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Exact 3D Delaunay tetrahedralization (Bowyer–Watson, ghost hull, BRIO/Hilbert, verifier) (#14)
* mesh: exact Bowyer-Watson Delaunay tetrahedralization with ghost tets, BRIO/Hilbert order and a verifier Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm * mesh: Delaunay benchmark, hash-paired cavity faces, mesh.delaunay.points debug command Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm * mesh: simplify Delaunay helpers, share predicates and parsers, halve verifier work Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm --------- Co-authored-by: jyviko <2501648+jyviko@users.noreply.github.com> Co-authored-by: Claude Opus 5.5 <noreply@anthropic.com>
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//! Exact, deterministic 3D Delaunay tetrahedralization of a point set.
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//!
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//! # Method
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//! - **Incremental cavity insertion** (A. Bowyer, *Computing Dirichlet tessellations*, Comput. J.
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//! 24(2):162–166, 1981; D. F. Watson, *Computing the n-dimensional Delaunay tessellation with
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//! application to Voronoi polytopes*, Comput. J. 24(2):167–172, 1981). Each point finds the
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//! tetrahedra whose circumsphere contains it (the cavity, grown by adjacency from one of them),
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//! deletes them and joins the cavity's boundary faces to the point.
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//! - **Insertion order**: BRIO rounds with Hilbert order inside each round (Amenta, Choi, Rote,
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//! SoCG 2003; Skilling 2004). See [`order`].
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//! - **Point location**: the stochastic visibility walk with memory (O. Devillers, S. Pion,
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//! M. Teillaud, *Walking in a triangulation*, IJFCS 13(2):181–199, 2002, §2–§3), starting at
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//! the last tetrahedron created. In a Delaunay triangulation the walk cannot cycle
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//! (H. Edelsbrunner, *An acyclicity theorem for cell complexes in d dimensions*, Combinatorica
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//! 10(3):251–260, 1990); it is still capped at the number of tetrahedra, after which a linear
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//! scan locates the point.
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//! - **The infinite vertex.** The convex hull is closed by *ghost* tetrahedra: each hull face is
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//! joined to the vertex [`GHOST`], so every face has exactly two tetrahedra and the adjacency is
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//! that of a triangulated 3-sphere (no boundary cases). A ghost tetrahedron's "circumsphere" is
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//! the open half-space beyond its hull face; a point on the plane of the face is decided by the
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//! finite tetrahedron on the other side of that face (the limit of spheres through the face
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//! whose centre runs off to infinity), so the decision is the same perturbed one
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//! (O. Devillers, M. Teillaud, *Perturbations for Delaunay and weighted Delaunay 3D
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//! triangulations*, CGTA 44(3):160–168, 2011, §3 and the infinite-vertex discussion).
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//! - **Compact adjacency arrays**: per tetrahedron four vertex ids and four neighbour ids, where
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//! neighbour `i` is across the face opposite vertex `i`. Deleted slots are recycled LIFO.
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//!
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//! # Exactness and degeneracies
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//! Every decision is an exact predicate from [`crate::predicates`]: `orient3d` for the walk and
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//! for the hull, `insphere_sos` (Devillers–Teillaud symbolic perturbation) for the cavity. There
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//! are no tolerances. The perturbation is a function of the coordinates only (it lifts each point
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//! by an infinitesimal that depends on its lexicographic rank), so the result is *the* Delaunay
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//! triangulation of the perturbed points: unique, independent of the insertion order, and with
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//! no flat tetrahedra, even on grids where every eight neighbours are cospherical.
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//!
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//! - **Duplicates are merged.** Points with equal coordinates (`-0.0 == 0.0`) become one vertex:
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//! the one with the lowest input index. The others are reported by
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//! [`Triangulation::duplicate_of`] and appear in no tetrahedron.
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//! - **Lower-dimensional inputs** (all points collinear or coplanar, a single point, none) have
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//! no tetrahedra: [`Triangulation::dimension`] reports their affine dimension and the
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//! tetrahedron lists are empty. Points arriving later that leave the plane start the 3D
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//! triangulation.
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//! - **Hostile coordinates** are rejected at the boundary by [`Point3::new`] (NaN, infinities,
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//! magnitudes outside `[2⁻¹²⁰, 2¹⁶⁰]`); at most [`MAX_POINTS`] points.
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//!
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//! # Determinism
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//! Same points in the same order → same tetrahedra in the same slots, bit for bit: the order is
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//! a fixed function of the input, the walk's "random" face choice comes from a fixed-seed
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//! generator, sorting has total keys, and nothing iterates a hash map.
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//!
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//! # Verification
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//! [`Triangulation::verify`] checks the adjacency invariants, the orientation of every
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//! tetrahedron, the convexity of the hull and the empty-circumsphere property of every interior
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//! face (exact and perturbed), which by Delaunay's lemma makes every circumsphere empty. Tests
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//! also use [`Triangulation::verify_empty_spheres`], the brute-force check of every point against
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//! every circumsphere.
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mod order;
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mod verify;
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#[cfg(test)]
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mod tests;
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use crate::predicates::{Axis, GenericPoint, Point3, Sign, generic_orient2d, insphere_sos, orient3d, tet_volume_sign};
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use crate::{Error, Result};
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/// The infinite vertex of the ghost tetrahedra that close the convex hull.
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pub const GHOST: u32 = u32::MAX;
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/// Marks a deleted tetrahedron slot (first vertex).
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const DEAD: u32 = u32::MAX - 1;
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/// No tetrahedron / no vertex.
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const NONE: u32 = u32::MAX;
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/// An empty slot of the edge table (no edge has two equal endpoints).
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const EMPTY_EDGE: u64 = u64::MAX;
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/// The most points a triangulation takes (vertex ids stay far below [`GHOST`] and the ~6.7
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/// tetrahedra per point fit `u32` slot ids).
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pub const MAX_POINTS: usize = 1 << 24;
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/// Where a point lies relative to the triangulation ([`Triangulation::locate`]).
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#[derive(Debug, Clone, Copy, PartialEq, Eq)]
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pub enum Location {
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/// In the closed finite tetrahedron `t` and not one of its vertices.
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Inside(u32),
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/// Strictly beyond the hull face of the ghost tetrahedron `t`.
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Outside(u32),
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/// Equal to this vertex.
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Vertex(u32),
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}
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/// The outcome of [`Triangulation::insert`].
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#[derive(Debug, Clone, Copy, PartialEq, Eq)]
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pub enum Insertion {
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/// A new vertex.
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Inserted(u32),
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/// The point equals this existing vertex; it was recorded as a duplicate (its own id is the
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/// last point index).
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Duplicate(u32),
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}
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/// A Delaunay tetrahedralization with ghost tetrahedra closing the hull.
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///
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/// Vertex ids are input point indices. Tetrahedron ids are slots: [`Triangulation::slots`]
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/// counts them, and [`Triangulation::tet`] returns `None` for a recycled slot. A tetrahedron's
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/// vertices `[a, b, c, d]` have positive volume, `det[b − a, c − a, d − a] > 0` (the tet4 node
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/// order). A ghost tetrahedron has [`GHOST`] in one slot; putting a point in place of the ghost
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/// gives positive volume exactly when that point is beyond the hull face.
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#[derive(Debug, Clone)]
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pub struct Triangulation {
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points: Vec<Point3>,
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/// `NONE`, or the vertex this point duplicates.
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duplicate_of: Vec<u32>,
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tets: Vec<[u32; 4]>,
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/// Neighbour `i` is across the face opposite vertex `i`.
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adj: Vec<[u32; 4]>,
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free: Vec<u32>,
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/// A live tetrahedron incident to each vertex (`NONE` until it is in one).
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vertex_tet: Vec<u32>,
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/// While the points span less than 3D: affinely independent vertices, and the others.
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basis: Vec<u32>,
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pending: Vec<u32>,
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/// Cavity marks: `epoch` = in the cavity, `epoch + 1` = tested and outside.
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mark: Vec<u32>,
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epoch: u32,
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/// Where the next walk starts.
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hint: u32,
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/// State of the walk's face-order generator.
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rng: u64,
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scratch: Scratch,
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}
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#[derive(Debug, Clone, Default)]
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struct Scratch {
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cavity: Vec<u32>,
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stack: Vec<u32>,
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/// `(cavity tet, face slot, outside neighbour)`.
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boundary: Vec<(u32, usize, u32)>,
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/// `(new tet vertices, slot of the outside neighbour, neighbour, the neighbour's slot back)`.
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new_tets: Vec<([u32; 4], usize, u32, usize)>,
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/// Open-addressing table `(edge key, tet, slot)` linking the new tetrahedra to each other.
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edges: Vec<(u64, u32, usize)>,
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}
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/// A Delaunay triangulation is corrupt: this never happens with exact predicates, and is
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/// reported instead of panicking.
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fn corrupt(what: &str) -> Error {
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Error::Failed(format!("Delaunay triangulation is inconsistent: {what}"))
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}
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/// The sign of the volume `det[b − a, c − a, d − a]` (the tet4 convention).
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#[inline]
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fn volume_sign(p: [&Point3; 4]) -> Sign {
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let [a, b, c, d] = p;
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tet_volume_sign(a, b, c, d)
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}
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/// Whether three points are collinear: all three axis projections are (exactly).
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fn collinear(a: &Point3, b: &Point3, c: &Point3) -> bool {
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let [a, b, c] = [a, b, c].map(|p| GenericPoint::from(*p));
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[Axis::X, Axis::Y, Axis::Z].into_iter().all(|axis| generic_orient2d(&a, &b, &c, axis).is_zero())
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}
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/// A coordinate triple as a [`Point3`], with `-0.0` made `+0.0` so equal points have equal bits.
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fn admit([x, y, z]: [f64; 3]) -> Result<Point3> {
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Point3::new(x + 0.0, y + 0.0, z + 0.0)
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}
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impl Triangulation {
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fn empty() -> Self {
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Triangulation {
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points: Vec::new(),
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duplicate_of: Vec::new(),
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tets: Vec::new(),
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adj: Vec::new(),
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free: Vec::new(),
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vertex_tet: Vec::new(),
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basis: Vec::new(),
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pending: Vec::new(),
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mark: Vec::new(),
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epoch: 0,
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hint: NONE,
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rng: 0x2545_F491_4F6C_DD1D,
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scratch: Scratch::default(),
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}
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}
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/// The Delaunay tetrahedralization of `points`. Duplicates are merged into the lowest
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/// index; fewer than four affinely independent points give no tetrahedra.
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pub fn new(points: &[[f64; 3]]) -> Result<Self> {
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if points.len() > MAX_POINTS {
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return Err(Error::Limit(format!("{} points; a triangulation takes at most {MAX_POINTS}", points.len())));
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}
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let mut tri = Self::empty();
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tri.points =
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points.iter().enumerate().map(|(i, &c)| admit(c).map_err(|e| Error::Invalid(format!("point {i}: {e}")))).collect::<Result<_>>()?;
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let n = tri.points.len();
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tri.duplicate_of = vec![NONE; n];
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tri.vertex_tet = vec![NONE; n];
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// Merge duplicates: sort by coordinates, ties by index, keep the first of each run.
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let mut by_coords: Vec<u32> = (0..n as u32).collect();
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let key = |v: u32| tri.points.get(v as usize).map(|p| p.coords().map(f64::to_bits)).unwrap_or_default();
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// Equal values have equal bits after `admit`; the bit order is total and deterministic.
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by_coords.sort_unstable_by_key(|&v| (key(v), v));
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for run in by_coords.chunk_by(|&a, &b| key(a) == key(b)) {
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if let [keep, rest @ ..] = run {
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for &v in rest {
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if let Some(d) = tri.duplicate_of.get_mut(v as usize) {
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*d = *keep;
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}
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}
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}
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}
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let unique: Vec<u32> = (0..n as u32).filter(|&v| tri.duplicate_of.get(v as usize) == Some(&NONE)).collect();
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let expected = unique.len().saturating_mul(7);
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tri.tets.reserve(expected);
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tri.adj.reserve(expected);
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tri.mark.reserve(expected);
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for v in order::brio_hilbert(&tri.points, &unique) {
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tri.add(v, true)?;
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}
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Ok(tri)
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}
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/// Insert one more point (a Steiner point, say). Its vertex id is its index in
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/// [`Triangulation::points`]. A point equal to an existing vertex is recorded as a duplicate.
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pub fn insert(&mut self, p: [f64; 3]) -> Result<Insertion> {
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if self.points.len() >= MAX_POINTS {
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return Err(Error::Limit(format!("a triangulation takes at most {MAX_POINTS} points")));
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}
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let v = self.points.len() as u32;
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self.points.push(admit(p)?);
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self.duplicate_of.push(NONE);
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self.vertex_tet.push(NONE);
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self.add(v, false)
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}
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/// The points, by vertex id (including duplicates).
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pub fn points(&self) -> &[Point3] {
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&self.points
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}
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/// The vertex a duplicate point was merged into.
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pub fn duplicate_of(&self, v: u32) -> Option<u32> {
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self.duplicate_of.get(v as usize).copied().filter(|&d| d != NONE)
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}
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/// The affine dimension of the distinct points: 3 when there are tetrahedra, otherwise 0
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/// (none or one point), 1 (collinear) or 2 (coplanar).
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pub fn dimension(&self) -> usize {
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if self.tets.is_empty() { self.basis.len().saturating_sub(1) } else { 3 }
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}
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/// Tetrahedron slots, live or recycled, finite or ghost.
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pub fn slots(&self) -> usize {
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self.tets.len()
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}
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/// The vertices of the tetrahedron in slot `t` (one may be [`GHOST`]); `None` for a recycled
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/// or out-of-range slot.
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pub fn tet(&self, t: u32) -> Option<[u32; 4]> {
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self.tets.get(t as usize).copied().filter(|v| v[0] != DEAD)
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}
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/// The neighbours of tetrahedron `t`: neighbour `i` shares the face opposite vertex `i`.
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pub fn neighbors(&self, t: u32) -> Option<[u32; 4]> {
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self.tet(t)?;
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self.adj.get(t as usize).copied()
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}
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/// Whether `t` is a ghost tetrahedron (one vertex is [`GHOST`]).
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pub fn is_ghost(&self, t: u32) -> bool {
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self.tet(t).is_some_and(|v| v.contains(&GHOST))
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}
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/// A live tetrahedron incident to vertex `v` (finite or ghost), once `v` is triangulated.
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pub fn incident_tet(&self, v: u32) -> Option<u32> {
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self.vertex_tet.get(v as usize).copied().filter(|&t| t != NONE)
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}
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/// The finite tetrahedra with their slot ids, in slot order; vertices in tet4 order.
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pub fn finite_tets(&self) -> impl Iterator<Item = (u32, [u32; 4])> + '_ {
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self.tets.iter().enumerate().filter(|(_, v)| v[0] != DEAD && !v.contains(&GHOST)).map(|(t, v)| (t as u32, *v))
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}
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/// The number of finite tetrahedra.
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pub fn finite_tet_count(&self) -> usize {
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self.finite_tets().count()
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}
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/// Locate `p`: the walk from the last tetrahedron created.
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pub fn locate(&self, p: [f64; 3]) -> Result<Option<Location>> {
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if self.tets.is_empty() {
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return Ok(None);
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}
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let point = admit(p)?;
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let mut rng = self.rng;
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self.walk(&point, self.hint, &mut rng).map(Some)
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}
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// ---- internals ----
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fn pt(&self, v: u32) -> Result<&Point3> {
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self.points.get(v as usize).ok_or_else(|| corrupt("vertex id"))
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}
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fn tet_at(&self, t: u32) -> Result<[u32; 4]> {
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self.tet(t).ok_or_else(|| corrupt("tetrahedron id"))
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}
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fn adj_at(&self, t: u32) -> Result<[u32; 4]> {
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self.adj.get(t as usize).copied().ok_or_else(|| corrupt("tetrahedron id"))
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}
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/// The four points of a finite tetrahedron.
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fn pts(&self, [a, b, c, d]: [u32; 4]) -> Result<[&Point3; 4]> {
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Ok([self.pt(a)?, self.pt(b)?, self.pt(c)?, self.pt(d)?])
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}
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/// The four points of `vs` with slot `k` replaced by `p` (`k` may hold the ghost).
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fn with_point<'a>(&'a self, vs: [u32; 4], k: usize, p: &'a Point3) -> Result<[&'a Point3; 4]> {
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let mut out = [p; 4];
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for (i, (o, &v)) in out.iter_mut().zip(&vs).enumerate() {
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if i != k {
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*o = self.pt(v)?;
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}
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}
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Ok(out)
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}
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/// Add vertex `v` (already in `points`). A duplicate is detected and recorded: in 3D by the
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/// walk, below 3D by comparing with the vertices so far unless the caller knows `v` is
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/// `unique` (the sort-based merge of [`Triangulation::new`]).
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fn add(&mut self, v: u32, unique: bool) -> Result<Insertion> {
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if !self.tets.is_empty() {
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return self.insert_vertex(v);
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}
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let p = *self.pt(v)?;
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if !unique && let Some(&u) = self.basis.iter().chain(&self.pending).find(|&&u| self.points.get(u as usize) == Some(&p)) {
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if let Some(d) = self.duplicate_of.get_mut(v as usize) {
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*d = u;
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}
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return Ok(Insertion::Duplicate(u));
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}
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match self.basis.as_slice() {
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[] | [_] => self.basis.push(v),
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&[a, b] => {
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if collinear(self.pt(a)?, self.pt(b)?, &p) {
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self.pending.push(v);
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} else {
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self.basis.push(v);
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}
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}
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&[a, b, c, ..] => {
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if orient3d(self.pt(a)?, self.pt(b)?, self.pt(c)?, &p).is_zero() {
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self.pending.push(v);
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} else {
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self.start([a, b, c, v])?;
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for u in std::mem::take(&mut self.pending) {
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self.insert_vertex(u)?;
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}
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}
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}
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}
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Ok(Insertion::Inserted(v))
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}
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/// The first tetrahedron and its four ghosts.
|
||||
fn start(&mut self, [a, b, c, d]: [u32; 4]) -> Result<()> {
|
||||
let t0 = if volume_sign(self.pts([a, b, c, d])?) == Sign::Positive { [a, b, c, d] } else { [b, a, c, d] };
|
||||
let mut all = vec![t0];
|
||||
for j in 0..4 {
|
||||
let mut g = t0;
|
||||
g[j] = GHOST;
|
||||
// Swapping two other slots turns the ghost to face away from the hull.
|
||||
g.swap((j + 1) % 4, (j + 2) % 4);
|
||||
all.push(g);
|
||||
}
|
||||
let mut faces: Vec<([u32; 3], u32, usize)> = Vec::with_capacity(20);
|
||||
for vs in all {
|
||||
let t = self.alloc(vs)?;
|
||||
for k in 0..4 {
|
||||
faces.push((verify::face(vs, k), t, k));
|
||||
}
|
||||
self.hint = t;
|
||||
}
|
||||
faces.sort_unstable();
|
||||
for pair in faces.chunks(2) {
|
||||
match pair {
|
||||
[(f, t, i), (g, u, j)] if f == g => {
|
||||
self.set_adj(*t, *i, *u)?;
|
||||
self.set_adj(*u, *j, *t)?;
|
||||
}
|
||||
_ => return Err(corrupt("initial tetrahedron faces")),
|
||||
}
|
||||
}
|
||||
self.basis.clear();
|
||||
Ok(())
|
||||
}
|
||||
|
||||
fn set_adj(&mut self, t: u32, i: usize, n: u32) -> Result<()> {
|
||||
let slot = self.adj.get_mut(t as usize).and_then(|a| a.get_mut(i)).ok_or_else(|| corrupt("tetrahedron id"))?;
|
||||
*slot = n;
|
||||
Ok(())
|
||||
}
|
||||
|
||||
fn alloc(&mut self, vs: [u32; 4]) -> Result<u32> {
|
||||
let t = match self.free.pop() {
|
||||
Some(t) => {
|
||||
*self.tets.get_mut(t as usize).ok_or_else(|| corrupt("free slot"))? = vs;
|
||||
*self.adj.get_mut(t as usize).ok_or_else(|| corrupt("free slot"))? = [NONE; 4];
|
||||
t
|
||||
}
|
||||
None => {
|
||||
if self.tets.len() >= (DEAD - 1) as usize {
|
||||
return Err(Error::Limit("too many tetrahedra".into()));
|
||||
}
|
||||
self.tets.push(vs);
|
||||
self.adj.push([NONE; 4]);
|
||||
self.mark.push(0);
|
||||
(self.tets.len() - 1) as u32
|
||||
}
|
||||
};
|
||||
for v in vs {
|
||||
if let Some(s) = self.vertex_tet.get_mut(v as usize) {
|
||||
*s = t;
|
||||
}
|
||||
}
|
||||
Ok(t)
|
||||
}
|
||||
|
||||
fn kill(&mut self, t: u32) -> Result<()> {
|
||||
*self.tets.get_mut(t as usize).ok_or_else(|| corrupt("tetrahedron id"))? = [DEAD; 4];
|
||||
self.free.push(t);
|
||||
Ok(())
|
||||
}
|
||||
|
||||
/// Whether `p` is in conflict with tetrahedron `t` (inside its perturbed circumsphere). Only
|
||||
/// `dig` calls it: it reads the current cavity marks.
|
||||
fn conflict(&self, t: u32, p: &Point3) -> Result<bool> {
|
||||
let vs = self.tet_at(t)?;
|
||||
let Some(g) = vs.iter().position(|&v| v == GHOST) else {
|
||||
let [a, b, c, d] = self.pts(vs)?;
|
||||
// `insphere` wants Shewchuk-positive orientation: swap the first two vertices.
|
||||
return Ok(insphere_sos(b, a, c, d, p) == Sign::Positive);
|
||||
};
|
||||
Ok(match volume_sign(self.with_point(vs, g, p)?) {
|
||||
Sign::Positive => true,
|
||||
Sign::Negative => false,
|
||||
// On the hull face's plane: the finite tetrahedron across that face decides.
|
||||
Sign::Zero => {
|
||||
let o = self.adj_at(t)?.get(g).copied().ok_or_else(|| corrupt("slot"))?;
|
||||
// During `dig` the finite tetrahedron may already be decided.
|
||||
match self.mark_of(o) {
|
||||
m if m == self.epoch => return Ok(true),
|
||||
m if m == self.epoch + 1 => return Ok(false),
|
||||
_ => {}
|
||||
}
|
||||
let ws = self.tet_at(o)?;
|
||||
if ws.contains(&GHOST) {
|
||||
return Err(corrupt("two ghost tetrahedra share a hull face"));
|
||||
}
|
||||
let [a, b, c, d] = self.pts(ws)?;
|
||||
insphere_sos(b, a, c, d, p) == Sign::Positive
|
||||
}
|
||||
})
|
||||
}
|
||||
|
||||
/// The visibility walk towards `p` from `start` (Devillers, Pion, Teillaud 2002).
|
||||
fn walk(&self, p: &Point3, start: u32, rng: &mut u64) -> Result<Location> {
|
||||
let mut t = start;
|
||||
let vs = self.tet_at(t)?;
|
||||
if let Some(g) = vs.iter().position(|&v| v == GHOST) {
|
||||
t = self.adj_at(t)?.get(g).copied().ok_or_else(|| corrupt("slot"))?;
|
||||
}
|
||||
let mut prev = NONE;
|
||||
for _ in 0..self.tets.len().saturating_add(16) {
|
||||
let vs = self.tet_at(t)?;
|
||||
if vs.contains(&GHOST) {
|
||||
return Ok(Location::Outside(t));
|
||||
}
|
||||
let adj = self.adj_at(t)?;
|
||||
let ps = self.pts(vs)?;
|
||||
// xorshift64 (Marsaglia 2003): which face to try first.
|
||||
*rng ^= *rng << 13;
|
||||
*rng ^= *rng >> 7;
|
||||
*rng ^= *rng << 17;
|
||||
let first = (*rng >> 61) as usize;
|
||||
let mut next = NONE;
|
||||
for i in 0..4 {
|
||||
let k = (first + i) % 4;
|
||||
let n = adj.get(k).copied().unwrap_or(NONE);
|
||||
if n == prev {
|
||||
continue;
|
||||
}
|
||||
let mut q = ps;
|
||||
if let Some(slot) = q.get_mut(k) {
|
||||
*slot = p;
|
||||
}
|
||||
if volume_sign(q) == Sign::Negative {
|
||||
next = n;
|
||||
break;
|
||||
}
|
||||
}
|
||||
if next == NONE {
|
||||
return self.inside(vs, t, p);
|
||||
}
|
||||
prev = t;
|
||||
t = next;
|
||||
}
|
||||
self.locate_by_scan(p)
|
||||
}
|
||||
|
||||
/// `p` lies in the closed finite tetrahedron `t`: one of its vertices, or inside.
|
||||
fn inside(&self, vs: [u32; 4], t: u32, p: &Point3) -> Result<Location> {
|
||||
for v in vs {
|
||||
if self.pt(v)? == p {
|
||||
return Ok(Location::Vertex(v));
|
||||
}
|
||||
}
|
||||
Ok(Location::Inside(t))
|
||||
}
|
||||
|
||||
/// Point location by testing every tetrahedron (the walk's fallback).
|
||||
fn locate_by_scan(&self, p: &Point3) -> Result<Location> {
|
||||
let mut outside = None;
|
||||
for (t, &vs) in self.tets.iter().enumerate() {
|
||||
if vs[0] == DEAD {
|
||||
continue;
|
||||
}
|
||||
match vs.iter().position(|&v| v == GHOST) {
|
||||
Some(g) => {
|
||||
if outside.is_none() && volume_sign(self.with_point(vs, g, p)?) == Sign::Positive {
|
||||
outside = Some(t as u32);
|
||||
}
|
||||
}
|
||||
None => {
|
||||
let mut inside = true;
|
||||
for k in 0..4 {
|
||||
if volume_sign(self.with_point(vs, k, p)?) == Sign::Negative {
|
||||
inside = false;
|
||||
break;
|
||||
}
|
||||
}
|
||||
if inside {
|
||||
return self.inside(vs, t as u32, p);
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
outside.map(Location::Outside).ok_or_else(|| corrupt("a point is neither inside nor outside the hull"))
|
||||
}
|
||||
|
||||
fn next_epoch(&mut self) {
|
||||
if self.epoch >= u32::MAX - 3 {
|
||||
self.mark.fill(0);
|
||||
self.epoch = 0;
|
||||
}
|
||||
self.epoch += 2;
|
||||
}
|
||||
|
||||
fn mark_of(&self, t: u32) -> u32 {
|
||||
self.mark.get(t as usize).copied().unwrap_or(0)
|
||||
}
|
||||
|
||||
fn set_mark(&mut self, t: u32, m: u32) {
|
||||
if let Some(s) = self.mark.get_mut(t as usize) {
|
||||
*s = m;
|
||||
}
|
||||
}
|
||||
|
||||
/// Insert vertex `v` into the 3D triangulation.
|
||||
fn insert_vertex(&mut self, v: u32) -> Result<Insertion> {
|
||||
let p = *self.pt(v)?;
|
||||
let mut rng = self.rng;
|
||||
let located = self.walk(&p, self.hint, &mut rng)?;
|
||||
self.rng = rng;
|
||||
let seed = match located {
|
||||
Location::Vertex(u) => {
|
||||
if let Some(d) = self.duplicate_of.get_mut(v as usize) {
|
||||
*d = u;
|
||||
}
|
||||
return Ok(Insertion::Duplicate(u));
|
||||
}
|
||||
Location::Inside(t) | Location::Outside(t) => t,
|
||||
};
|
||||
let mut s = std::mem::take(&mut self.scratch);
|
||||
let result = self.dig(seed, &p, &mut s).and_then(|()| self.fill(v, &mut s));
|
||||
self.scratch = s;
|
||||
result.map(|()| Insertion::Inserted(v))
|
||||
}
|
||||
|
||||
/// The cavity of `p`: every tetrahedron in conflict, grown by adjacency from `seed`, and the
|
||||
/// faces between it and the rest.
|
||||
fn dig(&mut self, seed: u32, p: &Point3, s: &mut Scratch) -> Result<()> {
|
||||
self.next_epoch();
|
||||
let (inside, outside) = (self.epoch, self.epoch + 1);
|
||||
s.cavity.clear();
|
||||
s.stack.clear();
|
||||
s.boundary.clear();
|
||||
self.set_mark(seed, inside);
|
||||
s.cavity.push(seed);
|
||||
s.stack.push(seed);
|
||||
while let Some(t) = s.stack.pop() {
|
||||
for (j, o) in self.adj_at(t)?.into_iter().enumerate() {
|
||||
let m = self.mark_of(o);
|
||||
if m == inside {
|
||||
continue;
|
||||
}
|
||||
if m != outside && self.conflict(o, p)? {
|
||||
self.set_mark(o, inside);
|
||||
s.cavity.push(o);
|
||||
s.stack.push(o);
|
||||
} else {
|
||||
self.set_mark(o, outside);
|
||||
s.boundary.push((t, j, o));
|
||||
}
|
||||
}
|
||||
}
|
||||
Ok(())
|
||||
}
|
||||
|
||||
/// Replace the cavity by the tetrahedra joining its boundary faces to `v`.
|
||||
fn fill(&mut self, v: u32, s: &mut Scratch) -> Result<()> {
|
||||
s.new_tets.clear();
|
||||
// Everything is read before any slot is recycled: a new tetrahedron may reuse the id of
|
||||
// a cavity tetrahedron that another outside neighbour still points to.
|
||||
for &(t, j, o) in &s.boundary {
|
||||
let mut vs = self.tet_at(t)?;
|
||||
*vs.get_mut(j).ok_or_else(|| corrupt("slot"))? = v;
|
||||
let back = self.adj_at(o)?.iter().position(|&n| n == t).ok_or_else(|| corrupt("one-sided adjacency"))?;
|
||||
s.new_tets.push((vs, j, o, back));
|
||||
}
|
||||
for &t in &s.cavity {
|
||||
self.kill(t)?;
|
||||
}
|
||||
// Each edge of the cavity boundary is shared by exactly two boundary faces; their new
|
||||
// tetrahedra meet across the face spanned by that edge and `v`. Pair them through a
|
||||
// small open-addressing table keyed by the edge (lookups only, so no order leaks out).
|
||||
// 1.5 edges per new tetrahedron: at most 75% load.
|
||||
let size = (s.new_tets.len() * 2).next_power_of_two().max(16);
|
||||
s.edges.clear();
|
||||
s.edges.resize(size, (EMPTY_EDGE, NONE, 0));
|
||||
let mask = size - 1;
|
||||
let mut open = 0usize;
|
||||
for &(vs, j, o, back) in &s.new_tets {
|
||||
let t = self.alloc(vs)?;
|
||||
self.set_adj(t, j, o)?;
|
||||
self.set_adj(o, back, t)?;
|
||||
for k in (0..4).filter(|&k| k != j) {
|
||||
let mut e = vs.iter().enumerate().filter(|&(i, _)| i != j && i != k).map(|(_, &u)| u);
|
||||
let (a, b) = (e.next().unwrap_or(NONE), e.next().unwrap_or(NONE));
|
||||
let key = (u64::from(a.min(b)) << 32) | u64::from(a.max(b));
|
||||
let mut h = (key.wrapping_mul(0x9E37_79B9_7F4A_7C15) >> 32) as usize & mask;
|
||||
let mut placed = false;
|
||||
for _ in 0..size {
|
||||
let entry = s.edges.get_mut(h).ok_or_else(|| corrupt("edge table"))?;
|
||||
if entry.0 == EMPTY_EDGE {
|
||||
*entry = (key, t, k);
|
||||
open += 1;
|
||||
placed = true;
|
||||
break;
|
||||
}
|
||||
if entry.0 == key {
|
||||
let (_, u, l) = *entry;
|
||||
if u == NONE {
|
||||
return Err(corrupt("an edge of the cavity boundary has more than two faces"));
|
||||
}
|
||||
entry.1 = NONE;
|
||||
open -= 1;
|
||||
self.set_adj(t, k, u)?;
|
||||
self.set_adj(u, l, t)?;
|
||||
placed = true;
|
||||
break;
|
||||
}
|
||||
h = (h + 1) & mask;
|
||||
}
|
||||
if !placed {
|
||||
return Err(corrupt("edge table full"));
|
||||
}
|
||||
}
|
||||
self.hint = t;
|
||||
}
|
||||
if open != 0 {
|
||||
return Err(corrupt("the cavity boundary is not a closed surface"));
|
||||
}
|
||||
Ok(())
|
||||
}
|
||||
}
|
||||
Reference in new issue
Block a user