Boundary-Representation Distance Functions

• Boundary-Representation Distance Functions (BR-DFs) are mathematical constructs that compute the infimum of gauged distances from interior points to their boundaries, essential in computational geometry.
• They are constructed through diverse methods including convolutional schemes, PDE-based formulations, and physics-informed neural networks, ensuring robust numerical approximations.
• BR-DFs play a critical role in CAD, mesh generation, collision detection, and shape analysis, providing a rigorous framework for both computational and inverse geometric problems.

Boundary-Representation Distance Functions (BR-DFs) are a class of mathematical constructs that encode the distance from a point in a domain to its boundary. These functions are foundational in computational geometry, partial differential equations, geometric inverse problems, shape analysis, solid modeling, and numerical methods involving boundary data. The formalism and applications of BR-DFs span classical Euclidean settings, Finsler and Minkowski generalizations, polyhedral and smooth boundaries, PDE-based constructions, and data-driven contexts.

1. Definitions and Foundational Formulations

A Boundary-Representation Distance Function on an open subset $U \subset \mathbb{R}^n$ with boundary $\partial U$ is any function of the form

$$d(x) = \inf_{y \in \partial U} \gamma(x - y), \qquad x \in U,$$

where $\gamma$ is a (possibly asymmetric) gauge, norm, or Minkowski functional. In the canonical Euclidean case, $\gamma(z) = \|z\|_2$ and $d(x) = \min_{y \in \partial U} |x - y|$ gives the unsigned distance to the boundary.

Generalizations include:

• Asymmetric/Minkowski gauges: $\gamma_K(x) = \inf\{\lambda>0: x \in \lambda\,K\}$, where $K \subset \mathbb{R}^n$ is a compact, convex set with $0$ in its interior. This leads to the asymmetric, possibly non-reversible, Finsler context (Safdari, 2016).
• Polytope-based BR-DFs: If $P \subset \mathbb{R}^n$ is a convex polytope, then $\partial U$0 in the interior of $\partial U$1, where $\partial U$2 is the polytope’s Minkowski functional (Safdari, 10 Dec 2025).
• Signed distance function: $\partial U$3 if $\partial U$4; $\partial U$5 otherwise, where $\partial U$6 is the unsigned Euclidean distance to $\partial U$7 (Nikolov et al., 2024).

In Riemannian manifolds with boundary, the BR-DF becomes $\partial U$8 for $\partial U$9, $$d(x) = \inf_{y \in \partial U} \gamma(x - y), \qquad x \in U,$$0 (Pavlechko et al., 2022, Ivanov, 2020).

BR-DFs are also concretely constructed in numerical frameworks via PDE-state variables or neural network-based representations, as in the context of meshfree physics-informed neural networks or elliptic PDE-based approximations (Hasebe et al., 2024, Abgrall, 2022, Sukumar et al., 2021).

2. Regularity, Singularities, and Ridge Sets

The regularity of BR-DFs depends fundamentally on the regularity of the domain boundary, the convexity/smoothness of the gauge, and local geometric features such as corners or facets.

• Bootstrap (Differentiability $$d(x) = \inf_{y \in \partial U} \gamma(x - y), \qquad x \in U,$$1 $$d(x) = \inf_{y \in \partial U} \gamma(x - y), \qquad x \in U,$$2): If $$d(x) = \inf_{y \in \partial U} \gamma(x - y), \qquad x \in U,$$3 is differentiable in a neighborhood of a boundary point $$d(x) = \inf_{y \in \partial U} \gamma(x - y), \qquad x \in U,$$4, then both $$d(x) = \inf_{y \in \partial U} \gamma(x - y), \qquad x \in U,$$5 and $$d(x) = \inf_{y \in \partial U} \gamma(x - y), \qquad x \in U,$$6 are $$d(x) = \inf_{y \in \partial U} \gamma(x - y), \qquad x \in U,$$7 nearby. Moreover, if $$d(x) = \inf_{y \in \partial U} \gamma(x - y), \qquad x \in U,$$8 is $$d(x) = \inf_{y \in \partial U} \gamma(x - y), \qquad x \in U,$$9, then $\gamma$0 is $\gamma$1 in a tubular neighborhood (Nikolov et al., 2024).
• Asymmetric/Finsler and Polyhedral Regularity: When $\gamma$2 is strictly convex and smooth, $\gamma$3 is $\gamma$4 away from the ridge (singular) locus. For polytopal $\gamma$5, $\gamma$6 is explicitly piecewise linear/affine and often fails to be $\gamma$7 (or even $\gamma$8) on the ridge, with the singular set carrying positive measure (Safdari, 10 Dec 2025, Safdari, 2016).

The ridge (or cut locus) is the set where the BR-DF fails to be $\gamma$9; this includes:

• Points with multiple closest boundary points;
• Points where a generalized curvature condition is met: $\gamma(z) = \|z\|_2$0 ($\gamma(z) = \|z\|_2$1 unique minimizer) (Safdari, 2016).
• For polyhedral cases, the ridge can be a high-dimensional set (not measure zero), and even at points with multiple minimizers, $\gamma(z) = \|z\|_2$2 may remain differentiable (Safdari, 10 Dec 2025).

In the Euclidean/Finsler scenario with corners, explicit Hessian formulas hold in the regular set; at or near the ridge, the Hessian diverges (Safdari, 2016).

3. Computational and PDE-Based Constructions

A variety of numerical and analytic strategies exist for constructing or approximating BR-DFs:

• Integral (Convolutional) Schemes: Replace the $\gamma(z) = \|z\|_2$3 by Laplace (log-sum-exp) or soft-min approximations over the boundary (Belyaev et al., 2024). These yield fast, accurate BR-DFs directly from boundary samples, with error scaling $\gamma(z) = \|z\|_2$4 for composite estimators.
• Elliptic PDE-based Methods: Solutions $\gamma(z) = \|z\|_2$5 to $\gamma(z) = \|z\|_2$6 or similar (Helmholtz/heat-type equations) are used to reconstruct BR-DFs via $\gamma(z) = \|z\|_2$7, with uniform convergence rates and tractable numerics (Hasebe et al., 2024, Abgrall, 2022).
• Physics-Informed Neural Networks (PINNs): BR-DFs are encoded via analytic "approximate distance functions" (ADF) constructed with R-functions or mean-value potential fields, guaranteeing pointwise Dirichlet, Neumann, or Robin boundary condition satisfaction in meshfree settings (Sukumar et al., 2021, Deguchi et al., 25 Apr 2025).
• Differential (Taylor/Heat-Method) Approaches: For the heat-method, $\gamma(z) = \|z\|_2$8; set $\gamma(z) = \|z\|_2$9. Higher-order Taylor extrapolations (including up to second derivatives with respect to $d(x) = \min_{y \in \partial U} |x - y|$0) improve convergence rate to $d(x) = \min_{y \in \partial U} |x - y|$1 (Belyaev et al., 2024). Gradient normalization ("normalize-and-Poisson" steps) further enforce $d(x) = \min_{y \in \partial U} |x - y|$2 and reduce absolute error.

4. Generalizations and Metric Variants

Numerous variations and generalizations of the BR-DF exist:

• Boundary Distance Difference Functions: Maps of the form $d(x) = \min_{y \in \partial U} |x - y|$3 for $d(x) = \min_{y \in \partial U} |x - y|$4 in a manifold $d(x) = \min_{y \in \partial U} |x - y|$5 in the boundary, enable unique recovery of the Riemannian structure without convexity assumptions (Ivanov, 2020).
• Barrlund's Metrics: For points $d(x) = \min_{y \in \partial U} |x - y|$6, the Barrlund $d(x) = \min_{y \in \partial U} |x - y|$7-metric is $d(x) = \min_{y \in \partial U} |x - y|$8, a genuine metric with controlled distortion properties under quasiconformal maps (Fujimura et al., 2019).
• Shape Dissimilarity Metrics: For straight-edge figures, a convex combination of angular and edge-length disproportion measures, computable via iterative proportional fitting and feature projection to $d(x) = \min_{y \in \partial U} |x - y|$9 (angles) or $\gamma_K(x) = \inf\{\lambda>0: x \in \lambda\,K\}$0 (lengths), provides a rigorous boundary-shape "distance" (Roopa et al., 2016).

5. Applications in Modeling, Computation, and Analysis

BR-DFs are central in:

• Computer-Aided Design (CAD): Volumetric BR-DFs represent surface and face geometry as collections of signed and unsigned distance fields, enabling error-free conversion to faceted B-Rep models via extensions of marching cubes algorithms, with guarantees of watertightness and correct topology (Zhang et al., 18 Nov 2025).
• Mesh Generation/Adaptivity: Explicit formulas for the Hessian and singularity structure facilitate anisotropic mesh adaptation, level-set methods, and interface tracking (Safdari, 2016, Safdari, 10 Dec 2025).
• Collision Detection: Minkowski functional-based BR-DFs reduce collision queries for polytopes to thresholding the BR-DF value (Safdari, 10 Dec 2025).
• Numerical PDEs: Accurate BR-DFs enable robust imposition of essential and natural boundary conditions in PINNs, replacing penalty terms, simplifying loss landscapes, and increasing reliability, including for inverse problems and non-convex domains (Sukumar et al., 2021, Deguchi et al., 25 Apr 2025).
• Geophysical Inverse Problems: The boundary distance and difference function framework allows unique recovery of interior metrics from travel-time or arrival-time data, with or without strict convexity assumptions (Pavlechko et al., 2022, Ivanov, 2020).

6. Limitations, Singularities, and Open Problems

• Boundary Regularity: High regularity of both boundary and gauge is needed for maximal regularity of BR-DFs. Non-smoothness, corners, or polyhedral faces lead to thick singular sets, where the function may be only $\gamma_K(x) = \inf\{\lambda>0: x \in \lambda\,K\}$1 or $\gamma_K(x) = \inf\{\lambda>0: x \in \lambda\,K\}$2 (Safdari, 10 Dec 2025, Safdari, 2016).
• Ridge Structure: In polytopal settings, the measure and combinatorics of the ridge remain incompletely characterized. Understanding and handling the second-derivative jumps across ridges is an open analytic and computational challenge (Safdari, 10 Dec 2025).
• Numerical Error and Stability: For convolutional/differential approaches, parameter tuning (e.g., Laplace kernel $\gamma_K(x) = \inf\{\lambda>0: x \in \lambda\,K\}$3, grid resolution, normalization strategies) is essential for optimal accuracy (Belyaev et al., 2024). In PINN frameworks, the reliability of BR-DF encoding and the interaction with adaptive weight tuning are active research areas (Deguchi et al., 25 Apr 2025).
• Generative Models: Volumetric BR-DFs enable unconditional B-Rep shape generation with 100% valid output rate, but limitations appear at high face count, fine grid scales, or when generative models create high-noise latents (Zhang et al., 18 Nov 2025). Tight integration with CAD parametric representations remains an open development.

7. Connections to Broader Geometric and Analytical Frameworks

BR-DFs are deeply connected to the theory of viscosity solutions of Hamilton–Jacobi equations, the structure and regularity of cut loci, and boundary rigidity in Riemannian geometry. The explicit connection between differentiability of the signed distance and $\gamma_K(x) = \inf\{\lambda>0: x \in \lambda\,K\}$4 boundary regularity unifies several older regularity theorems (Nikolov et al., 2024). In geometric inverse problems, BR-DFs serve as function-space embeddings, enabling constructive recovery of complex geometric structures (Pavlechko et al., 2022, Ivanov, 2020).

From meshfree computational mechanics to inverse analysis and geometric data science, BR-DFs offer a rigorous, compact, and algorithmically favorable foundation for encoding, manipulating, and reconstructing geometric and physical data from boundary information.