Given a set of points $\mathbf V$ in $\Bbb R^3$ and edges $\mathbf E = \{(v_1,v_2) | v_1,v_2 \in \mathbf V\}$ we can define the Gaussian curvature $\mathbf K = 2\pi - \sum\theta$ for angles $\theta$ between edges adjacent to that vertex. We can define a mean curvature normal $$2\mathbf H\mathbf N = \frac 1{2\mathbf A}\sum(cot\alpha_{ij} + cot\beta_{ij})(\mathbf x_i - x_j)$$ Half the magnitude of this vector is the mean curvature $\mathbf H$.
Given $\mathbf G$ and $\mathbf H$ at every point in $\mathbf V$, can we reconstruct the original surface up to a translation and rotation? If so, what is the procedure for generating the vertex positions from the curvature values, given an initial point's position and the orientation of 2 adjacent edges? If not, what additional information needs to be stored at each vertex to guarantee the mapping back into position space is unique up to rotation and translation? (Assume the adjacency of the surface is grid-like, so every non-boundary vertex has valence 4. Does the result also hold in the general case?)