On-the-Fly Delaunay Tetrahedralization

• On-the-fly Delaunay tetrahedralization is a dynamic method that incrementally builds and refines 3D mesh structures while preserving the Delaunay condition.
• It supports efficient real-time updates via point insertions, deletions, and repositioning using techniques like the Bowyer–Watson process and local retriangulation.
• Its integration with adaptive sampling, parallel processing, and GPU acceleration supports scalable applications in SLAM, 3D reconstruction, and computational geometry.

On-the-fly Delaunay tetrahedralization refers to the dynamic construction and refinement of Delaunay tetrahedralizations of a point set in $\mathbb{R}^3$, where points are incrementally inserted, deleted, or repositioned, typically as part of real-time 3D surface extraction, shape optimization, or adaptive sampling frameworks. Unlike classical, static algorithms, on-the-fly approaches support streaming data, accommodate evolving or non-stationary point sets, and interleave tetrahedralization steps with numerical optimization, geometric refinement, or function evaluation to enable efficient, data-driven mesh structures for scientific computing, computer vision, and computational geometry.

1. Foundations of On-the-Fly Delaunay Tetrahedralization

A Delaunay tetrahedralization of a point set $P=\{p_1, \dots, p_n\} \subset \mathbb{R}^3$ is a partition of the convex hull of $P$ into tetrahedra such that the circumsphere of each tetrahedron contains no other point of $P$ in its interior. This property, formalized via the $\mathrm{inSphere}$ predicate, is central to all incremental and adaptive schemes (Binninger et al., 7 May 2025, Takikawa et al., 5 May 2026, Romanoni et al., 2015).

Dynamic or on-the-fly Delaunay tetrahedralization expands classical methods (such as Bowyer–Watson incremental insertion) to admit not only insertions but also dynamic removal, movement, and attribute updates to mesh vertices during online meshing or optimization. Streaming point sources (e.g. visual SFM/SLAM, adaptive reconstruction, or occupancy field evaluation) require the mesh to maintain its Delaunay property throughout, while supporting efficient retriangulation and local updates (Romanoni et al., 2015, Takikawa et al., 5 May 2026, Marot et al., 2018).

2. Data Structures and Key Predicates

Efficient on-the-fly tetrahedralization relies on explicit mesh representations and specialized data structures:

• Tetrahedral Mesh Arrays: Points (as $\texttt{point3d}_t$) are stored in contiguous, cache-optimized arrays. Tetrahedra maintain four vertex indices and adjacency links ($\texttt{neighbor\_IDs}$), allowing $O(1)$ access to neighbors for cavity expansion and boundary face tracking (Marot et al., 2018, Binninger et al., 7 May 2025).
• Spatial Acceleration: To quickly locate the containing tetrahedron for point insertions, spatial indices such as bounding-volume hierarchies or recent-insertion hints are used (Binninger et al., 7 May 2025, Takikawa et al., 5 May 2026).
• Steiner Points: To avoid large empty tetrahedra and ensure spatial locality of retriangulation, uniform "dummy" points are injected; these ensure the mesh remains well-behaved when points are sparsely distributed (Romanoni et al., 2015).
• Formal Predicates:
  • The 3D Delaunay condition is enforced via

  $\mathrm{inSphere}(p_1, p_2, p_3, p_4; q) < 0,$

  for every tetrahedron $\Delta = (p_1,\dots,p_4)$ and every $P=\{p_1, \dots, p_n\} \subset \mathbb{R}^3$0 (Romanoni et al., 2015, Binninger et al., 7 May 2025, Takikawa et al., 5 May 2026). - The local conflict region (cavity) for insertion/removal is precisely the set of tetrahedra whose circumsphere contains the new/deleted point.

3. Algorithmic Strategies: Insertion, Deletion, and Refinement

Incremental Insertion and Refinement

Point insertion uses the Bowyer–Watson process: locate a containing tetrahedron, discover the conflict cavity (set of tetrahedra whose circumspheres contain the point), remove this cavity, and retriangulate the resulting polyhedral hole by connecting the new point to all faces on its boundary. Local flips restore the Delaunay property as necessary (Binninger et al., 7 May 2025, Marot et al., 2018, Takikawa et al., 5 May 2026).

Adaptive refinement, as in Adaptive Delaunay Scaffolding (ADS), leverages on-the-fly DT to drive mesh sampling towards regions of high signal (e.g. zero crossings of an occupancy function): crossing edges are detected, then split by batched point insertions, with local reassessment of Delaunay cavities and adjacency pointers. Each insertion and retriangulation is $P=\{p_1, \dots, p_n\} \subset \mathbb{R}^3$1 amortized (Takikawa et al., 5 May 2026).

Deletion and Movement

Vertex deletion carves out the star-shaped cavity of incident tetrahedra and retriangulates the resulting hole. Movement of points (key in SLAM/SFM contexts) is decomposed as deletion at the old position followed by insertion at the new one, with backward/forward updates for all associated attributes (e.g. visibility weights) (Romanoni et al., 2015, Marot et al., 2018).

Moving-point update steps (Romanoni et al., 2015):

1. Backward ray-casting: decrement affected tetrahedra's weights along visibility rays from previous frames.
2. Remove old vertex and retriangulate its star cavity.
3. Insert new vertex and locally retriangulate insertion cavity.
4. Forward ray-casting: increment tetrahedra weights based on new visibility.
5. Transfer weights to new tetrahedra by associating each with the nearest deleted predecessor’s centroid.

Parallelization and Acceleration

For large-scale or real-time processing, parallel insertion is enabled by space-filling curve (e.g. Moore/Hilbert) partitioning, batching independent insertion tasks to threads, and lock-free memory management of new tetrahedra regions (Marot et al., 2018). Streaming and deletion tasks exploit the same infrastructure for consistent high-throughput updates.

4. Integration with Sampling, Optimization, and Surface Extraction

On-the-fly Delaunay tetrahedralization serves as an adaptive scaffold for a range of geometric and differentiable tasks:

• Implicit Function Sampling: In ADS, on-the-fly DT is interleaved with occupancy function queries for isosurface extraction: all long crossing edges are batched, safe midpoints are inserted, and the mesh scaffolds converges towards $P=\{p_1, \dots, p_n\} \subset \mathbb{R}^3$2, the desired surface (Takikawa et al., 5 May 2026).
• Gradient-based Mesh Optimization: TetWeave employs online DT for mesh optimization, jointly updating grid point locations and per-vertex signed distances (including spherical harmonic coefficients for directionality), allowing for adaptive meshing and differentiable surface extraction under user-specified loss functionals (Binninger et al., 7 May 2025).
• Surface Extraction: Marching tetrahedra is performed directly on the Delaunay mesh, with each tetrahedron contributing triangles where its edges cross the target surface. This ensures watertight, intersection-free manifolds and supports near-linear memory scaling relative to output size (Binninger et al., 7 May 2025, Takikawa et al., 5 May 2026).
• Adaptive Resampling: Mesh density adapts by dropping passive points and resampling in high-error regions based on voxelized error statistics, further refined by barycentric interpolation and re-insertion into the evolving DT (Binninger et al., 7 May 2025).

5. Complexity, Memory Usage, and Experimental Insights

The efficiency and scalability of on-the-fly Delaunay tetrahedralization approaches are characterized by the following complexity and empirical metrics:

Method/Scenario	Complexity per Operation	Empirical Memory (per vertex)	Throughput/Performance
Serial (sorted, random)	$P=\{p_1, \dots, p_n\} \subset \mathbb{R}^3$3 total	32–332 bytes	3M–65M tetrahedra/s (up to 3B in 53s)
Online adaptive insertion	$P=\{p_1, \dots, p_n\} \subset \mathbb{R}^3$4 per insertion	Near-linear in $P=\{p_1, \dots, p_n\} \subset \mathbb{R}^3$5	Real-time, 10fps SFM frame rate
Moving-point update	$P=\{p_1, \dots, p_n\} \subset \mathbb{R}^3$6 per move ($P=\{p_1, \dots, p_n\} \subset \mathbb{R}^3$7 const)	-	50× faster than prior, $P=\{p_1, \dots, p_n\} \subset \mathbb{R}^3$80.04s/move
Mesh optimization (TetWeave)	$P=\{p_1, \dots, p_n\} \subset \mathbb{R}^3$9 (DT), $P$0 Marching Tets	$P$1 output	Adaptive, memory-efficient

Key experimental findings include real-time mesh updating at $P$2 fps for moving points (Romanoni et al., 2015), linear speedup in parallel implementations up to memory/core limits (Marot et al., 2018), and order-of-magnitude reductions in required function evaluations for surfacing implicit models (Takikawa et al., 5 May 2026). The adaptive nature of on-the-fly DTs ensures attention is focused on geometrically or numerically salient regions, leading to efficient use of computational and memory resources.

6. Application Domains and Recent Extensions

On-the-fly Delaunay tetrahedralization underpins modern approaches in multi-view 3D reconstruction, mesh compression, geometric texture generation, and differentiable isosurface extraction (Binninger et al., 7 May 2025, Takikawa et al., 5 May 2026). In SLAM/SFM, dynamic point management and online mesh updating are critical for manifold surface tracking in non-static scenes (Romanoni et al., 2015). In geometry processing and graphics, the framework supports adaptive detail placement and real-time mesh refinement.

Recent advances include:

• Batching and parallelization schemes for high-throughput, fine-grained streaming of input points (Marot et al., 2018).
• Hybrid approaches that leverage GPU acceleration for function querying, inSphere predicates, and batched mesh updates (Takikawa et al., 5 May 2026, Romanoni et al., 2015).
• Extensions to non-rigid and multi-body scenarios via integration of motion fields and per-point trajectory management (Romanoni et al., 2015).

7. Limitations and Outlook

Despite significant advances, limitations persist:

• The global-local trade-off governed by batch/window sizes impacts both reconstruction accuracy and real-time responsiveness (Romanoni et al., 2015).
• Uniform Steiner grids, while robust, may be inefficient for highly inhomogeneous point clouds; adaptive grid insertion, such as octree-based schemes, can improve spatial efficiency (Romanoni et al., 2015).
• The worst-case retriangulation can approach $P$3 in degenerate configurations, though spatial sorting and cache-aware memory layouts mitigate this in practice (Marot et al., 2018).

Ongoing research focuses on further reducing synchronization overheads, extending frameworks to support dynamically moving bodies, bridging to differentiable programming models, and exploiting hardware accelerators for massive-scale dynamic meshing (Binninger et al., 7 May 2025, Takikawa et al., 5 May 2026, Romanoni et al., 2015, Marot et al., 2018).