If we have a polygon in the plane, the Riemann mapping theorem tells us that a conformal map from the unit disk to that polygon exists, but it doesn't say how to find it. It turns out that we can actually construct one, in the following way:

This is called the Schwarz-Christoffel mapping.

Without loss of generality, let us assume that the polygon is given in terms of its vertices, $w_1,w_2,\cdots ,w_n$$w_1, w_2, \cdots, w_n$, in counterclockwise winding.

The values ${\alpha }_k$$\alpha_k$ that appear in the expression are the interior angles of the polygon, in multiples of $\pi$$\pi$.

In order to find a conformal mapping of the unit disk to this polygon, we need to find values of the parameters $a,c\in \mathbb{C}$${a, c} \in \mathbb{C}$ and ${\theta }_1,{\theta }_2,...,{\theta }_n\in \mathbb{R}$${\theta_1, \theta_2, ... , \theta_n} \in \mathbb{R}$ that satisfy:

The parameters ${\theta }_1,{\theta }_2,...,{\theta }_n$$\theta_1, \theta_2, ... , \theta_n$ control the (relative) sidelengths of the polygon, $a$$a$ controls the translation degrees of freedom and $c$$c$ controls the overall scale and rotation.

Although this can be done numerically, click the image below to try specifying a polygon and seeing if you determine values for ${\theta }_j$$\theta_j$ (blue dots on the left) that map to the target polygon (dashed red) on the right!