Coordinates #
Theorem 1. The vertices of an \(n\)-sided polygon can be represented as the \(n\)th roots of unity:
$$ r e^{\frac{2\pi ik}{n}} = r\cos\left({\frac{2\pi k}{n}}\right) + i r\sin\left({\frac{2\pi k}{n}}\right),k=[1,2 \dots n] $$Lemma 2. The form of a line passing through two points \((x_1,y_1)\) and \((x_2,y_2)\):
$$ y=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\left(x-x_{1}\right) + y_{1} $$Proof. Take standard point-slope and substitute \(m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) □
Line Form #
To form an envelope from a polygon’s edges, like the one above, we take the form of a line passing through any two points. Let the angle formed by the first point be \(\alpha\) and let its successor be \(\beta\):
$$ \begin{align} & \left(x_{1},y_{1}\right)=\left(r\cos\alpha,r\sin\alpha\right) \\ & \left(x_{1},y_{1}\right)=\left(r\cos\beta,r\sin\beta\right) \\ \end{align} $$Then by Lemma 1.2, the line connecting any two adjacent vertices is represented by:
$$ \begin{align} y &=\frac{r\sin\beta-r\sin\alpha}{r\cos\beta-r\cos\alpha}\left(x-r\cos\alpha\right)+r\sin\alpha \\ & =\frac{\sin\beta-\sin\alpha}{\cos\beta-\cos\alpha}\left(x-r\cos\alpha\right)+r\sin\alpha \end{align} $$Using the standard sum-to-product identities, we can rewrite both the numerator and denominator. Then, we can cancel the common sine term along with the leading coefficients.
$$ \begin{align} y &=\frac{2\cos\left(\frac{\beta+\alpha}{2}\right)\sin\left(\frac{\beta-\alpha}{2}\right)}{-2\sin\left(\frac{\beta+\alpha}{2}\right)\sin\left(\frac{\beta-\alpha}{2}\right)}\left(x-r\cos\alpha\right)+r\sin\alpha \\ &= -\frac{\cos\left(\frac{\beta+\alpha}{2}\right)}{\sin\left(\frac{\beta+\alpha}{2}\right)}\left(x-r\cos\alpha\right)+r\sin\alpha\\ &=-\cot\left(\frac{\beta+\alpha}{2}\right)\left(x-r\cos\alpha\right)+r\sin\alpha \end{align} $$Since \(\alpha\) and \(\beta\) must be consecutive angles inside the polygon, \(\alpha = \frac{2\pi k}{n}\) and \(\beta = \frac{2\pi (k+1)}{n}\):
$$ \begin{aligned} y&=\left(\frac{\frac{2\pi\left(k+1\right)}{n}+\frac{2\pi k}{n}}{2}\right)\left(x-r\cos\left(\frac{2\pi k}{n}\right)\right)+r\sin\left(\frac{2\pi k}{n}\right)\\ & =\boxed{\boldsymbol{-\cot\left(\frac{2\pi k+\pi}{n}\right)\left(x-r\cos\left(\frac{2\pi k}{n}\right)\right)+r\sin\left(\frac{2\pi k}{n}\right)}} \end{aligned} $$