Single-Pass Plane Sweeping Strategy

• Single-Pass Plane Sweeping Strategy is a method that traverses a spatial domain once, processing each element in order to manage dependencies efficiently.
• It employs an advancing sweepline and active data structures to update local regions, ensuring each object is processed only a single time.
• The strategy is applied in computational geometry, computer graphics, and robotics for tasks such as tessellation, intersection detection, and mesh traversal.

A single-pass plane sweeping strategy is a geometric or algorithmic approach wherein a plane (or sweepline) systematically traverses a domain, dataset, or arrangement, processing or updating entities in the order they are encountered, and ensuring that each object is handled exactly once per pass. This strategy is foundational in computational geometry, computer graphics, computational physics, and robotics, supporting efficient algorithms for visibility, tessellation, mesh traversal, intersection, path planning, and geometric arrangements. The single-pass constraint guarantees that the process is not revisited; data is updated or computed precisely as the sweep progresses, often leveraging local dependencies, spatial orderings, or analytic invariants.

1. Core Principles and Mathematical Foundations

Underlying all variants of the single-pass plane sweeping strategy is the idea of advancing a plane (or sweepline) through a spatial domain—typically along a chosen axis or principal direction—while maintaining an active data structure that tracks entities (points, edges, faces, intersection candidates, etc.) interacting with the current plane position. Fundamental models include:

• Ordered processing according to sorted coordinates (e.g., sorting points along a principal component for Delaunay tessellation (Trencséni et al., 2012)).
• Dependency graphs dictating update order, such as cell connectivity in discrete ordinates sweeps for neutron transport (where acyclicity is established for regular triangle meshes (Camminady et al., 2018)).
• Transport of scalar or vector quantities across a grid leveraging numerical PDE discretization, where a solution is propagated by resolving local updates in the sweep direction (as in PDE-based visibility maps (Ibrahim et al., 11 Mar 2024)).

Mathematically, these approaches employ constructs such as rotation matrices for cell dependencies, circumsphere intersection tests for tessellation, and partial differential equations for information transport. The sweep order ensures every cell or entity is updated precisely once, satisfying local or global conditions needed for correctness and efficiency.

2. Algorithmic Variants

Several concrete algorithmic instantiations exist:

• Memory-Efficient Incremental Sweeping: Data is organized along a sorting axis (typically the first principal component) and processed incrementally. Completed regions—where no future input can affect the active data—are "offlined" or purged from memory. This technique restricts in-memory usage to a thin slab about the sweep plane, achieving memory usage proportional to the dataset's "thickness" in the sweep direction (Trencséni et al., 2012).
• Space-Efficient Geometric Algorithms: Problems are partitioned into cells or strips that fit within available workspace. The input is batch-processed with careful streaming and merging of local solutions, optimizing both running time and storage (Elmasry et al., 2015).
• Procedural Sweeps for Solid Envelope Construction: For solid sweeps in CAD, sweeping is performed analytically with a procedural model backed by Newton–Raphson iterative methods. The envelope's parametrization is determined through seed surfaces and point-wise refinement, informed by local geometric invariants (Adsul et al., 2012).
• Dynamic Programming in Grid-Based Sweeps: Single-pass grid sweeps solve visibility and path planning problems by transporting scalar quantities (e.g., visibility level) using discrete PDE upwinds, requiring only a fixed number of operations per cell without any preprocessing (Ibrahim et al., 11 Mar 2024).
• Plane Sweeping in Multi-View 3D Reasoning: The strategy is adapted for 3D object or plane reconstruction by warping feature volumes across planes—either fronto-parallel or slanted—using differentiable homographies. This yields pixel-wise or voxel-wise depth or geometry maps in a single traversal through the hypothesis space, regularized via soft probabilistic mechanisms (Liu et al., 2022, Xu et al., 28 Oct 2024).

3. Dependency Management and Update Ordering

A central challenge in single-pass sweeping is the establishment and exploitation of acyclic dependencies:

• Streaming Operator Inversion: In discrete ordinates solvers, the physical flow of information dictates the update order. The domain (e.g., triangle mesh) is translated into a dependency graph, where directed edges embody the directional coupling (normal vectors with positive inner product with the sweep direction). Topological sorting guarantees a valid single-pass update sequence if the graph is acyclic (Camminady et al., 2018).
• Self-Intersection Detection in Sweeps: For geometric envelopes, local anomalies (self-intersections, singularities) are detected via continuous invariants. For a point $p=(u,v,t)$, a function $\theta(p)$ evaluates rank deficiencies and self-penetration, guiding real-time trimming or correction within the sweep (Adsul et al., 2012).
• Analytic Formulas and Intrinsic Metrics: In pursuit-evasion or coverage, dependencies are encoded by geodesic Fréchet distances with winding number constraints, directly linking the sweep cost to intrinsic domain width for convex regions, and dictating endpoint pairings for sensor curves (Adams et al., 2016).

4. Computational and Resource Efficiency

Single-pass sweeping achieves major gains in efficiency and scalability due to:

• Offlining / Workspace Batching: Only a local slab of the data must be in-memory at any time (tessellation (Trencséni et al., 2012); geometric intersection (Elmasry et al., 2015)). When the sweep passes beyond a cell’s or simplex’s influence (as prescribed by circumsphere–sweep plane tests, or data partition boundaries), affected data is serialized to disk or merged into global output.
• Dynamic Programming / Entropy-Satisfying Schemes: Grid-based visibility sweeps use upwind discretization of the transport equation ($\mathcal{O}(n)$ complexity), eliminating the need for repeated ray casting or visibility polygon computation (Ibrahim et al., 11 Mar 2024).
• Procedural Parametrizations: When closed-form representations are unavailable, iterative seed-and-refinement strategies yield accurate geometric output using only local queries and derivative information (Adsul et al., 2012).

5. Applications Across Domains

Single-pass plane sweeping is foundational in:

• Computational Geometry: Delaunay tessellation, Voronoi partitioning, intersection reporting, Klee’s measure, hidden surface removal in graphics [(Trencséni et al., 2012), 9910017, (Elmasry et al., 2015)].
• Numerical Physics/Engineering: Neutron transport, radiative transfer, contamination clearing, sensor-driven coverage (Camminady et al., 2018, Adams et al., 2016).
• Computer Vision and Robotics: Multi-view 3D reconstruction, object segmentation, path planning in grid environments, visibility-based heuristics (Liu et al., 2022, Xu et al., 28 Oct 2024, Ibrahim et al., 11 Mar 2024).
• Arrangement Sweeping in Topology: Continuous deformation and maintenance of intersection properties for families of curves (generalization to non-piercing or pseudo-disc arrangements) (Dalal et al., 25 Mar 2024).

6. Generalizations, Limitations, and Extensions

While powerful, single-pass plane sweeping strategies are constrained by:

• Mesh Topology: Guaranteeing acyclic dependencies requires careful mesh design—regular triangles and no hanging nodes in 2D. Extension to higher dimensions or non-conforming discretizations may introduce cycles or ambiguities (Camminady et al., 2018).
• Resolution and Sampling: Adaptive sampling rates are necessary for handling local complexity—finite-resolution sweeps may require density adjustments to capture fine-grained features [9910017].
• Parametric and Intrinsic Complexity: Analytic metrics (e.g., geodesic Fréchet) provide optimality guarantees in continuous domains, but may complicate computation in nonconvex or obstacle-laden environments (Adams et al., 2016).
• Generalization to Arbitrary Arrangements: Extensions from 2-intersecting curves to non-piercing curves preserve sweeping capability but may necessitate more sophisticated combinatorial or coloring arguments for algorithmic tractability (Dalal et al., 25 Mar 2024).

7. Impact and Outlook

By uniting geometric insight, dependency management, and computational efficiency, single-pass plane sweeping strategies continue to drive advances across large-scale data processing, fast geometric search, real-time graphics, and optimal coverage theory. Their adaptability to diverse domains—anchored by strict guarantees of update order, resource bounds, and output correctness—renders them essential to both theoretical and applied research programs in computational mathematics, physics-based simulation, and intelligent systems.