# API Design: Two-Track Approach
This document explains the dual API design for working with Delaunay triangulations:
the **Builder API** for constructing and maintaining Delaunay triangulations, and
the **Edit API** for explicit topology editing via bistellar flips.
## Overview
The library provides two distinct APIs for different use cases:
1. **Builder API** (`DelaunayTriangulation::insert` / `::remove_vertex`)
- High-level construction and maintenance of Delaunay triangulations
- Automatically maintains the Delaunay property (empty circumsphere)
- Designed for building triangulations from point sets
- Uses cavity-based insertion and fan retriangulation
2. **Edit API** (`prelude::triangulation::flips::BistellarFlips` trait)
- Low-level topology editing via bistellar (Pachner) flips
- Explicit control over individual topology operations
- Does **not** automatically restore the Delaunay property
- Designed for topology manipulation, research, and custom algorithms
## When to Use Each API
### Use the Builder API when
- Building a Delaunay triangulation from a set of points
- Adding/removing vertices while maintaining the Delaunay property
- You need automatic geometric property preservation
- Working with standard computational geometry workflows
**Example use cases:**
- Computing convex hulls
- Nearest-neighbor queries
- Voronoi diagram construction
- Mesh generation
- Scientific simulations requiring Delaunay meshes
### Use the Edit API when
- Implementing custom topological algorithms
- Researching bistellar flip sequences
- Building non-Delaunay triangulations with specific properties
- Experimenting with topology transformations
- You need explicit control over topology changes
**Example use cases:**
- Implementing custom Delaunay repair strategies
- Topology optimization algorithms
- Research on triangulation properties
- Building triangulations with non-Delaunay constraints
- Educational demonstrations of bistellar flip theory
## Builder API Reference
### Simple Construction: `DelaunayTriangulation::new()`
For most use cases, the simple constructor is sufficient:
```rust
use delaunay::prelude::triangulation::*;
// Simple construction from vertices (Euclidean space, default options)
let vertices = vec![
vertex!([0.0, 0.0, 0.0]),
vertex!([1.0, 0.0, 0.0]),
vertex!([0.0, 1.0, 0.0]),
vertex!([0.0, 0.0, 1.0]),
];
let mut dt: DelaunayTriangulation<_, (), (), 3> =
DelaunayTriangulation::new(&vertices).unwrap();
// Incremental insertion (maintains Delaunay property)
let new_vertex = vertex!([0.5, 0.5, 0.5]);
dt.insert(new_vertex).unwrap();
// Vertex removal (maintains Delaunay property via fan retriangulation)
let vertex_key = /* ... */;
dt.remove_vertex(vertex_key).unwrap();
```
### Advanced Construction: `DelaunayTriangulationBuilder`
For advanced configuration (toroidal topology, custom validation policies, etc.),
use `DelaunayTriangulationBuilder`:
```rust
use delaunay::prelude::triangulation::*;
use delaunay::core::triangulation::{TopologyGuarantee, ValidationPolicy};
// Toroidal (periodic) triangulation in 2D
let vertices = vec![
vertex!([0.1, 0.1]),
vertex!([0.9, 0.9]),
vertex!([0.5, 0.5]),
];
let mut dt = DelaunayTriangulationBuilder::new(&vertices)
.toroidal([1.0, 1.0]) // Phase 1: canonicalized toroidal construction
.topology_guarantee(TopologyGuarantee::PLManifoldStrict)
.build::<()>()
.unwrap();
dt.set_validation_policy(ValidationPolicy::Always);
// Works like any other DelaunayTriangulation
dt.insert(vertex!([0.25, 0.75])).unwrap();
```
**When to use the Builder:**
- **Toroidal/periodic triangulations**: Use `.toroidal()` (Phase 1 canonicalized) or
`.toroidal_periodic()` (Phase 2 periodic image-point) with explicit domain periods
- **Custom topology guarantees**: Set stricter or more relaxed manifold checks
- **Custom validation policies**: Configure `ValidationPolicy` via
`dt.set_validation_policy(...)` before or after build; the active policy controls
automatic topology validation for subsequent insert/remove and other modification
operations
- **Custom repair policies**: Configure Delaunay repair behavior
See `docs/topology.md` for more on toroidal triangulations and `docs/validation.md`
for topology guarantee and validation policy details.
### Key Characteristics
- **Automatic property preservation**: The Delaunay empty-circumsphere property is maintained automatically
- **Cavity-based insertion**: New vertices are inserted by identifying conflicting cells, removing them, and filling the cavity
- **Fan retriangulation**: Vertex removal uses fan-based retriangulation of the vertex star
- **Auxiliary data**: Vertices and cells carry optional user data (`U` / `V`). Read via `vertex.data()` /
`cell.data()`, write via `dt.set_vertex_data(key, data)` / `dt.set_cell_data(key, data)` (O(1),
invariant-preserving). See [`workflows.md`](workflows.md) for examples.
- **Error handling**: Operations fail gracefully if they would violate invariants (see
[`invariants.md`](invariants.md)).
- **Validation**: The active `ValidationPolicy` (set with
`dt.set_validation_policy(...)`) governs automatic topology validation for
subsequent construction/modification operations
## Edit API Reference
The Edit API is exposed through the `BistellarFlips` trait in `prelude::triangulation::flips`:
```rust
use delaunay::prelude::triangulation::*;
use delaunay::prelude::triangulation::flips::*;
// Start with a valid triangulation
let vertices = vec![
vertex!([0.0, 0.0, 0.0]),
vertex!([1.0, 0.0, 0.0]),
vertex!([0.0, 1.0, 0.0]),
vertex!([0.0, 0.0, 1.0]),
];
let mut dt: DelaunayTriangulation<_, (), (), 3> =
DelaunayTriangulation::new(&vertices).unwrap();
// k=1 move: Insert a vertex into a cell (splits cell into D+1 cells)
let cell_key = dt.cells().next().unwrap().0;
let info = dt.flip_k1_insert(cell_key, vertex!([0.25, 0.25, 0.25])).unwrap();
// k=1 inverse: Remove a vertex (collapses its star)
let vertex_key = info.inserted_face_vertices[0];
dt.flip_k1_remove(vertex_key).unwrap();
// k=2 move: Flip a facet (2 cells ↔ D cells)
let facet = /* FacetHandle */;
let info = dt.flip_k2(facet).unwrap();
// k=2 inverse: Flip from an edge star (D cells ↔ 2 cells)
let edge = EdgeKey::new(info.inserted_face_vertices[0], info.inserted_face_vertices[1]);
dt.flip_k2_inverse_from_edge(edge).unwrap();
// k=3 move: Flip a ridge (3 cells ↔ D-1 cells, requires D ≥ 3)
let ridge = /* RidgeHandle */;
let info = dt.flip_k3(ridge).unwrap();
// k=3 inverse: Flip from a triangle star (D-1 cells ↔ 3 cells)
let triangle = TriangleHandle::new(
info.inserted_face_vertices[0],
info.inserted_face_vertices[1],
info.inserted_face_vertices[2],
);
dt.flip_k3_inverse_from_triangle(triangle).unwrap();
```
### Available Flip Operations
#### k=1 Moves (Cell Split/Merge)
- **Forward (`flip_k1_insert`)**: Insert a vertex into a cell, splitting it into D+1 cells
- Valid for D ≥ 1
- Replaces 1 cell with D+1 cells
- Removed face: the entire cell (D-simplex)
- Inserted face: the new vertex (0-simplex)
- **Inverse (`flip_k1_remove`)**: Remove a vertex, collapsing its star
- Requires the vertex star to be collapsible (star of D+1 cells forming a ball)
- Replaces D+1 cells with 1 cell
#### k=2 Moves (Facet Flip)
- **Forward (`flip_k2`)**: Flip a facet shared by 2 cells
- Valid for D ≥ 2
- Replaces 2 cells with D cells
- Removed face: the shared facet ((D-1)-simplex)
- Inserted face: an edge (1-simplex)
- **Inverse (`flip_k2_inverse_from_edge`)**: Flip from an edge star
- Requires an edge with star of D cells
- Replaces D cells with 2 cells
#### k=3 Moves (Ridge Flip)
- **Forward (`flip_k3`)**: Flip a ridge
- Valid for D ≥ 3
- Replaces 3 cells with D-1 cells
- Removed face: a ridge ((D-2)-simplex)
- Inserted face: a triangle (2-simplex)
- **Inverse (`flip_k3_inverse_from_triangle`)**: Flip from a triangle star
- Requires a triangle with star of D-1 cells
- Replaces D-1 cells with 3 cells
### Key Characteristics
- **Explicit control**: You specify exactly which flip to perform
- **No automatic property preservation**: The Delaunay property is **not** maintained automatically
- **Reversible**: Each forward move has a corresponding inverse
- **Geometric validation**: Flips check for degeneracy and manifold preservation
- **Flexible**: Can be used to build custom repair or optimization algorithms
### Important Caveats
⚠️ **The Edit API does not preserve the Delaunay property automatically.**
After applying flips, you should:
1. Manually verify the Delaunay property if needed:
```rust
dt.is_valid().unwrap(); ```
2. Consider running a repair pass if you need the Delaunay property again (requires `K: ExactPredicates`):
- `dt.repair_delaunay_with_flips()` (flip-based repair)
- `dt.repair_delaunay_with_flips_advanced(DelaunayRepairHeuristicConfig::default())` (includes a heuristic rebuild fallback)
## Combining Both APIs
You can mix both APIs in the same workflow:
```rust
use delaunay::prelude::triangulation::*;
use delaunay::prelude::triangulation::flips::*;
// 1. Build initial triangulation (Builder API)
let vertices = vec![
vertex!([0.0, 0.0, 0.0]),
vertex!([1.0, 0.0, 0.0]),
vertex!([0.0, 1.0, 0.0]),
vertex!([0.0, 0.0, 1.0]),
];
let mut dt: DelaunayTriangulation<_, (), (), 3> =
DelaunayTriangulation::new(&vertices).unwrap();
// 2. Add vertices using Builder API (maintains Delaunay)
dt.insert(vertex!([0.5, 0.5, 0.5])).unwrap();
// 3. Make custom topology edits (Edit API)
let facet = /* ... */;
dt.flip_k2(facet).unwrap();
// 4. Verify Delaunay property if needed
if let Err(e) = dt.is_valid() {
eprintln!("Warning: Delaunay property violated after manual edit: {}", e);
// Optionally restore using Builder API or custom repair
}
```
## Validation and Guarantees
Both APIs work with the same validation framework but have different guarantees:
### Builder API Guarantees
- ✅ Maintains **structural invariants** (Level 1-2)
- ✅ Maintains **manifold topology** (Level 3, controlled by `TopologyGuarantee`)
- ✅ Designed to maintain **Delaunay property** (Level 4)
- ✅ Fails gracefully if invariants cannot be maintained
### Edit API Guarantees
- ✅ Maintains **structural invariants** (Level 1-2)
- ✅ Checks **geometric degeneracy** (prevents degenerate flips)
- ⚠️ Does **not** automatically maintain Delaunay property
- ⚠️ User is responsible for property preservation
### Validation Levels
Use the appropriate validation level for your needs:
```rust
// Level 2: Structural only (fast)
dt.tds().is_valid().unwrap();
// Level 3: + Manifold topology
dt.as_triangulation().is_valid().unwrap();
// Level 4: + Delaunay property (most comprehensive)
dt.is_valid().unwrap();
// Full diagnostic report
let report = dt.validation_report();
```
## Implementation Details
### Internal Organization
- **Builder API**: Implemented in `core::delaunay_triangulation` and `core::algorithms::incremental_insertion`
- **Edit API**: Implemented in `triangulation::flips` (public trait) and `core::algorithms::flips` (internal implementation)
- **Low-level primitives**: Context builders and flip application functions are `pub(crate)` in `core::algorithms::flips`
### Design Rationale
The separation serves several purposes:
1. **Clear contracts**: Builder API guarantees Delaunay property; Edit API does not
2. **Safety**: Low-level flip primitives are not exposed to prevent accidental misuse
3. **Flexibility**: Edit API enables research and custom algorithms without restricting the design
4. **Documentation**: Clear distinction between "construction" and "manipulation" workflows
## Examples
See the following examples for practical demonstrations:
- `examples/topology_editing_2d_3d.rs` - 2D+3D example showing both APIs
- `examples/pachner_roundtrip_4d.rs` - Advanced 4D example with all flip types
- `examples/triangulation_3d_100_points.rs` - Builder API usage for construction
## Further Reading
- **Bistellar flip theory**: See `REFERENCES.md` for academic citations
- **Validation framework**: See `docs/validation.md` for detailed validation guide
- **Invariant rationale**: See [`invariants.md`](invariants.md) for theory and implementation pointers
- **Topology analysis**: See `docs/topology.md` for topological concepts
- **API implementation**: See `triangulation::flips` module documentation