Length of a Full Steiner Tree as a Function of Terminal Coordinates

Given the coordinates of the terminals $\{(x_j,y_j)\}_{j=1}^n$ of the full Euclidean Steiner tree, its length equals $\left| \sum_{j=1}^n z_j U_j \right| \, ,$ where $\{z_j:=x_j+ \mathbf i y_j\}_{j=1}^n$ and $\{U_j\}_{j=1}^n$ are suitably chosen $6$th roots of unity. We also extend this result for the cost of the optimal Weber networks which are topologically equivalent to some full Steiner trees.

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