Shape-Based Tracing of Singularities

• Shape-based tracing of singularities is a method that analyzes the geometric structures of singular points using explicit data such as curvature and polygonal approximations.
• It leverages analytic and numerical strategies, including the use of shape operators and numerical continuation, to robustly classify and track singularity evolution.
• This approach integrates techniques from PDE analysis, algebraic geometry, and optical field theory to provide stable, topologically correct representations of complex singular phenomena.

Shape-based tracing of singularities refers to the extraction, analysis, and tracking of the geometric or morphological structure of singular points or loci within mathematical objects, PDE solutions, optical fields, or algebraic varieties. Unlike strictly topological or spectral methods, shape-based approaches leverage explicit geometric data—such as curve or surface arrangements, curvature, torsion, or polygonal approximation—to characterize, follow, and interpret singularities and their transformations under various contexts. Modern developments span analytic, numerical, and physically motivated frameworks, yielding robust techniques for practical computation, geometric inference, and singularity classification.

1. Geometric and Analytical Foundations

The shape-based perspective treats singularities as objects whose geometry or differential structure can be analyzed and traced throughout ambient spaces or parameter domains. In the context of congruence of solutions to second-order PDEs, the "shape map" is constructed on jet spaces associated with a connection, capturing the infinitesimal deformation transverse to solution families. The trace of the shape map governs the evolution of congruence volumes, serving as an indicator for singularity (collapse) formation via a Raychaudhuri-type differential equation. Given a congruence embedding $Z$, the shape operator $A_Z$ evolves according to

$$L_{Z_v}A_Z = -A_Z^2 - i_{Z_v}(Z^\varphi \Phi_+ + Z^\varphi r_+),$$

and singularity (collapse) is dictated by divergence of the integrated trace $\theta = \operatorname{Tr}A_Z$ to $-\infty$ (Rossi et al., 2015).

In real algebraic geometry, shape-based tracing considers the polygonal or numerical approximation of implicit curves and surfaces, with singularities identified as points where Jacobians or gradients drop rank. The geometric structure of neighborhoods around these points is replaced or encoded by clusters, facilitating robust visualization and analysis (Chen et al., 2019).

2. Algorithmic Strategies for Tracing Singularities

Numerical and computational shape-based methods involve determining and tracking key points and structure in the vicinity of singularities. In the visualization of planar and space algebraic curves, the algorithm proceeds through:

• Exact and pseudo-singular point computation: solving for $f(x,y)=0$ and $\nabla f(x,y)=0$ in a bounding box;
• Clustering singularities into "natural clusters" based on spatial proximity;
• Fencing neighborhoods using spheres and computing intersection points (fencing points);
• Seeding with "witness points" for closed components via critical point methods;
• Performing tangent-directed numerical continuation with robust predictor-corrector schemes, step control based on Hausdorff error, and local singular value monitoring to prevent branch jumping.

Polygonal traces are then constructed, forming $\varepsilon$-Hausdorff approximations of the underlying structures. The approach guarantees topological correctness outside singular neighborhoods, collapsing complex local singular geometry to a representative point for stability and computational tractability (Chen et al., 2019).

3. Shape-Based Recovery of Singularities via Projection and Reconstruction

Reconstruction of surfaces with singularities from observable geometric data utilizes the encoded shape of silhouette curves under projective projections. The process involves:

• Factoring the silhouette discriminant $\Delta$ into singular and regular image components to separate types of singularities;
• Classifying special points (nodes, cusps, tangencies) by local multiplicity patterns;
• Computing conductor ideals at these points to capture local-to-global singular structure;
• Using adjoint forms to lift the 2D silhouette to the 3D fat contour (branched cover), enabling the recovery of the full surface equation.

This analytic mechanism demonstrates that the planar silhouette's local and global shape contains complete information about the surface's singular locus, and the construction of the surface equation is unique up to projective automorphisms fixing the projection center (Gallet et al., 2018).

4. Physical and Applied Contexts: Optical Vortex Structures

In optical field theory, phase singularities manifest as vortex lines with characteristic 3D morphology. Shape-based tracing in this context proceeds by:

• Extracting vortex lines-by evaluating the phase field on a 3D grid to obtain occupancy volumes;
• Parameterizing lines as curves $r(s)$ and computing geometric descriptors—tangent, normal, binormal, curvature $\kappa(s)$, and torsion $\tau(s)$—with discrete finite differences;
• Comparing shapes through voxel-based $L^2$ distances, Hausdorff distances, or elastic shape metrics.

Comparative studies demonstrate that shape-based descriptors are significantly more robust than topological invariants (e.g., crossing/linking number) or spectral fingerprints in turbulence. Classification of highly-structured "flower beams" through 3D CNNs on voxelized singularity shapes achieves accuracies exceeding 90% in weak turbulence, outperforming all competing methodologies (Tsvetkov et al., 11 Nov 2025).

5. Trace Singularities in Obstacle Scattering and Inverse Problems

In analytic settings, such as obstacle scattering, singularities in the trace of certain operators encode geometric information about underlying obstacles. For pairs of obstacles, the Fourier transform of the relative spectral shift function,

$$\hat{\xi}_{rel}(t) = \int_{-\infty}^{\infty}e^{-it\lambda}\xi_{rel}(\lambda)d\lambda,$$

contains singularities whose locations and residues are in direct correspondence with the geometric parameters of the configuration—specifically, lengths and stability properties of billiard orbits between obstacles. The Poisson relation equates the distributional singularities of $\hat{\xi}_{rel}(t)$ with sums over periodic bouncing-ball orbits, with the shortest orbit inducing the leading singular term. The exponential decay rate of auxiliary spectral functions (such as $\Xi(\lambda)$) as $\Im\lambda\to+\infty$ is sharply determined by the length of this shortest orbit and the wave-trace invariant, directly relating spectral singularities to geometric shape data (Fang et al., 2021).

6. Shape-Based Deformations and Creation/Flattening of Singularities

Geometric Function Theory addresses the deformation of domains to create or flatten boundary singularities (e.g., cusps or slits) via homeomorphisms of controlled (bi-)conformal energy and integrable inner distortion. A canonical shape-based result is the sharp dichotomy for exponential cusps $u_\alpha(t) = e/\exp(1/t)^\alpha$ in $\mathbb{R}^n$:

• For $\alpha < n$, bi-conformal mappings exist mapping the unit ball to a domain with such a cusp, with finite energy and the inverse being Lipschitz.
• For $\alpha \geq n$, no such mapping with finite bi-conformal energy exists.

Explicit deformation constructions support the positive direction, while modulus-of-continuity arguments preclude mappings for sharper cusps or slit domains beyond the energy threshold. Shape-based tracing here refers both to the explicit control of deformations in singular geometry, and to the theoretical detection of when such objects can or cannot be mapped (Iwaniec et al., 2019).

7. Theoretical Implications and Comparative Advantages

Shape-based tracing of singularities is particularly effective in regimes where purely topological or spectral invariants are unstable or coarse. For instance, turbulence in optical fields disrupts linkages, but large-scale geometric features may persist; shape-based analysis captures and classifies such information with higher fidelity. In computational algebraic geometry, the replacement of local singular neighborhoods by geometric clusters yields stable, topologically correct representations for intersection and connectivity tasks. Analytically, trace singularities and wave-trace invariants directly reveal geometric features not accessible via spectral data alone.

A plausible implication is that shape-based strategies provide a unifying geometric language that links analysis, topology, and computation, yielding robust tools for diverse singularity phenomena across mathematics and physics.