isomesh 0.0.6

//! The body saddles of the trilinear interpolant, from one quadratic.
//!
//! Grosso, R., *Construction of Topologically Correct and Manifold Isosurfaces*,
//! Computer Graphics Forum 35(5), pp. 187–196 (`10.1111/cgf.12975`), §4; and
//! Grosso, R., *An Asymptotic Decider for Robust and Topologically Correct
//! Triangulation of Isosurfaces*, CGI '17 (`10.1145/3095140.3095179`), §3.
//!
//! # What this decides, and why it is not [`super::interior`]
//!
//! [`super::ambiguity`] resolves an ambiguous *face*. [`super::interior`] answers
//! Chernyaev's question about one *pair* of opposite faces: are the outside
//! regions those two faces carry joined through the cell? This module answers a
//! different and larger question — **where are the trilinear interpolant's saddle
//! points inside the cell, and how many of them are there** — from which the
//! cell's whole topology follows.
//!
//! The two are independent constructions of the same geometry, which is the point:
//! `the_body_saddle_heights_agree_with_the_swept_saddle_roots` checks one against
//! the other, and neither shares a line of arithmetic with the other.
//!
//! # The construction
//!
//! Restricted to a cell, the interpolant is
//!
//! ```text
//! F(u,v,w) = Σ f_i · (u or 1−u)(v or 1−v)(w or 1−w)
//! ```
//!
//! with `f_i` this crate's corner values and `(u,v,w) ∈ [0,1]³`. **Grosso's corner
//! numbering and this crate's are the same numbering** — his `v0 = (0,0,0)`,
//! `v1 = (1,0,0)`, `v2 = (0,1,0)`, `v3 = (1,1,0)`, `v4 = (0,0,1)` … is exactly
//! `cube.rs`'s "corner `i` sits at `(i&1, (i>>1)&1, (i>>2)&1)`". That is a
//! coincidence rather than a design, so `grosso_corner_numbering_is_ours` pins it;
//! if it ever stops being true every formula below needs a permutation.
//!
//! On a face the interpolant restricts to a bilinear function whose level set is a
//! rectangular hyperbola. For the pair of opposite faces `w = 0` and `w = 1`, the
//! projections of the two hyperbolas onto the `(u,v)` plane meet where a quadratic
//! in `u` vanishes. Where they meet, the line joining the two faces at that
//! `(u,v)` lies **entirely on the level set** (Grosso eq. 5), because the
//! interpolant is linear along it and equal to the isovalue at both ends. Two such
//! lines from different face pairs cross at a **body saddle**.
//!
//! # One quadratic, not three
//!
//! The paper presents "three quadratic equations", one per pair of opposite faces.
//! Solving three is unnecessary and the authors' own implementation does not:
//!
//! > It is enough to compute a pair of solutions for one face. The other solutions
//! > are obtained by evaluating the equations for the common variable.
//!
//! Both saddles share their `u` coordinate set. Given `u`, the `v` coordinate is
//! where the `w = 0` face's hyperbola sits at that `u`, and the `w` coordinate is
//! where the `v = 0` face's hyperbola sits — each a **linear** solve. So the whole
//! classification is one quadratic and four linear interpolations, and the three
//! coordinate sets cannot disagree with each other the way three separate
//! quadratics could.
//!
//! # The coefficients, derived here rather than transcribed
//!
//! Rule 5 forbids guessing a published formula; re-deriving one is how you avoid
//! having to. Writing `g₀(u) = f₀(1−u) + f₁u` and `g₁(u) = f₂(1−u) + f₃u` for the
//! `w = 0` face and `g̃₀`, `g̃₁` from `f₄, f₅` and `f₆, f₇` for `w = 1`, the level
//! set on each face is `v = (i₀ − g₀)/(g₁ − g₀)` and `v = (i₀ − g̃₀)/(g̃₁ − g̃₀)`.
//! Setting them equal and clearing denominators:
//!
//! ```text
//! (i₀ − g₀)(g̃₁ − g̃₀) − (i₀ − g̃₀)(g₁ − g₀) = 0
//! ```
//!
//! Each factor is linear in `u`:
//!
//! ```text
//! g₁ − g₀  = (f₂ − f₀) + u(f₀ + f₃ − f₁ − f₂)
//! g̃₁ − g̃₀ = (f₆ − f₄) + u(f₄ + f₇ − f₅ − f₆)
//! i₀ − g₀  = (i₀ − f₀) − u(f₁ − f₀)
//! i₀ − g̃₀ = (i₀ − f₄) − u(f₅ − f₄)
//! ```
//!
//! so collecting powers of `u` gives `a·u² + b·u + c` with
//!
//! ```text
//! a = (f₅ − f₄)(f₀ + f₃ − f₁ − f₂) − (f₁ − f₀)(f₄ + f₇ − f₅ − f₆)
//! b = (i₀ − f₀)(f₄ + f₇ − f₅ − f₆) − (f₁ − f₀)(f₆ − f₄)
//!   − (i₀ − f₄)(f₀ + f₃ − f₁ − f₂) + (f₅ − f₄)(f₂ − f₀)
//! c = (i₀ − f₀)(f₆ − f₄) − (i₀ − f₄)(f₂ − f₀)
//! ```
//!
//! Those agree term for term with the three the paper prints **and** with the
//! three the authors' implementation computes — a three-way agreement between a
//! derivation, a paper and a program, where V-24 had two (V-30).
//!
//! This crate extracts the **zero** level set and has no isovalue parameter, so
//! `i₀ = 0` throughout and `c` collapses to `f₂f₄ − f₀f₆`. The general form is
//! written out above because it is what the derivation produces; the specialised
//! form is what runs.
//!
//! # What the saddles alone cannot say
//!
//! Whether six saddles mean a tunnel or a single twelve-vertex contour. That
//! distinction **cannot be made from the saddles' positions alone**, and the paper
//! states an asymptote-side criterion for it — Proposition 1 and Corollary 1.
//!
//! V-31 recorded that criterion as one the prose never pins down, and following
//! the authors' contour-count shortcut instead as rule 5 working as intended.
//! **Half of that was wrong, and A-020 corrected it (M-230).** The criterion *is*
//! derivable from the paper's own normal form for the face bilinear, it is
//! implemented here as [`BodySaddles::same_asymptote_side`], and it agrees with
//! the contour count on every cell measured — false exactly when there is one
//! contour. So the shortcut was sound for what it decides, and the reason to have
//! the predicate is that a derivation checked against a program is worth more than
//! either alone (V-30's shape).
//!
//! It does **not** settle everything the contour count was being asked to settle.
//! A case-13 cell with contours of nine and three passes the criterion and is not
//! a tunnel; Corollary 6's length bound is what excludes it. See
//! [`Contours::topology`].

#[cfg(test)]
mod tests;

use crate::cube::{
    EDGE_AXIS, EDGE_CORNERS, EDGE_COUNT, corner_offset, edge_crossing, face_corners,
};
use crate::real::Real;

use super::table::{NO_EDGE, face_bit, segment_links};

/// The number of candidate body saddles a cell can have. Two, from one quadratic.
pub const SADDLE_COUNT: usize = 2;

/// Bit `k` of [`BodySaddles::inside_mask`], by coordinate and solution.
///
/// The layout matches the authors' implementation so that the two can be compared
/// directly: `u₀, u₁, v₀, v₁, w₀, w₁` in bits `0..6`.
const fn coordinate_bit(axis: usize, solution: usize) -> u8 {
    1 << (2 * axis + solution)
}

/// All six coordinate bits set — the configuration that has an inner hexagon.
pub const ALL_INSIDE: u8 = 0b0011_1111;

/// The trilinear interpolant's body saddles within one cell.
///
/// Constructed by [`BodySaddles::of`]. The two candidates share one `u` set, one
/// `v` set and one `w` set; solution `k` is the point `(u[k], v[k], w[k])`.
///
/// A coordinate is **usable** only when [`BodySaddles::inside_mask`] marks it.
/// Coordinates that are not marked may be non-finite: the linear solve for `v` or
/// `w` divides by a difference that vanishes when the interpolant does not vary
/// along that axis, and no sentinel is substituted for the result. The mask is the
/// single place that says which numbers mean anything, which is why every accessor
/// that returns a *point* is gated on it.
#[derive(Clone, Copy, Debug, PartialEq)]
pub struct BodySaddles<R> {
    coordinate: [[R; SADDLE_COUNT]; 3],
    inside: u8,
}

impl<R: Real> BodySaddles<R> {
    /// Locate the body saddles of the interpolant on this cell.
    ///
    /// `corner` is the eight corner values in `cube.rs`'s numbering, which
    /// is also Grosso's. The zero level set is assumed, as everywhere in this
    /// crate.
    #[must_use]
    pub fn of(corner: &[R; 8]) -> Self {
        let [a, b, c] = Self::coefficients(corner);
        let (u, roots) = Self::roots(a, b, c);

        // `v` is where the `w = 0` face's hyperbola sits at this `u`, and `w` is
        // where the `v = 0` face's hyperbola sits. Both are linear, which is what
        // makes one quadratic enough for all three coordinates.
        let mut v = [R::ZERO; SADDLE_COUNT];
        let mut w = [R::ZERO; SADDLE_COUNT];
        for ((vk, wk), &u) in v.iter_mut().zip(w.iter_mut()).zip(u.iter()).take(roots) {
            *vk = Self::level_crossing(corner[0], corner[1], corner[2], corner[3], u);
            *wk = Self::level_crossing(corner[0], corner[1], corner[4], corner[5], u);
        }
        let coordinate = [u, v, w];

        let mut inside = 0u8;
        for (axis, values) in coordinate.iter().enumerate() {
            for (k, &value) in values.iter().enumerate().take(roots) {
                if value > R::ZERO && value < R::ONE {
                    inside |= coordinate_bit(axis, k);
                }
            }
        }

        Self { coordinate, inside }
    }

    /// The quadratic's coefficients `[a, b, c]`, at the zero level set.
    ///
    /// Separate from [`BodySaddles::of`] so that
    /// `the_coefficients_reproduce_the_face_hyperbola_difference` can check them
    /// against a direct evaluation of the expression they were derived from,
    /// rather than against themselves.
    #[must_use]
    pub fn coefficients(corner: &[R; 8]) -> [R; 3] {
        // The two faces' bilinear "twist" terms, and the two edge differences the
        // derivation pairs them with.
        let twist_lo = (corner[0] + corner[3]) - (corner[1] + corner[2]);
        let twist_hi = (corner[4] + corner[7]) - (corner[5] + corner[6]);
        let du_lo = corner[1] - corner[0];
        let du_hi = corner[5] - corner[4];
        let dv_lo = corner[2] - corner[0];
        let dv_hi = corner[6] - corner[4];

        let a = du_hi * twist_lo - du_lo * twist_hi;
        // `i₀ = 0`, so `(i₀ − f₀)` is `−f₀` and `(i₀ − f₄)` is `−f₄`.
        let b = (corner[4] * twist_lo - corner[0] * twist_hi) + (du_hi * dv_lo - du_lo * dv_hi);
        let c = corner[2] * corner[4] - corner[0] * corner[6];
        [a, b, c]
    }

    /// The real roots of `a·u² + b·u + c`, and how many there are.
    ///
    /// # Why not the textbook formula
    ///
    /// The authors' implementation uses `(−b ± √d)/2a`, which cancels
    /// catastrophically in whichever branch subtracts `√d` from a near-equal `|b|`
    /// — and `a` is a difference of near-equal products with nothing keeping it
    /// away from zero. [`super::interior`] rejected the same formula for the same
    /// reason at A-002c. Kahan's form adds magnitudes instead, so `q` is accurate
    /// to rounding; `q/a` is the large root and, by Vieta, `c/q` the small one.
    ///
    /// # Why `a == 0` is a root count and not an absence
    ///
    /// With `a` zero the equation is linear and has **one** root, `−c/b`. The
    /// textbook formula loses it — it divides by `2a` and yields infinities — and
    /// the authors' implementation inherits that loss. Solving the smaller
    /// polynomial as a smaller polynomial keeps the count honest, which matters
    /// because Grosso §5.3 selects the cell's interior vertex by *how many* face
    /// pairs have a single solution. The divergence is deliberate and measured;
    /// see M-207.
    fn roots(a: R, b: R, c: R) -> ([R; SADDLE_COUNT], usize) {
        let mut roots = [R::ZERO; SADDLE_COUNT];
        if a == R::ZERO {
            if b == R::ZERO {
                return (roots, 0);
            }
            roots[0] = -c / b;
            return (roots, 1);
        }

        let discriminant = b * b - R::TWO * R::TWO * a * c;
        if discriminant < R::ZERO {
            return (roots, 0);
        }
        if discriminant == R::ZERO {
            // A double root is **one** intersection point, not two: the two
            // hyperbolas touch rather than cross, and Proposition 1 counts points.
            // Reporting two here would let a degenerate zero-area "hexagon" claim
            // six saddles.
            roots[0] = -b / (R::TWO * a);
            return (roots, 1);
        }

        // `signum(0)` is `+1` for `+0.0`; either sign keeps this a sum of
        // magnitudes, because `b` is then zero and adds nothing. `q` cannot be
        // zero here: that would need `|b| = −|b|`, hence `b == 0` *and*
        // `discriminant == 0`, which the branch above has already taken.
        let q = -(b + b.signum() * discriminant.sqrt()) * R::HALF;
        roots[0] = q / a;
        roots[1] = c / q;
        (roots, SADDLE_COUNT)
    }

    /// Where the level set crosses the segment from `lo` to `hi` at parameter `u`.
    ///
    /// `lo` interpolates `lo0 → lo1` across `u` and `hi` likewise; the return value
    /// is the coordinate at which the interpolation between them reaches zero. Not
    /// guarded: the caller's mask records whether the result is usable, and an
    /// epsilon here would move a saddle rather than reject it.
    fn level_crossing(lo0: R, lo1: R, hi0: R, hi1: R, u: R) -> R {
        let s = R::ONE - u;
        let lo = lo0 * s + lo1 * u;
        let hi = hi0 * s + hi1 * u;
        -lo / (hi - lo)
    }

    /// Which of the six coordinates lie strictly inside the cell.
    ///
    /// Bits `0..6` are `u₀, u₁, v₀, v₁, w₀, w₁`. Strict on both ends: a coordinate
    /// of exactly `0` or `1` places the saddle *on* a face rather than inside the
    /// cell, which is the singular configuration A-002i owns and not a body saddle.
    #[must_use]
    pub const fn inside_mask(&self) -> u8 {
        self.inside
    }

    /// How many of the six coordinates lie strictly inside the cell.
    #[must_use]
    pub const fn inside_count(&self) -> u32 {
        self.inside.count_ones()
    }

    /// Does this cell have all six saddle coordinates, and therefore an inner
    /// hexagon?
    ///
    /// **This is not the same as "has a tunnel".** Six saddles mean the cell is
    /// either a tunnel or a single twelve-vertex contour; which one it is follows
    /// from the number of closed contours the cut edges form, not from the saddles.
    /// See this module's header.
    #[must_use]
    pub const fn has_inner_hexagon(&self) -> bool {
        self.inside == ALL_INSIDE
    }

    /// The coordinates along one axis, whether or not they are usable.
    ///
    /// `axis` is `0` for `u`, `1` for `v`, `2` for `w`. Consult
    /// [`inside_mask`](Self::inside_mask) before using a value.
    ///
    /// # Panics
    ///
    /// If `axis` is not less than three.
    #[must_use]
    pub fn axis(&self, axis: usize) -> [R; SADDLE_COUNT] {
        self.coordinate[axis]
    }

    /// The six vertices of the inner hexagon, in order around it.
    ///
    /// `None` unless every coordinate is inside the cell, since the hexagon is not
    /// otherwise defined. Consecutive vertices differ in exactly one coordinate,
    /// so every edge of the hexagon is parallel to an axis — which is what
    /// Grosso's Proposition 2 says, and
    /// `hexagon_edges_are_axis_parallel_and_close` checks.
    ///
    /// # The order is the reference implementation's, not the paper's
    ///
    /// The paper's own listing of these six points is corrupt in the copy this
    /// project holds — its first branch assigns `p₁` and `p₂` the same triple,
    /// which cannot be a hexagon. Guessing the intended point is exactly what
    /// rule 5 forbids, so the order below is the authors' program's (V-31).
    #[must_use]
    pub fn inner_hexagon(&self) -> Option<[[R; 3]; 6]> {
        if !self.has_inner_hexagon() {
            return None;
        }
        let [u, v, w] = self.coordinate;
        Some([
            [u[0], v[0], w[0]],
            [u[0], v[0], w[1]],
            [u[1], v[0], w[1]],
            [u[1], v[1], w[1]],
            [u[1], v[1], w[0]],
            [u[0], v[1], w[0]],
        ])
    }
}

/// Which of a cell's six faces are **singular**, as a [`face_bit`] mask.
///
/// A face is singular when the bilinear interpolant's saddle sits exactly *on*
/// the level set — Grosso 2017 §3, where the asymptotic decider's coefficient `α`
/// equals the isovalue and *"there are no asymptotes"*. This crate extracts the
/// zero level set, so in the face's normal form `G = α + η(u−u₀)(v−v₀)` the
/// condition `α = 0` becomes the two diagonal products being equal:
///
/// ```text
/// v₀·v₂ == v₁·v₃
/// ```
///
/// # Why this is detection and not resolution, and why it is here
///
/// [`super::ambiguity::face_is_joined`] already gives a tie a *defined* answer —
/// it resolves to **separated** — so nothing is undefined today and no mesh has a
/// hole because of this. What is missing is topological correctness: Grosso 2017's
/// Definition 3.2 says *"a topologically correct triangulation across singular cell
/// faces will not divide the surface into two branches"*, and resolving to
/// separated is exactly dividing it into two.
///
/// So `ambiguity` is deliberately untouched — changing it would move every mesh
/// this crate has ever produced, including the committed golden hashes — and the
/// singular case belongs to the opt-in `trilinear` path instead. This function is
/// the first half of it: **say where the configuration is**, which is what makes
/// the second half measurable rather than asserted (A-002i).
///
/// # It reads a face's own four values and nothing else
///
/// The same locality [`super::ambiguity`] relies on: two cells sharing a face
/// compute this from the same four numbers in a possibly different order, and the
/// predicate is invariant under the rotations and reflections that relabelling can
/// produce, because it compares the two *diagonal* products and a face's diagonals
/// are the same two pairs whichever corner is called first. That is what lets
/// neighbours agree without communicating.
#[must_use]
pub fn singular_face_mask<R: Real>(corner_value: &[R; 8]) -> u8 {
    let mut mask = 0u8;
    for axis in 0..3usize {
        for side in 0..2u8 {
            let c = face_corners(axis, side);
            let v = [
                corner_value[c[0] as usize],
                corner_value[c[1] as usize],
                corner_value[c[2] as usize],
                corner_value[c[3] as usize],
            ];
            if v[0] * v[2] == v[1] * v[3] {
                mask |= face_bit(axis, side);
            }
        }
    }
    mask
}

/// The longest contour a **tunnel** can have, from Grosso's Corollary 6.
///
/// > If two contours build a tunnel, one of the contours can have at most 6
/// > vertices and the other 3 vertices.
///
/// Only the bound is used, as a *necessary* condition read contrapositively: a
/// contour longer than this cannot belong to a tunnel. The corollary's other half
/// — that the second contour has exactly three vertices — **is falsified by
/// measurement** and is not used: `[4,4]` and `[3,3,6]` tunnels are common
/// (M-230), and both have a second contour that is not a triangle.
pub const MAX_TUNNEL_CONTOUR: usize = 6;

/// Most closed contours one cell's cut edges can form.
///
/// Twelve cut edges and three to a ring is four, and four is reached — by the
/// configuration whose four inside corners are the corners of one tetrahedron,
/// each isolated from the others.
pub const MAX_CONTOURS: usize = 4;

/// The closed rings a cell's cut edges form, in order around each ring.
///
/// Grosso's Algorithm 1 step 2: *"the intersection of the isosurface with the
/// cell is computed at the cell edges. From these intersection points a set of up
/// to four contours is generated **using the asymptotic decider to preserve
/// consistency across cells borders**."*
///
/// # Crack-freeness is inherited here rather than argued
///
/// The rings come from [`super::table::segment_links`], which is the *same*
/// function the table and the face decider already use, driven by the *same*
/// `joined` mask [`super::ambiguity::joined_mask`] computes. So a cell's face
/// connectivity is untouched by anything in this module, and ✗11's face-locality
/// property — a face's segments are a function of that face's own four corner
/// signs and nothing else — keeps covering it, as does A-002's
/// [`validate_decider_table`](super::validate_decider_table) over all 16,384
/// `(case, mask)` pairs. **The interior classification cannot move a face
/// segment**, which is the whole reason this construction needs no grid
/// subdivision pass where Custodio's does (V-29).
///
/// # Why the tunnel test lives here and not in [`BodySaddles`]
///
/// Six body saddles mean the cell is *either* a tunnel *or* one twelve-vertex
/// contour, and the saddles cannot tell you which. The number of contours can,
/// and that is what the authors' own implementation branches on. See
/// [`Contours::topology`].
#[derive(Clone, Copy, Debug, PartialEq, Eq)]
pub struct Contours {
    /// Every ring's edges, laid end to end.
    edges: [u8; EDGE_COUNT],
    /// Where each ring starts in `edges`, plus a final sentinel at the end.
    start: [u8; MAX_CONTOURS + 1],
    count: u8,
}

impl Contours {
    /// Walk a cell's cut edges into closed rings.
    ///
    /// `case` is the corner-sign index and `joined` the per-face resolution mask,
    /// exactly as [`super::table::segment_links`] takes them.
    #[must_use]
    pub fn of(case: u8, joined: u8) -> Self {
        let next = segment_links(case, joined);
        let mut edges = [NO_EDGE; EDGE_COUNT];
        let mut start = [0u8; MAX_CONTOURS + 1];
        let mut count = 0usize;
        let mut filled = 0usize;
        let mut visited = 0u16;

        for first in 0..EDGE_COUNT as u8 {
            if next[first as usize] == NO_EDGE || visited & (1 << first) != 0 {
                continue;
            }
            debug_assert!(count < MAX_CONTOURS, "more rings than a cell can hold");
            start[count] = filled as u8;
            count += 1;

            let mut current = first;
            while visited & (1 << current) == 0 {
                visited |= 1 << current;
                edges[filled] = current;
                filled += 1;
                current = next[current as usize];
            }
        }
        start[count] = filled as u8;

        Self {
            edges,
            start,
            count: count as u8,
        }
    }

    /// How many closed rings this cell has. Zero for an empty or full cell.
    #[must_use]
    pub const fn count(&self) -> usize {
        self.count as usize
    }

    /// One ring's cut edges, in order around it.
    ///
    /// # Panics
    ///
    /// If `index` is not less than [`count`](Self::count).
    #[must_use]
    pub fn ring(&self, index: usize) -> &[u8] {
        assert!(index < self.count(), "contour {index} does not exist");
        let (lo, hi) = (self.start[index] as usize, self.start[index + 1] as usize);
        &self.edges[lo..hi]
    }

    /// The length of the longest ring, or zero if there are none.
    #[must_use]
    pub fn longest(&self) -> usize {
        (0..self.count())
            .map(|i| self.ring(i).len())
            .max()
            .unwrap_or(0)
    }

    /// What the cell's surface is, once the saddles and the rings are both known.
    ///
    /// **This is the decision six body saddles cannot make alone.**
    ///
    /// - fewer than six saddles → every ring is a disk;
    /// - six saddles and **one** ring → that ring has twelve vertices and is
    ///   still a disk, just a long one;
    /// - six saddles, **two or more** rings, none longer than
    ///   [`MAX_TUNNEL_CONTOUR`] → two of them are the two ends of a tunnel, and a
    ///   third, if present, is a separate disk;
    /// - six saddles, two or more rings, and one of them **longer** than that →
    ///   [`Topology::SeparateDisks`], which is not a tunnel and has no
    ///   triangulation rule. See that variant.
    ///
    /// # The ring count alone is not the answer, and the length bound is the fix
    ///
    /// The authors' implementation branches on the ring count and nothing else,
    /// and V-31 recorded that as making Grosso's asymptote-side criterion
    /// (Proposition 1, Corollary 1) unnecessary. **The first half of that is
    /// right and the second is not** (M-230). The criterion is derivable — see
    /// [`BodySaddles::same_asymptote_side`] — and it agrees with the ring count
    /// exactly, on every cell measured: it is false precisely when there is one
    /// ring. So it settles the twelve-vertex split and says nothing about this
    /// one.
    ///
    /// What the ring count misses is a case-13 cell whose rings come out **nine
    /// and three**. It has two rings, so the count calls it a tunnel; its inside
    /// region is two separate blobs rather than one cylinder (M-229); and it is
    /// the only shape that then reaches the contour edge Grosso's tunnel
    /// triangulation has no rule for (M-228). **Corollary 6 excludes it and was
    /// being read as a description rather than a test** — hence the length bound
    /// here, which is that corollary used contrapositively.
    #[must_use]
    pub fn topology<R: Real>(&self, saddles: &BodySaddles<R>) -> Topology {
        if !saddles.has_inner_hexagon() {
            return Topology::Disks;
        }
        match self.count() {
            0 | 1 => Topology::TwelveVertexContour,
            _ if self.longest() <= MAX_TUNNEL_CONTOUR => Topology::Tunnel,
            _ => Topology::SeparateDisks,
        }
    }
}

/// What the trilinear interpolant does inside one cell.
#[derive(Clone, Copy, Debug, PartialEq, Eq, Hash)]
pub enum Topology {
    /// Every contour bounds a disk. The overwhelming majority of cells.
    Disks,
    /// Two contours are the ends of a cylinder through the cell.
    Tunnel,
    /// A single contour through all twelve cut edges, still a disk. Only
    /// reachable from Marching Cubes' case 13, and rare even there — Grosso
    /// counts **7** in a 512²×641 CT skull, against 2,057 tunnels.
    TwelveVertexContour,
    /// Six body saddles, two or more contours, and one of them longer than
    /// [`MAX_TUNNEL_CONTOUR`]. **Not a tunnel, and Grosso gives no rule for it.**
    ///
    /// Reachable on Marching Cubes' case 13 at particular face resolutions, where
    /// the contours come out **nine and three**. The ring count calls such a cell
    /// a tunnel; Corollary 6 excludes it; and flood-filling its inside region puts
    /// its inside corners in **two** components rather than one, so the two
    /// contours bound two separate disks rather than the ends of one cylinder
    /// (M-229).
    ///
    /// The cell therefore needs the *disk* triangulation — but §5.3 selects that
    /// path's interior vertex from face pairs whose quadratic has a **single**
    /// solution, and a six-saddle cell has none. So there is no rule to apply and
    /// none to derive without guessing, which rule 5 forbids;
    /// [`MarchingCubes::extract`](super::MarchingCubes::extract) returns
    /// [`Error::UnresolvedSixSaddle`](crate::Error::UnresolvedSixSaddle) rather
    /// than emitting a hole. A-020b owns deriving it.
    SeparateDisks,
}

/// One of the three pairs of opposite faces, named by the axis its lines run
/// along.
///
/// Pair `U` is the faces `u = 0` and `u = 1`, joined by lines parallel to `u`,
/// and so on.
#[derive(Clone, Copy, Debug, PartialEq, Eq, Hash)]
pub enum Pair {
    /// Faces `u = 0` and `u = 1`.
    U = 0,
    /// Faces `v = 0` and `v = 1`.
    V = 1,
    /// Faces `w = 0` and `w = 1`.
    W = 2,
}

impl Pair {
    /// All three, in `u, v, w` order.
    pub const ALL: [Pair; 3] = [Pair::U, Pair::V, Pair::W];
}

impl<R: Real> BodySaddles<R> {
    /// Which of a pair's two lines lie inside the cell, as a two-bit mask.
    ///
    /// A line joining one pair of opposite faces is fixed by the *other two*
    /// coordinates, and it lies inside the cell exactly when both of them do.
    ///
    /// # The `u` pair's two solutions are crossed, and this is derived not copied
    ///
    /// The obvious reading is that solution `k` contributes the line at
    /// `(v[k], w[k])`, and for the `v` and `w` pairs that is right. **For the `u`
    /// pair it is wrong**, and the inner hexagon is what shows it. Consecutive
    /// hexagon vertices differ in exactly one coordinate, so a pair of vertices
    /// differing *only* in `u` is a `u`-parallel line; reading them off
    /// [`inner_hexagon`](Self::inner_hexagon)'s ring gives
    ///
    /// ```text
    /// (u₀,v₀,w₁) ↔ (u₁,v₀,w₁)   →   the u-line at (v₀, w₁)
    /// (u₀,v₁,w₀) ↔ (u₁,v₁,w₀)   →   the u-line at (v₁, w₀)
    /// ```
    ///
    /// — indices `0,1` and `1,0`, **crossed**. The same reading gives `(u₀,v₀)`
    /// and `(u₁,v₁)` for the `w` pair and `(u₀,w₀)` and `(u₁,w₁)` for the `v`
    /// pair, both uncrossed. The authors' implementation agrees, and it was the
    /// disagreement with the obvious reading that made this worth deriving:
    /// `the_line_inventory_agrees_with_the_inner_hexagon` checks all three pairs
    /// against the ring rather than against the transcription (M-215).
    #[must_use]
    pub fn lines(&self, pair: Pair) -> u8 {
        let inside = self.inside;
        // `(axis, solution)` index pairs, per line, per pair of opposite faces.
        let needed: [[(usize, usize); 2]; 2] = match pair {
            // A u-line is fixed by (v, w) — crossed.
            Pair::U => [[(1, 1), (2, 0)], [(1, 0), (2, 1)]],
            // A v-line is fixed by (u, w).
            Pair::V => [[(0, 0), (2, 0)], [(0, 1), (2, 1)]],
            // A w-line is fixed by (u, v).
            Pair::W => [[(0, 0), (1, 0)], [(0, 1), (1, 1)]],
        };
        let mut mask = 0u8;
        for (line, coordinates) in needed.iter().enumerate() {
            if coordinates
                .iter()
                .all(|&(axis, k)| inside & coordinate_bit(axis, k) != 0)
            {
                mask |= 1 << line;
            }
        }
        mask
    }

    /// How many of a pair's two lines lie inside the cell.
    #[must_use]
    pub fn line_count(&self, pair: Pair) -> u32 {
        self.lines(pair).count_ones()
    }

    /// Do both solutions lie on the same side of every face's asymptotes?
    ///
    /// **This is Grosso's Proposition 1, and it is the half of it that
    /// [`has_inner_hexagon`](Self::has_inner_hexagon) does not cover.** The
    /// proposition is an *if and only if* with two clauses:
    ///
    /// > ...is homeomorphic to a cylinder, i.e. it is a tunnel, if and only if for
    /// > all three pairs of opposite faces the corresponding hyperbolas have two
    /// > intersection points within the range `[0,1]²` **and the intersection
    /// > points are always at the same side of the corresponding asymptotes**.
    ///
    /// `has_inner_hexagon` is the first clause. This is the second, and Corollary 1
    /// states its contrapositive: six solutions that are *not* all on the same side
    /// give a twelve-vertex contour rather than a tunnel.
    ///
    /// # Derived from the paper's own normal form, not guessed
    ///
    /// Rule 5 forbids inventing a predicate the prose leaves vague, and V-31
    /// recorded this one as vague. It is not: §3 gives the face bilinear its normal
    /// form `G(s,t) = α + η(s − s_c)(t − t_c)` and says *"the lines `s = s_c` and
    /// `t = t_c` are the asymptotes"*, which names them exactly. Matching
    /// coefficients against `G = a + (b−a)s + (c−a)t + (a+d−b−c)st` for a face whose
    /// corners are `a, b, c, d` at `(0,0), (1,0), (0,1), (1,1)` gives
    ///
    /// ```text
    /// η = a + d − b − c        η·s_c = a − c        η·t_c = a − b
    /// ```
    ///
    /// A point on the hyperbola has `η(s − s_c)(t − t_c)` equal to a fixed non-zero
    /// constant, so `sign(s − s_c)` and `sign(t − t_c)` determine each other and
    /// **one coordinate per face decides it** — "same side of the asymptotes" is
    /// "same branch of the hyperbola", which is what the proposition's proof argues
    /// geometrically: *"otherwise it is not possible to move from one intersection
    /// point to the other within the empty region corresponding to the tunnel."*
    ///
    /// The comparison is written as `η·s − (a − c)` rather than `s − s_c` so that
    /// no division happens. The common factor `η` cancels between the two solutions,
    /// and a face with `η == 0` — a bilinear with no twist, whose level set is a
    /// straight line with no asymptotes — yields the same constant for both and is
    /// reported as same-side, which is the only answer a line admits.
    ///
    /// # The crossing in [`lines`](Self::lines) does not reach here
    ///
    /// Pair `U`'s two solutions are crossed, `(v₁,w₀)` and `(v₀,w₁)`. This is a
    /// predicate over the *set* of two points, so which pairing they take is
    /// immaterial: `{v[0], v[1]}` is the same set either way.
    ///
    /// Returns `false` when the cell has no inner hexagon, since Proposition 1's
    /// first clause has already failed and there is no second question to ask.
    #[must_use]
    pub fn same_asymptote_side(&self, corner: &[R; 8]) -> bool {
        if !self.has_inner_hexagon() {
            return false;
        }
        let [u, v, _] = self.coordinate;
        // Per face: its four corner indices as `a, b, c, d` at `(0,0), (1,0),
        // (0,1), (1,1)` in that face's own `(s, t)`, and which saddle coordinate
        // plays `s`. Faces `w = *` use `(u, v)` and faces `v = *` use `(u, w)`, so
        // `s` is `u` for four of the six; faces `u = *` use `(v, w)`.
        const FACE: [([usize; 4], usize); 6] = [
            ([0, 1, 2, 3], 0), // w = 0, (s, t) = (u, v)
            ([4, 5, 6, 7], 0), // w = 1
            ([0, 1, 4, 5], 0), // v = 0, (s, t) = (u, w)
            ([2, 3, 6, 7], 0), // v = 1
            ([0, 2, 4, 6], 1), // u = 0, (s, t) = (v, w)
            ([1, 3, 5, 7], 1), // u = 1
        ];

        for ([a, b, c, d], axis) in FACE {
            let twist = (corner[a] + corner[d]) - (corner[b] + corner[c]);
            let offset = corner[a] - corner[c];
            let s = if axis == 0 { u } else { v };
            let lo = twist * s[0] - offset;
            let hi = twist * s[1] - offset;
            if lo * hi < R::ZERO {
                return false;
            }
        }
        true
    }

    /// The point where a pair's line `line` crosses the cell, given the value of
    /// the coordinate it runs along.
    ///
    /// The line is fixed in the other two coordinates; `along` supplies the third.
    ///
    /// Test-only: nothing in the extractor needs a point *on* a line, only the
    /// saddles the lines cross at. It exists so
    /// `the_line_inventory_agrees_with_the_inner_hexagon` can check the pairing
    /// against the hexagon's ring rather than against the transcription it agrees
    /// with. A-002h may promote it when the hexagon is meshed.
    #[cfg(test)]
    pub(super) fn point_on_line_for_test(&self, pair: Pair, line: usize, along: R) -> [R; 3] {
        let [u, v, w] = self.coordinate;
        match pair {
            Pair::U if line == 0 => [along, v[1], w[0]],
            Pair::U => [along, v[0], w[1]],
            Pair::V if line == 0 => [u[0], along, w[0]],
            Pair::V => [u[1], along, w[1]],
            Pair::W if line == 0 => [u[0], v[0], along],
            Pair::W => [u[1], v[1], along],
        }
    }
}

impl<R: Real> BodySaddles<R> {
    /// The one interior vertex an ambiguous disk contour is fanned from, in the
    /// cell's own `[0,1]³` coordinates.
    ///
    /// Grosso §5.3. A contour of six or fewer vertices that touches no ambiguous
    /// face is a flat polygon and needs nothing; a contour of six to nine that
    /// crosses an ambiguous face **twice** cannot be triangulated manifoldly
    /// without one, *"otherwise a third edge in one of the ambiguous faces has to
    /// be introduced, violating manifoldness across cells borders."*
    ///
    /// `None` when every in-range line belongs to a single pair of opposite faces,
    /// because then no two lines cross and there is no interior saddle to use —
    /// and none is needed, since such a contour does not double back.
    ///
    /// # This branch structure is the reference implementation's, and the check
    /// on it is geometric rather than textual
    ///
    /// The paper says only *"if there are only two such lines, the additional
    /// vertex is the intersection point. If there are three such lines, the
    /// additional inner vertex is the midpoint between the two saddle points"* —
    /// which does not determine the four-line case, and does not say which of
    /// three lines' three pairwise intersections the "two saddle points" are. The
    /// index selection below is therefore transcribed from the authors' program
    /// (V-31), and **verified by the one property that a transcription error
    /// cannot survive**: the point it returns must lie *on* the level set.
    /// `the_interior_vertex_lies_on_the_level_set` checks exactly that against an
    /// independent evaluation of the interpolant, which is the same instrument
    /// that validated the inner hexagon.
    #[must_use]
    pub fn interior_vertex(&self) -> Option<[R; 3]> {
        let set = |axis: usize, k: usize| self.inside & coordinate_bit(axis, k) != 0;
        // **Selection, not a 0/1 weight, and the difference is a real defect
        // (M-221).** The reference multiplies each candidate by a 0/1 flag and
        // sums, which is a selection *for finite values only*: an out-of-range
        // coordinate can be `NaN` — the linear solve for `v` or `w` divides by a
        // difference that vanishes where the interpolant does not vary along that
        // axis — and `0 × NaN` is `NaN`, not zero. The reference dodges it by
        // substituting `-1` for every non-finite coordinate; this crate keeps the
        // mask as the single authority on which numbers mean anything, so the
        // selection has to be a branch.
        let f = |axis: usize, k: usize, x: R| if set(axis, k) { x } else { R::ZERO };
        let one = |axis: usize, k: usize| if set(axis, k) { R::ONE } else { R::ZERO };
        let count = |a: (usize, usize), b: (usize, usize)| u8::from(set(a.0, a.1) && set(b.0, b.1));

        // Lines per pair of opposite faces. The `u` pair is crossed; see
        // [`lines`](Self::lines) for why that is derived and not copied.
        let fc_w = count((0, 0), (1, 0)) + count((0, 1), (1, 1));
        let fc_v = count((0, 0), (2, 0)) + count((0, 1), (2, 1));
        let fc_u = count((1, 0), (2, 1)) + count((1, 1), (2, 0));
        let total = fc_w + fc_v + fc_u;

        // Every line in one pair — parallel lines never cross, so there is no
        // saddle. Also covers `total == 0`.
        if total == fc_w || total == fc_v || total == fc_u {
            return None;
        }

        let [u, v, w] = self.coordinate;
        let (uc, vc, wc) = match total {
            2 if fc_w == 0 => (
                f(0, 0, u[0]) + f(0, 1, u[1]),
                f(1, 0, v[0]) + f(1, 1, v[1]),
                f(1, 0, w[1]) + f(1, 1, w[0]),
            ),
            2 if fc_v == 0 => (
                f(0, 0, u[0]) + f(0, 1, u[1]),
                f(0, 0, v[0]) + f(0, 1, v[1]),
                f(0, 0, w[1]) + f(0, 1, w[0]),
            ),
            2 => (
                f(1, 0, u[0]) + f(1, 1, u[1]),
                f(1, 0, v[0]) + f(1, 1, v[1]),
                f(1, 0, w[0]) + f(1, 1, w[1]),
            ),
            // Three lines: the mean of the two solutions on each axis, which is
            // the paper's "midpoint between the two saddle points" written so that
            // an axis with only one solution contributes that one unchanged.
            3 => (
                (f(0, 0, u[0]) + f(0, 1, u[1])) / (one(0, 0) + one(0, 1)),
                (f(1, 0, v[0]) + f(1, 1, v[1])) / (one(1, 0) + one(1, 1)),
                (f(2, 0, w[0]) + f(2, 1, w[1])) / (one(2, 0) + one(2, 1)),
            ),
            4 => {
                let on = |axis: usize| one(axis, 0) + one(axis, 1);
                if on(2) == R::ONE {
                    (
                        f(2, 0, u[0]) + f(2, 1, u[1]),
                        f(2, 1, v[0]) + f(2, 0, v[1]),
                        f(2, 0, w[0]) + f(2, 1, w[1]),
                    )
                } else if on(1) == R::ONE {
                    (
                        f(1, 0, u[0]) + f(1, 1, u[1]),
                        f(1, 0, v[0]) + f(1, 1, v[1]),
                        f(1, 1, w[0]) + f(1, 0, w[1]),
                    )
                } else {
                    (
                        f(0, 0, u[0]) + f(0, 1, u[1]),
                        f(0, 0, v[0]) + f(0, 1, v[1]),
                        f(0, 0, w[0]) + f(0, 1, w[1]),
                    )
                }
            }
            // Five or six lines is the hexagon's territory, and a cell there is a
            // tunnel or a twelve-vertex contour rather than a fanned disk. A-002h
            // owns it; `the_disk_path_never_sees_five_or_six_lines` asserts this
            // arm is unreachable from the disk path rather than assuming it.
            _ => return None,
        };
        Some([uc, vc, wc])
    }
}

/// The code a triangle uses to name the cell's one interior vertex, as opposed
/// to one of the twelve cut edges.
///
/// Deliberately the same value as [`super::table::CENTROID_BASE`]: both name a
/// **cell-local** vertex that no other cell can reach, which is exactly why
/// neither is cached and why A-015's index-space budget already accounts for the
/// slot. What differs is where the position comes from — a centroid averages its
/// cycle's edge vertices, this one is a saddle of the interpolant
/// ([`BodySaddles::interior_vertex`]).
pub const INTERIOR: u8 = EDGE_COUNT as u8;

impl Contours {
    /// Fan every ring into triangles, naming cut edges by index and the cell's
    /// interior vertex by [`INTERIOR`].
    ///
    /// `interior` says whether [`BodySaddles::interior_vertex`] produced one. When
    /// it did, rings of more than three vertices fan from it — `k` triangles for a
    /// ring of `k` — because a ring that crosses an ambiguous face twice cannot be
    /// closed manifoldly from one of its own vertices without laying a third edge
    /// in that face. A ring of exactly three is a triangle either way and takes
    /// the direct emission, as the reference implementation does.
    ///
    /// When there is none, every ring fans from its own first vertex: `k − 2`
    /// triangles, and no interior vertex is created at all.
    ///
    /// Winding follows the ring order, which [`super::table::segment_links`]
    /// already orients — so this inherits A-001's counter-clockwise-from-outside
    /// convention rather than choosing one.
    pub fn fan(&self, interior: bool, mut emit: impl FnMut([u8; 3])) {
        for r in 0..self.count() {
            let ring = self.ring(r);
            if ring.len() == 3 {
                emit([ring[0], ring[1], ring[2]]);
            } else if interior {
                for k in 0..ring.len() {
                    emit([ring[k], ring[(k + 1) % ring.len()], INTERIOR]);
                }
            } else {
                for k in 1..ring.len() - 1 {
                    emit([ring[0], ring[k], ring[k + 1]]);
                }
            }
        }
    }

    /// How many triangles [`fan`](Self::fan) will emit.
    #[must_use]
    pub fn triangle_count(&self, interior: bool) -> usize {
        (0..self.count())
            .map(|r| {
                let k = self.ring(r).len();
                if k == 3 {
                    1
                } else if interior {
                    k
                } else {
                    k - 2
                }
            })
            .sum()
    }
}

/// Most cell-local interior vertices any one cell can need.
///
/// Six, and only a tunnel or a twelve-vertex contour reaches it: those name every
/// vertex of the inner hexagon. A fanned disk uses one, [`INTERIOR`] itself.
///
/// # Six rather than three, on the authors' own later advice
///
/// Grosso 2016 §5.1 collapses the hexagon to **three** inner vertices by taking
/// the midpoint of each consecutive pair (its equation 7). Grosso 2017 §4.1 does
/// not: *"We triangulate the contours with the method proposed in \[6\], but
/// **instead of three we use all six vertices of the inner hexagon**."* The
/// reference implementation is the 2017 one, and six is what this follows — the
/// collapse loses the geometry of a saddle to save two vertices in a case that
/// occurs in roughly one cell in two hundred.
pub const MAX_INTERIOR_VERTICES: usize = 6;

/// Where the surface crosses one cut edge, in the cell's own `[0,1]³`
/// coordinates.
///
/// The edge runs along one axis, so two of the three coordinates are the corner's
/// own `0` or `1` and the third is the crossing parameter.
#[must_use]
pub fn local_crossing<R: Real>(edge: u8, corner: &[R; 8]) -> [R; 3] {
    let [lo, hi] = EDGE_CORNERS[edge as usize];
    let axis = EDGE_AXIS[edge as usize] as usize;
    let o = corner_offset(lo);
    let mut p = [
        R::from_f64(f64::from(o[0])),
        R::from_f64(f64::from(o[1])),
        R::from_f64(f64::from(o[2])),
    ];
    p[axis] = edge_crossing(corner[lo as usize], corner[hi as usize]);
    p
}

/// Squared distance in the cell's local coordinates.
fn distance_squared<R: Real>(a: [R; 3], b: [R; 3]) -> R {
    let d = [a[0] - b[0], a[1] - b[1], a[2] - b[2]];
    d[0] * d[0] + d[1] * d[1] + d[2] * d[2]
}

/// Steps between two hexagon indices around the six-ring, never more than three.
const fn ring_distance(a: usize, b: usize) -> usize {
    let r = a.abs_diff(b);
    if r > 2 { 6 - r } else { r }
}

/// The hexagon index halfway between two that are two steps apart.
const fn ring_midpoint(a: usize, b: usize) -> usize {
    let (lo, hi) = if a < b { (a, b) } else { (b, a) };
    if (hi + 2) % 6 == lo {
        (hi + 1) % 6
    } else {
        (hi + lo) / 2
    }
}

impl Contours {
    /// Which ring, if any, is *not* part of the tunnel.
    ///
    /// Corollary 5: with a tunnel and three contours, the one that does not belong
    /// to it has three vertices. Corollary 7 says how to tell which — its vertices
    /// lie to one side of the quadratic's solutions — and the test is whether the
    /// ring's span in `u` misses the interval between the two `u` roots entirely.
    ///
    /// # One transcribed line is a typo and is fixed here
    ///
    /// The reference computes the ring's `u` span with
    /// `umax = (u_e2 > umax) ? u_e1 : umax` — comparing the third vertex and then
    /// storing the **second**. It is plainly meant to be `u_e2`, it is the only
    /// one of six such lines written that way, and a max that can return a value
    /// smaller than the one it just compared is not a max. Written correctly here,
    /// and `the_detached_ring_is_the_one_outside_the_u_span` measures how often
    /// the two disagree (M-219).
    #[must_use]
    pub fn detached_ring<R: Real>(
        &self,
        saddles: &BodySaddles<R>,
        corner: &[R; 8],
    ) -> Option<usize> {
        if self.count() != 3 {
            return None;
        }
        let u = saddles.axis(0);
        let (u_lo, u_hi) = if u[0] < u[1] {
            (u[0], u[1])
        } else {
            (u[1], u[0])
        };

        for r in 0..self.count() {
            let ring = self.ring(r);
            if ring.len() != 3 {
                continue;
            }
            let mut lo = R::INFINITY;
            let mut hi = R::NEG_INFINITY;
            for &e in ring {
                let x = local_crossing(e, corner)[0];
                lo = lo.min(x);
                hi = hi.max(x);
            }
            if u_lo > hi || u_hi < lo {
                return Some(r);
            }
        }
        None
    }

    /// Triangulate a tunnel or a twelve-vertex contour against the inner hexagon.
    ///
    /// Grosso 2016 §5.1 and §5.2 with 2017 §4.1's six-vertex refinement. Each ring
    /// vertex is assigned to its nearest hexagon vertex; each ring edge then spans
    /// zero, one or two steps around the hexagon and closes with one, two or three
    /// triangles accordingly. A detached three-vertex ring is emitted directly.
    ///
    /// Hexagon vertex `k` is named by the code `INTERIOR + k`.
    ///
    /// Returns the number of ring edges whose endpoints landed **three** steps
    /// apart on the hexagon — a case the construction has no rule for and emits
    /// nothing at. Zero on every configuration measured; returned rather than
    /// asserted so a caller cannot mistake a hole for a closed patch.
    pub fn fan_tunnel<R: Real>(
        &self,
        saddles: &BodySaddles<R>,
        corner: &[R; 8],
        mut emit: impl FnMut([u8; 3]),
    ) -> usize {
        let Some(hexagon) = saddles.inner_hexagon() else {
            return 0;
        };
        // Buffered rather than streamed, because the closing fan's winding is not
        // known until the ring walk has been done: a twelve-vertex contour can
        // circle the hexagon either way, and the closure has to go the other way.
        // Twelve ring edges at three triangles each plus a four-triangle closure
        // is the bound, and `the_worst_case_tunnel_triangle_count_is_pinned`
        // measures what is actually reached.
        let mut buffer = [[0u8; 3]; 40];
        let mut n = 0usize;
        let detached = self.detached_ring(saddles, corner);
        let mut unresolved = 0usize;

        for r in 0..self.count() {
            let ring = self.ring(r);
            if Some(r) == detached {
                buffer[n] = [ring[0], ring[1], ring[2]];
                n += 1;
                continue;
            }

            // Nearest hexagon vertex for each ring vertex.
            let nearest = |edge: u8| -> usize {
                let p = local_crossing(edge, corner);
                let mut best = 0usize;
                let mut best_d = R::INFINITY;
                for (k, &h) in hexagon.iter().enumerate() {
                    let d = distance_squared(p, h);
                    if d < best_d {
                        best_d = d;
                        best = k;
                    }
                }
                best
            };

            for k in 0..ring.len() {
                let (a, b) = (ring[k], ring[(k + 1) % ring.len()]);
                let (ha, hb) = (nearest(a), nearest(b));
                match ring_distance(ha, hb) {
                    0 => {
                        buffer[n] = [a, b, INTERIOR + ha as u8];
                        n += 1;
                    }
                    1 => {
                        // Split the quad along its shorter diagonal.
                        let pa = local_crossing(a, corner);
                        let pb = local_crossing(b, corner);
                        if distance_squared(pa, hexagon[hb]) < distance_squared(pb, hexagon[ha]) {
                            {
                                buffer[n] = [a, b, INTERIOR + hb as u8];
                                n += 1;
                            }
                            {
                                buffer[n] = [a, INTERIOR + hb as u8, INTERIOR + ha as u8];
                                n += 1;
                            }
                        } else {
                            {
                                buffer[n] = [a, b, INTERIOR + ha as u8];
                                n += 1;
                            }
                            {
                                buffer[n] = [b, INTERIOR + hb as u8, INTERIOR + ha as u8];
                                n += 1;
                            }
                        }
                    }
                    2 => {
                        let hm = ring_midpoint(ha, hb);
                        {
                            buffer[n] = [a, b, INTERIOR + hm as u8];
                            n += 1;
                        }
                        {
                            buffer[n] = [a, INTERIOR + hm as u8, INTERIOR + ha as u8];
                            n += 1;
                        }
                        {
                            buffer[n] = [b, INTERIOR + hb as u8, INTERIOR + hm as u8];
                            n += 1;
                        }
                    }
                    _ => unresolved += 1,
                }
            }
        }

        // **§5.2's closing step, and the reason it exists.** A tunnel is an
        // annulus: its two contours pass the inner hexagon from opposite sides,
        // so every hexagon edge is covered twice and the patch closes itself. A
        // twelve-vertex contour is a *disk* — one ring, walking the hexagon once
        // — so it leaves the hexagon as a hexagonal hole, and every hexagon edge
        // is covered once. Grosso closes it by merging the six into three and
        // emitting one last triangle; keeping all six (2017 §4.1) the same hole
        // takes a four-triangle fan. Without this the patch has a six-edge
        // boundary that no neighbouring cell shares, which is a hole in the mesh
        // — and `the_tunnel_patch_is_manifold_inside_the_cell` is what found it
        // (M-218).
        if self.count() == 1 {
            // Which way round the hexagon did the ring walk go? Every hexagon-to-
            // hexagon edge it laid answers it, and they all agree; `+1` means it
            // ran `k → k+1`.
            let mut spin = 0i32;
            for t in &buffer[..n] {
                for k in 0..3 {
                    let (a, b) = (t[k], t[(k + 1) % 3]);
                    if a >= INTERIOR && b >= INTERIOR {
                        let (x, y) = ((a - INTERIOR) as usize, (b - INTERIOR) as usize);
                        spin += if y == (x + 1) % 6 { 1 } else { -1 };
                    }
                }
            }
            for k in 1..5u8 {
                if spin > 0 {
                    {
                        buffer[n] = [INTERIOR, INTERIOR + k + 1, INTERIOR + k];
                        n += 1;
                    }
                } else {
                    {
                        buffer[n] = [INTERIOR, INTERIOR + k, INTERIOR + k + 1];
                        n += 1;
                    }
                }
            }
        }
        for t in &buffer[..n] {
            emit(*t);
        }
        unresolved
    }
}

/// Most triangles one cell's patch can carry on the trilinear path.
///
/// Twenty-two, reached by a tunnel; a fanned disk tops out at twelve. Measured
/// exhaustively for the disk path and over 400,000 random cells for the tunnel —
/// `the_worst_case_triangle_count_is_pinned` and
/// `the_worst_case_tunnel_triangle_count_is_pinned` both pin it in both
/// directions. Larger than [`super::table::MAX_TRIANGLES`], which sizes the
/// 256-case table's own array and is untouched by this path.
pub const MAX_PATCH_TRIANGLES: usize = 24;