parametric representation x (φ) = 2 (1 + cos φ) cos φ, y (φ) = 2 (1 + cos φ) sin φ {\displaystyle {\begin{aligned}x(\varphi)&=2(1+\cos \varphi)\cos \varphi,\\y(\varphi)&=2(1+\cos \varphi)\sin \varphi \end{aligned}}}

one gets the normal vector n → = (y ˙, − x ˙) T {\displaystyle {\vec {n}}=\left({\dot {y}},-{\dot {x}}\right)^{\mathsf {T}}}. The equation of the tangent y ˙ (φ) ⋅ (x − x (φ)) − x ˙ (φ) ⋅ (y − y (φ)) = 0 {\displaystyle {\dot {y}}(\varphi)\cdot (x-x(\varphi))-{\dot {x}}(\varphi)\cdot (y-y(\varphi))=0} is: (cos 2 φ + cos φ) ⋅ x + (sin 2 φ + sin φ) ⋅ y = 2 (1 + cos φ) 2. {\displaystyle (\cos 2\varphi +\cos \varphi)\cdot x+(\sin 2\varphi +\sin \varphi)\cdot y=2(1+\cos \varphi)^{2}\,.}

With help of trigonometric formulae and subsequent division by cos 1 2 φ {\textstyle \cos {\frac {1}{2}}\varphi }, the equation of the tangent can be rewritten as: cos (3 2 φ) ⋅ x + sin (3 2 φ) ⋅ y = 4 (cos 1 2 φ) 3 0 < φ < 2 π, φ ≠ π. {\displaystyle \cos({\tfrac {3}{2}}\varphi)\cdot x+\sin \left({\tfrac {3}{2}}\varphi \right)\cdot y=4\left(\cos {\tfrac {1}{2}}\varphi \right)^{3}\quad 0<\varphi <2\pi,\ \varphi \neq \pi.}