delaunay

Rust crate providing D-dimensional Delaunay triangulations and convex hulls (2D through 5D explicitly tested) constructed with a PL-manifold (default) or pseudomanifold guarantee on finite point sets with Euclidean and toroidal global topologies. Uses exact predicates and Simulation of Simplicity for robustness and degeneracy handling, and Hilbert curves for deterministic insertion ordering and efficient spatial indexing. Provides an explicit 4-level validation hierarchy on individual elements, triangulation data structure validity, manifold topology, and Delaunay property adherence. Allows for the complete set of Pachner moves up to D=5 using bistellar flips, vertex insertion and deletion, and the conversion of non-Delaunay triangulations into Delaunay triangulations via bounded flip/rebuilds. Auxiliary data may be stored directly in vertices and simplices with external secondary maps provided for vertex- and simplex-keyed algorithm use, and the entire data structure is serializable/deserializable. Written in safe Rust with no unsafe code.

📐 Introduction

This library implements dimension-generic Delaunay triangulations in Rust for computational geometry and scientific workflows. It is inspired by CGAL, which is a C++ library for computational geometry, and Spade, a Rust library focused on 2D triangulations. The goal is to provide a lightweight but rigorous Rust-native option for workflows that need explicit topology settings, validation levels, deterministic construction controls, and repair behavior.

The core idea is an implemented and validated algorithmic framework for robust Delaunay construction, PL-manifold-aware local moves, and bistellar repair. The software artifact is part of the contribution: public APIs, examples, property-based tests, diagnostics, and generated performance summaries are kept together so the library can be inspected, reused, and benchmarked rather than treated as a one-off implementation.

Use this crate when you want:

• Delaunay triangulations or convex hulls in 2D through 5D.
• A Rust-native bistellar flip / Pachner move API for topological editing and Delaunay repair.
• Exact predicates and deterministic Simulation of Simplicity (SoS) handling for degenerate inputs.
• PL-manifold checks and explicit topology guarantees for triangulations that need more than a bag of simplices.
• Typed construction, insertion, validation, topology, and repair diagnostics instead of stringly error handling.
• Validation reports that separate element, structure, topology, and Delaunay failures.
• Batch construction controls for insertion order, deduplication, repair cadence, and retry behavior.
• Release-mode 3D construction through 10,000 vertices with final Levels 1–4 validation in the current large-scale characterization harness.
• PL-manifold-aware editing via bistellar flips.

This is not a replacement for full meshing packages such as CGAL, TetGen, or Gmsh when you need constrained Delaunay triangulations, out-of-core meshing, GPU/parallel meshing, or production-scale dynamic remeshing.

🧪 Scientific Basis

This crate models triangulations of finite point sets in R^d as oriented simplicial complexes with explicit combinatorial and geometric checks. The framework couples robust predicates, topology-aware local moves, repair policies, validation reports, and benchmarked workflows instead of treating these as separate utilities.

• Core operational invariant for editing/repair: coherent orientation + PL-manifold validity.
• Local move system: exposed bistellar flips (k = 1, 2, 3 and inverses), providing the supported Pachner moves set in dimensions up to 5D.
• Geometric convergence basis in finite-point workflows: Herbert Edelsbrunner and Nimish R. Shah, Incremental Topological Flipping Works for Regular Triangulations, Algorithmica (1996), https://doi.org/10.1007/BF01975867.
• Robust predicate basis: Shewchuk-style floating-point filters with exact fallback, plus deterministic Simulation of Simplicity for exact degeneracies.
• Topology basis: Level 3 validation checks PL-manifold structure, links, and Euler/topological consistency before Level 4 Delaunay predicates are trusted.
• Artifact basis: unit tests, integration tests, property tests, examples, release benchmark summaries, and docs.rs documentation exercise the same public APIs users copy.
• Scope of claims: guarantees apply to supported library workflows (construction, flip-based repair, and validation APIs), not arbitrary external mutation of internal structures.

See REFERENCES.md, Invariants, and the Numerical Robustness Guide for the complete technical background.

✨ Features

• Copyable data types associated with vertices and simplices (integers, floats, chars, custom enums), plus SimplexSecondaryMap and VertexSecondaryMap aliases for caller-owned key-indexed algorithm state
• D-dimensional Delaunay triangulations
• D-dimensional Convex hulls
• Bistellar flip / Pachner moves Edit API up to 5D: k-flips for k = 1, 2, 3 plus inverse moves
• Delaunay repair using bistellar flips for k=2/k=3 with inverse edge/triangle queues in 4D/5D
• Simulation of Simplicity (SoS) for deterministic handling of degenerate orientation and in-sphere configurations
• PL-manifold topology validation by default, with Pseudomanifold available as an explicit opt-out
• Toroidal (periodic) triangulations via DelaunayTriangulationBuilder: .toroidal(...) canonicalizes points into the fundamental domain, while .toroidal_periodic(...) builds a validated periodic image-point quotient in 2D and compact 3D
• Geometry quality metrics for simplices: radius ratio and normalized volume (dimension-agnostic)
• Serialization/deserialization of all data structures to/from JSON
• Tested for 2-, 3-, 4-, and 5-dimensional triangulations
• Focused public preludes for construction, insertion, repair, validation, topology, query, geometry, generators, ordering, collections, TDS, and diagnostics workflows
• Configurable predicate kernels: AdaptiveKernel (default; exact arithmetic + SoS), RobustKernel (exact, preserves degeneracy signals), FastKernel (raw floating-point predicates for well-conditioned exploratory work; not accepted by explicit repair APIs)
• Bulk insertion ordering (InsertionOrderStrategy): Hilbert curve (default) or input order
• Batch construction options (ConstructionOptions): optional deduplication, configurable local Delaunay repair cadence, and deterministic retries
• Incremental construction APIs: insertion, insertion statistics, and transactional vertex removal (remove_vertex)
• 4-level validation hierarchy (element validity → TDS structural validity → manifold topology → Delaunay property), including full diagnostics via validation_report
• Coherent combinatorial orientation validation/normalization for simplices, maintaining oriented simplicial complexes
• Typed construction, insertion, TDS, topology, validation, and repair errors that preserve source context for callers and diagnostics
• Reusable research software artifact: public examples, property tests, diagnostics, docs.rs landing-page documentation, and benchmark summaries generated from checked public workflows
• Release performance summaries generated from the public API benchmark contract, current Criterion run metadata, and generated simplex counts
• 10,000-vertex 3D large-scale characterization run: zero skipped vertices, final flip repair clean, and validation_report OK for Levels 1–4 in roughly 100 seconds on maintainer Apple M4 Max hardware
• Safe Rust: #![forbid(unsafe_code)]

See CHANGELOG.md for release details. Older releases are archived under docs/archive/changelog/.

🟢 Minimal Construction Example

The construction API has two entry points:

• DelaunayTriangulationBuilder - primary construction interface for the common case and advanced configuration (custom options, toroidal topology, custom kernels)
• DelaunayTriangulation::new(&vertices) - legacy convenience constructor

Add the library to your crate:

cargo add delaunay

Choose the smallest prelude that matches the task:

Task	Import
Construct/configure a Delaunay triangulation	use delaunay::prelude::construction::*
Read-only traversal, adjacency, convex hulls, and comparison helpers	use delaunay::prelude::query::*
Points, kernels, predicates, and geometric measures	use delaunay::prelude::geometry::*
Random points or triangulations for examples, tests, and benchmarks	use delaunay::prelude::generators::*
Low-level incremental insertion building blocks	use delaunay::prelude::insertion::*
Bistellar flips / Edit API	use delaunay::prelude::flips::*
Delaunay repair diagnostics and policies	use delaunay::prelude::repair::*
Delaunayize workflow	use delaunay::prelude::delaunayize::*
Construction telemetry diagnostics	use delaunay::prelude::diagnostics::*
Construction validation cadence/policy	use delaunay::prelude::validation::*
Hilbert ordering and quantization utilities	use delaunay::prelude::ordering::*
Low-level TDS simplices, facets, keys, and validation reports	use delaunay::prelude::tds::*
Collection aliases and small buffers	use delaunay::prelude::collections::*
Topology validation and Euler characteristic helpers	use delaunay::prelude::topology::validation::*
Topological spaces and topology traits	use delaunay::prelude::topology::spaces::*

use delaunay::prelude::* remains available for quick experiments, but examples and benchmarks in this repository prefer focused preludes so imports document intent. The broad delaunay::prelude::* import is retained for compatibility, but new docs and tests should prefer the narrow workflow preludes above.

Low-level imports

delaunay::core is an internal implementation namespace. Public low-level APIs are exposed through delaunay::tds, delaunay::collections, delaunay::algorithms, and delaunay::query, plus the matching focused preludes. Contributors should follow the namespace policy in CONTRIBUTING.md and docs/code_organization.md.

use delaunay::prelude::construction::{
    DelaunayTriangulationBuilder, DelaunayTriangulationConstructionError, vertex,
};

fn main() -> Result<(), DelaunayTriangulationConstructionError> {
    let vertices = vec![
        vertex!([0.0, 0.0, 0.0, 0.0]),
        vertex!([1.0, 0.0, 0.0, 0.0]),
        vertex!([0.0, 1.0, 0.0, 0.0]),
        vertex!([0.0, 0.0, 1.0, 0.0]),
        vertex!([0.0, 0.0, 0.0, 1.0]),
        vertex!([0.2, 0.2, 0.2, 0.2]),
    ];

    let dt = DelaunayTriangulationBuilder::new(&vertices).build::<()>()?;

    assert_eq!(dt.dim(), 4);
    assert_eq!(dt.number_of_vertices(), 6);

    // Optional verification:
    // - `dt.is_valid()` checks Level 4 only (Delaunay property).
    // - `dt.validate()` checks Levels 1-4 (elements + structure + topology + Delaunay).
    assert!(dt.validate().is_ok());
    Ok(())
}

Toroidal (Periodic) Triangulations

For periodic boundary conditions, use DelaunayTriangulationBuilder:

use delaunay::prelude::construction::{
    DelaunayTriangulationBuilder, DelaunayTriangulationConstructionError, TopologyKind, vertex,
};

fn main() -> Result<(), DelaunayTriangulationConstructionError> {
    let vertices = vec![
        vertex!([0.1, 0.2]),
        vertex!([0.8, 0.3]),
        vertex!([0.5, 0.7]),
        vertex!([1.2, 0.4]), // Wraps to [0.2, 0.4]
    ];

    let dt = DelaunayTriangulationBuilder::new(&vertices)
        .toroidal([1.0, 1.0])
        .build::<()>()?;

    assert_eq!(dt.topology_kind(), TopologyKind::Toroidal);
    Ok(())
}

For boundary-facet identification and periodic neighbor pointers, use .toroidal_periodic([..]) in 2D or compact 3D; see the toroidal construction workflow for the full recipe and current 4D/5D guardrails.

Need more control?

• Editing with flips (Edit API): see docs/workflows.md for a minimal example and docs/api_design.md for details.
• Flip-based Delaunay repair, including the heuristic rebuild fallback (repair_delaunay_with_flips*): see docs/workflows.md.
• Repair diagnostics and mutating-operation rollback: remove_vertex rolls back if post-removal repair or orientation canonicalization fails, and repair failures preserve typed source errors for debugging.
• Insertion outcomes and statistics (insert_with_statistics, insert_best_effort_with_statistics, InsertionOutcome, InsertionStatistics): see docs/workflows.md and docs/numerical_robustness_guide.md.
• Topology guarantees (TopologyGuarantee) and automatic topology validation (ValidationPolicy): see docs/validation.md and docs/topology.md.
• Release benchmark summaries: see benches/README.md and benches/PERFORMANCE_RESULTS.md.

✅ Validation and Guarantees

Level	What is validated	Primary API
1	Element validity (vertex/simplex primitives)	dt.validate() / dt.validation_report()
2	TDS structural validity (keys, incidences, neighbors)	dt.tds().is_valid()
3	Manifold topology (link checks, Euler/topological consistency)	dt.as_triangulation().is_valid()
4	Delaunay property (empty-circumsphere via local predicates)	dt.is_valid()
1-4	Cumulative checks with diagnostics	dt.validate() or dt.validation_report()

TopologyGuarantee controls which Level 3 manifold constraints are enforced, and ValidationPolicy controls when Level 3 checks run automatically during incremental insertion. Incompatible combinations are rejected by the fallible try_set_* policy setters; use ValidationPolicy::ExplicitOnly for caller-owned full-validation checkpoints with the default PL-manifold guarantee.

🔬 Reproducibility

The construction pipeline exposes deterministic controls for experiments and regression testing:

• Deterministic insertion ordering via InsertionOrderStrategy: Hilbert (default) or Input (use Input to preserve caller-provided order exactly)
• Deterministic preprocessing via DedupPolicy
• Deterministic retry behavior via RetryPolicy (including seeded shuffled retries) or RetryPolicy::Disabled
• Cadenced batch repair behavior via ConstructionOptions when large batch construction should repair local Delaunay fronts during insertion
• Explicit topology/validation configuration via TopologyGuarantee and ValidationPolicy

use delaunay::prelude::construction::{
    ConstructionOptions, DedupPolicy, DelaunayTriangulationBuilder, InsertionOrderStrategy,
    RetryPolicy, TopologyGuarantee, DelaunayTriangulationConstructionError, vertex,
};
use delaunay::prelude::validation::ValidationPolicy;

fn main() -> Result<(), DelaunayTriangulationConstructionError> {
    let vertices = vec![
        vertex!([0.0, 0.0]),
        vertex!([1.0, 0.0]),
        vertex!([0.0, 1.0]),
    ];

    let options = ConstructionOptions::default()
        .with_insertion_order(InsertionOrderStrategy::Input)
        .with_dedup_policy(DedupPolicy::Exact)
        .with_retry_policy(RetryPolicy::Disabled);

    let mut dt = DelaunayTriangulationBuilder::new(&vertices)
        .topology_guarantee(TopologyGuarantee::PLManifold)
        .construction_options(options)
        .build::<()>()?;

    dt.set_validation_policy(ValidationPolicy::Always);
    assert!(dt.validate().is_ok());
    Ok(())
}

For reproducible checks in CI/local runs, use just check, just test, just doc-check, or just ci.

⚠️ Limitations

• Dimension coverage: CI and property-test coverage target 2D–5D.
• Exact predicate limits: exact orientation is available through D=6; exact in-sphere is available through D=5. For D≥6, in-sphere classification relies on symbolic perturbation and deterministic tie-breaking because the (D+2)×(D+2) determinant exceeds the stack matrix limit.
• Periodic domains: .toroidal() canonicalizes coordinates into the fundamental domain. .toroidal_periodic() uses the periodic image-point method and is release-validated in 2D and compact 3D. 4D/5D periodic quotients fail fast pending scalable construction work in issue #416.
• Large 4D+ batches: thousands of 4D points can be expensive to investigate. Use release mode and the large-scale debug harness for characterization.
• 3D scale: the default just debug-large-scale-3d helper uses 7,500 vertices for the near-one-minute acceptance path. The 10,000-vertex run has also passed full Levels 1–4 validation as a heavier characterization probe; use just debug-large-scale-3d 10000 1 for local numbers.
• Feature gaps: Constrained Delaunay triangulations, Voronoi diagrams, built-in visualization, GPU/parallel meshing, and out-of-core construction are out of scope today.
• Validation/repair guarantees assume the library-managed construction/editing pipeline.

See Limitations and Scope for details and Roadmap for active follow-up work.

🚧 Project History

This crate was originally maintained at https://github.com/oovm/shape-rs through version 0.1.0. The original implementation provided basic Delaunay triangulation functionality.

Starting with version 0.3.4, maintenance transferred to this repository, which hosts a rewritten d-dimensional implementation focused on computational geometry research applications.

• 📚 Docs for the original implementation (0.1.0): https://docs.rs/delaunay/0.1.0/delaunay/
• 📚 Docs for the rewritten implementation (0.3.4+): https://docs.rs/delaunay

🤝 How to Contribute

We welcome contributions! Here's a quickstart:

# Clone and setup
git clone https://github.com/acgetchell/delaunay.git
cd delaunay

# Setup development environment (installs tools, builds project)
cargo install just
just setup            # Installs all development tools and dependencies

# Development workflow
just check            # All non-mutating lints/validators
just fix              # Apply formatters/auto-fixes (mutating)
just test             # Tests + benchmark/release compile smoke
just ci               # Comprehensive checks + tests + examples
just --list           # See all available commands
just help-workflows   # Show common workflow patterns

Benchmark commands that produce performance data use the perf Cargo profile for consistent ThinLTO settings. just ci remains the comprehensive validation path: it runs checks, the test workflow, and examples, but it does not pay the perf profile cost unless measuring performance.

For release performance documentation, run just bench-perf-summary from the release PR branch after version and documentation updates. It refreshes benches/PERFORMANCE_RESULTS.md from the public API Criterion suite, circumsphere predicate benchmarks, current run metadata, and generated simplex counts.

Try the examples:

just examples         # Run all examples
# Or run specific examples:
cargo run --release --example triangulation_and_hull
cargo run --release --example delaunayize_repair

📋 Examples

The examples/ directory contains several demonstrations:

• delaunayize_repair: Demonstrates the delaunayize_by_flips workflow (bounded topology repair + flip-based Delaunay repair + optional fallback)
• diagnostics: Opt-in structured diagnostics for validation and deliberately non-Delaunay TDS examples
• into_from_conversions: Demonstrates Into/From trait conversions and utilities
• numerical_robustness: Compares FastKernel, RobustKernel, and AdaptiveKernel on degenerate predicate inputs
• point_comparison_and_hashing: Demonstrates point comparison and hashing behavior
• topology_editing: Builder API vs Edit API in 2D/3D (bistellar flips and Delaunay preservation)
• triangulation_and_hull: Seeded 3D and 4D triangulations, boundary traversal, convex hull extraction, and hull containment/visibility queries

For detailed documentation, sample output, and usage instructions for each example, see examples/README.md.

For comprehensive guidelines on development environment setup, testing, benchmarking, performance analysis, and development workflow, please see CONTRIBUTING.md.

This includes information about:

• Building and testing the library
• Running benchmarks and performance analysis
• Code style and standards
• Submitting changes and pull requests
• Project structure and development tools

📖 Documentation

• API Design - Builder vs Edit API design (explicit bistellar flips)
• Benchmarks - Benchmark suites, perf-profile workflow, generated summaries, and release canaries
• Code Organization - Project structure and module patterns
• Diagnostics - Diagnostic helpers, structured reports, telemetry, and debug switches
• Invariants - Theoretical background and rationale for the topological and geometric invariants
• Numerical Robustness Guide - Robustness strategies, kernels, and retry/repair behavior
• Orientation Spec - Coherent combinatorial and geometric orientation rules
• Property Testing Summary - Property-based testing with proptest (where tests live, how to run)
• Limitations and Scope - Supported dimensions, predicate limits, and feature gaps
• Releasing - Release workflow (changelog + benchmarks + publish)
• Roadmap - Current follow-up work and deferred features
• Topology - Level 3 topology validation (manifoldness + Euler characteristic) and module overview
• Validation Guide - Comprehensive 4-level validation hierarchy guide (element → structural → manifold → Delaunay)
• Workflows - Happy-path construction plus practical Builder/Edit recipes (stats, repairs, and minimal flips)

📎 How to Cite

If you use this software in academic work, cite the Zenodo DOI and include the software metadata from CITATION.cff.

• DOI: https://doi.org/10.5281/zenodo.16931097
• Citation metadata: CITATION.cff

@software{getchell_delaunay,
  author = {Adam Getchell},
  title = {delaunay: A d-dimensional Delaunay triangulation library},
  doi = {10.5281/zenodo.16931097},
  url = {https://github.com/acgetchell/delaunay}
}

For release-specific fields (version, release date, ORCID), prefer CITATION.cff.

📚 References

For a comprehensive list of academic references and bibliographic citations used throughout the library, see REFERENCES.md.

🤖 AI Agents

This repository contains an AGENTS.md file, which defines the canonical rules and invariants for all AI coding assistants and autonomous agents working on this codebase.

AI tools (including ChatGPT, Claude, CodeRabbit, Codex, Cursor, GitHub Copilot, KiloCode, Warp, and CI repair agents) are expected to read and follow AGENTS.md when proposing or applying changes.

Portions of this library were developed with the assistance of these AI tools:

• ChatGPT
• Claude
• CodeRabbit
• Codex
• GitHub Copilot
• KiloCode
• WARP

All code was written and/or reviewed and validated by the author.