Geometric Feature Metric (GFM)

• GFM is an intrinsic, discrete metric framework that defines and computes fundamental geometric invariants such as Gaussian curvature and principal directions from triangulated surfaces.
• It leverages finite metric spaces and the Hausdorff–Gromov limit to capture accurate surface geometry without relying on traditional coordinate-based methods.
• GFM enhances computational reliability in mesh processing and geometric modeling by providing isometry-invariant, robust feature extraction even in noisy or irregular triangulations.

The Geometric Feature Metric (GFM) is an intrinsic, discrete metric framework for the analysis of triangulated surfaces, designed to define and compute fundamental geometric invariants such as Gaussian curvature, principal directions, and principal values directly from the metric space induced by the triangulation, as opposed to employing classical coordinate-based or differentiable approximations. This approach leverages finite metric spaces to model triangulations and interprets the smooth target surface as their Hausdorff–Gromov limit, enabling a coordinate-free, isometry-invariant characterization of surface geometry that improves both theoretical rigor and computational reliability [0401023].

1. Triangulated Surfaces as Finite Metric Spaces

The foundation of GFM in this context is the representation of a triangulated surface as a finite metric space. Consider a triangulation $\mathcal{T}$ comprising vertices $V$, edges $E$, and triangular faces $F$. Rather than using Euclidean coordinates or parametrizations, the metric perspective assigns to each point (vertex) a distance function induced by geodesics over the triangulation’s connectivity—i.e., distances are measured as shortest paths on the 1-skeleton of $\mathcal{T}$.

By viewing the smooth surface $S$ as the Gromov–Hausdorff limit of a sequence of such triangulations (with mesh size tending to zero), the metric approach captures the intrinsic geometry that the discrete object approximates. This abstraction provides:

• Isometry invariance, as metric properties do not depend on embedding.
• A pathway to define convergence and stability of geometric quantities as the triangulation is refined.
• Robustness to nonuniform sampling and mesh noise, since definitions are not tied to global coordinate frames.

2. Intrinsic and Discrete Metric Definitions

GFM employs intrinsic, discrete definitions of geometric quantities. For example, angles, distances, and associated geometric features are computed using metric data from within the triangulated mesh, such as discrete geodesic paths, rather than from ambient Euclidean space. For $u, v \in V$, the metric is typically defined as $d_{G}(u, v)$, the length of the shortest path along the edges.

In this context:

• Principal directions and values are extracted by analyzing variations in these intrinsic distances among neighboring points.
• Properties such as Gaussian curvature are determined not through local differential approximations but through metric deviations from the expected Euclidean relationships in geodesic triangles.

This discrete, intrinsic philosophy ensures that resultant geometric features are invariant under isometries and inherently tied to the geometry of the underlying surface.

3. Metric-Based Gaussian Curvature via Embedding Curvature

GFM computes Gaussian curvature discretely using notions of embedding curvature—in the sense of Wald and Menger—which exploit only metric information (distances among points):

Let $\Delta$ denote the (signed) area of a triangle formed by vertices with edge lengths $L_{ij}, L_{jk}, L_{ki}$. The embedding curvature $K$ of such a configuration is written as

$$K = \frac{1}{R^2} = \frac{16 \Delta^2}{L_{ij}^2 L_{jk}^2 L_{ki}^2}$$

where $R$ encodes curvature radii in the metric sense. This approach makes explicit how curvature is deduced from observed deviations between geodesic triangle areas and edge-length configurations versus corresponding flat (Euclidean) triangles.

Such definitions can be systematically applied to compute discrete curvature at vertices or within faces, and are readily adapted for irregular or nonuniform triangulations.

4. Principal Directions and Values from Metric Variation

Within the GFM paradigm, principal directions are determined through analysis of how the intrinsic geodesic distance function changes in local neighborhoods:

• Principal directions are those along which the second-order variation (or, in discrete terms, finite differences) of interpoint distances is extremal.
• Principal values correspond to the magnitude of these extreme variations, which serve as discrete proxies for principal curvatures.

Since these definitions use only internal metric measurements, they provide a coordinate-free analog of the classical shape operator, accessible even on polyhedral or noisy surfaces where derivatives are ill-defined.

5. Comparison with Classical and Numerical Approaches

Classical surface geometry computations typically require explicit differential structure—chart-based parametrizations, differentiability of the embedding, and computation of the second fundamental form. These methods necessitate accurate estimation of normals and tangents in the ambient space and are sensitive to mesh irregularity, discretization, and noise.

GFM, by contrast:

• Delivers purely intrinsic quantification, obviating the need for reliable global parametrizations, and directly relates surface invariants to observable mesh properties.
• Avoids a host of computational artifacts and conceptual ambiguities inherent to coordinate-based numerics, as outcomes depend only on interpoint distances.
• Presents a tradeoff: While conceptually and practically robust, metric computations (e.g., large numbers of shortest paths, many applications of Cayley–Menger determinants) can be more computationally intensive, especially for large meshes or when high-fidelity refinement is required.

6. Practical Applications and Implications

Adopting GFM for triangulated surfaces and polyhedral geometry has specific and wide-reaching consequences:

• Improved accuracy and stability in curvature estimation, enabling more reliable mesh processing, feature detection, and adaptive refinement strategies in computer graphics and geometric modeling.
• Isometry-invariant feature extraction, relevant for mesh compression, registration, and classification tasks where coordinate systems, scale, or orientation are ambiguous or unknown.
• Integration with finite element methods, providing mesh metrics that more faithfully encode physical domain geometry, which is critical when simulating phenomena that are sensitive to curvature and surface features.

In broader terms, GFM establishes a metric-theoretic foundation for discrete differential geometry, guiding how to discretize smooth geometric invariants in ways that are more robust under mesh perturbation and irregularity.

7. Limitations and Implementation Considerations

Despite its robustness and foundational appeal, metric-based approaches are not without challenges:

• For highly irregular or degenerate triangulations, metric computations (such as checking triangle rank, computing geodesic paths, or evaluating high-order determinants) may be ill-conditioned or computationally burdensome.
• In the presence of significant noise or outlier points, reliance on purely metric data may amplify certain errors relative to coordinate-based smoothing or regularization schemes.
• Applicability hinges on the quality of the underlying mesh: very sparse or poor-quality triangulations can lead to ambiguous or unstable metric feature computation.

Nonetheless, the intrinsic nature of GFM ensures that, when the mesh quality is sufficient and computational resources permit, the extracted geometric features are theoretically grounded and empirically resilient.

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In summary, the Geometric Feature Metric (GFM) formalizes a class of coordinate-free, intrinsic, and discrete metric-based constructions for quantifying geometric invariants of triangulated surfaces, most notably Gaussian curvature and principal directions. By building directly on the finite metric structure of the mesh and eschewing embedding-dependent or derivative-based definitions, GFM achieves robust, isometry-invariant characterization of surface geometry, with direct relevance to mesh processing, computational geometry, and discrete differential geometry methodologies [0401023].