Discrete-to-Continuum Metrics from Scalar Fields

We present a scalar-first framework for geometry processing in which geometric structure is generated from a scalar field $s:U\to\R$ through an explicit local rule $\mathcal{R}$. We call this scalar field a \emph{seam} (a potential/log-scale field; unrelated to UV/cut seams in parameterization). In the discrete setting, we study a conformal graph rule that assigns edge lengths using an endpoint quadrature of $e^{s}$, yielding an intrinsic shortest-path metric that is easy to optimize and differentiate. Our main results are (i) a quantitative discrete-to-continuum guarantee for the induced shortest-path metrics on graph families whose shortest-path distances approximate background geodesic distance up to an additive $O(h)$ error (e.g., sufficiently dense neighborhood graphs); (ii) a curvature sensitivity identity showing that first-order curvature variations are governed by the cotangent Laplacian; and (iii) a strictly convex inverse-design formulation for fitting target edge weights via a quadratic program in positive variables $X_u=e^{s(u)}$. These results position scalar-field parameterizations as a stable interface between differential-geometric objectives and practical optimization pipelines in mesh and graph processing.