Conformal mapping in shape representation

If a shape is considered to be a simple closed smooth curve in the complex plane, meaning that they have derivatives of all orders and that do not cross with themselves, then the Riemannian mapping theorem states that it is possible to map the unit disc conformally (meaning that infinitesimal angle between each two crossing curves is equal to the infinitesimal angle between the transformed curves) to the interior of any such shape. The conformal transformation is unique up to any preceding Möbius transformations mapping the unit disc to itself, this is, maps of the form
$$z  \rightarrow \frac{az + b}{\bar{b}z + \bar{a}}$$

If we identify $\nabla_{-}$ with the interior of the shape, and $\Gamma_{-}$ with the interior of the unit disk, then by the Riemannian mapping theorem there exists a map
$$\Phi_{-} : \nabla_{-} \rightarrow \Gamma_{-}$$

With the previous characteristics. Furthermore, if we identify $\nabla_{+}$ with the exterior of the shape, and $\Gamma_{+}$ with the exterior of the unit disk, the same conditions apply
$$\Phi_{+} : \nabla_{+} \rightarrow \Gamma_{+}$$

Now, we construct
$$\Psi =  \Phi_{+} \circ \Phi_{-}$$

This provides a fingerprint of the shape unique up to a Moebius transformation.

In practice, we use the Swarz-Christoffel Matlab toolbox, having a complex variable with the given vertices coordinates in $pol$, we compute the map $m$ by
pol=polygon(dots);
fc=diskmap(pol);
ex=extermap(pol);
iex=scmapinv(ex);
m=composite(fc,iex);

To extract the fingerprint, we need to evaluate the map in the $z$ coordinate given the angle, so that $z=exp(-i \theta)$, and the evaluation is $m(z)$ for all $\theta \in [0,2\pi)$. Now, the fingerprint is the angle of a given evaluation $m(z)$, this is, $\hat{\theta}=-\frac{log(m(z))}{i}$. plotting $(\theta,\hat{\theta})$, we see the fingerprint.

Checkout the paper here.