map_tile_toolkit/

polygon.rs

//! Polygon slicing for [`Slicer`](crate::Slicer): [`Slicer::polygon`](crate::Slicer::polygon)
//! yields every piece of a polygon in every tile it reaches, and
//! [`Slicer::tile_polygon`](crate::Slicer::tile_polygon) its piece in one tile.
//!
//! A polygon inside one tile's inner box is yielded whole, and one smaller than a tile with few vertices
//! (a building) by a walk over the few tiles it reaches, each given what
//! [`Slicer::tile_polygon`](crate::Slicer::tile_polygon) gives it: no tile it reaches can be covered, and
//! the sweep's setup would cost more than the clips. Both lend a lone ring that lies inside a tile's
//! buffered box as it is, without clipping it (most of OSM's polygons are rings of 4 to 8 vertices).
//! Any other polygon is sliced lazily by a row-by-row scanline:
//!
//! 1. **Prepare** — one pass over the polygon's rings, borrowing them in place: drop degenerate rings,
//!    take the bounding box, and record the `y`-range of every run of [`CHUNK`] consecutive edges. A
//!    polygon inside one tile's inner box is yielded whole, without the rest.
//! 2. **Scan rows** top to bottom. A row looks only at the chunks whose `y`-range reaches it: their
//!    edges that touch the row's buffered slab give the row's tile spans, and those crossing the row's
//!    center line `y = cy` give its crossings. The center line carries the winding ray of every tile in
//!    the row, so the crossings answer both remaining questions: how far an excursion outside a tile
//!    winds around it (the detour's shape), and whether a ring with no edge in a tile contains it.
//! 3. **Sweep** the row left to right over its edge tiles, closing every touching ring's arcs with
//!    the clip's shared core ([`ClipScratch::close_ring`]), and yielding the covered spans between
//!    them as [`FillRun`]s. A Fenwick tree over the row's crossings (in edge order) answers "signed
//!    crossings right of this tile among edges `i..j`" in `O(log n)`.
//!
//! Nothing outside the current row is kept, and a piece is built in one reused buffer, in the global
//! frame: [`Local`] moves it into the tile's frame as it is read.
//!
//! A piece is a valid polygon whenever the input is: the plain clip is kept if its outside segments
//! (those with an endpoint outside the box) are planar, which [`planar`](crate::planar) checks, and
//! otherwise rebuilt by the cheapest repair that is: [`repair`] runs the search for both slicers, each
//! building the attempts its own way (from the sweep's hits, or by clipping the rings).
//!
//! Unclipped ([`ClipMode::Unclipped`]), the sweep is the same but builds nothing: each edge tile that
//! would get a piece, and each tile of a fill run, gets the whole polygon, lent from the input.

use core::cmp::Ordering;
use core::iter;
use core::ops::{Range, RangeInclusive};

use geo_types::Coord;

use crate::TileError;
use crate::clip_polygon::{
    ClipScratch, Closed, PieceBuf, Probe, RingClip, RingPlan, arcs_points, clip_ring,
    clip_ring_planned, convex_hull, fill_box, keep_edge, lane_bounds, push_distinct,
    ring_meets_box, scan_arcs,
};
use crate::clip_polyline::to_local;
use crate::geom::{crossing_x_ceil, ring_orientation};
use crate::grid::{Grid, Seg, ring_within};
use crate::planar::{PAIR_BUDGET, PlanarScratch, Verdict, is_planar};
use crate::slicer::{ClipMode, Local};
use crate::tile::TileId;
use crate::vertex::{PolyVertex, Vertex};

/// Edges per chunk of the row index: a chunk's `y`-range decides which rows look at its edges.
const CHUNK: usize = 32;

/// How many tiles of one row an edge must touch to be kept as a single [`Span`] for the row rather
/// than one [`Hit`] per tile.
const LONG_EDGE: i64 = 4;

/// A horizontal run of tiles lying **entirely inside** the polygon: row `y`, columns `x.start..x.end`.
///
/// No ring edge touches these tiles' buffered boxes, so each is uniformly covered: in each of them,
/// [`Slicer::tile_polygon`](crate::Slicer::tile_polygon) yields this polygon as a single
/// all-synthetic box ring around the buffered box, which [`PolygonSlices::fill_ring`] (or
/// [`Slicer::fill_ring`](crate::Slicer::fill_ring)) returns — the same tile-local ring for every tile
/// of the run. Runs are maximal, and never overlap the polygon's edge tiles.
#[derive(Debug, Clone, PartialEq, Eq, Hash)]
#[non_exhaustive]
pub struct FillRun {
    /// The tile row.
    pub y: i32,
    /// The tile columns, end-exclusive.
    pub x: Range<i32>,
    /// The polygon's exterior winding, which its fill ring follows.
    orient: Ordering,
}

/// One slice of a polygon.
#[derive(Debug)]
pub enum PolygonSlice<'a, V, R> {
    /// One polygon's piece in one tile.
    Piece(PolygonPiece<'a, V, R>),
    /// Tiles a polygon covers entirely, whose content is [`PolygonSlices::fill_ring`] in each.
    Run(FillRun),
}

/// One polygon's rings in one tile: exterior first, then its holes that reach the tile (all of them,
/// and each whole, when unclipped).
#[derive(Debug)]
pub struct PolygonPiece<'a, V, R> {
    tile: TileId,
    rings: PieceRings<'a, V, R>,
}

#[derive(Debug)]
enum PieceRings<'a, V, R> {
    /// Clipped rings in the piece buffer (global, each closed), ending at `ends`, and the tile
    /// origin.
    Built {
        verts: &'a [V],
        ends: &'a [u32],
        origin: Coord<i32>,
    },
    /// The whole polygon — lying in the tile, or unclipped: its kept rings, lent from the input (or
    /// the copy of a ring that had duplicate vertices).
    Whole {
        rings: &'a [RingInfo],
        input: &'a [R],
        copies: &'a [V],
        origin: Coord<i32>,
    },
}

// Borrowed views: `Clone`/`Copy` without requiring them of the vertex or ring types.
impl<V, R> Clone for PieceRings<'_, V, R> {
    fn clone(&self) -> Self {
        *self
    }
}
impl<V, R> Copy for PieceRings<'_, V, R> {}
impl<V, R> Clone for PolygonPiece<'_, V, R> {
    fn clone(&self) -> Self {
        *self
    }
}
impl<V, R> Copy for PolygonPiece<'_, V, R> {}

impl<'a, V: Vertex, R: AsRef<[V]>> PolygonPiece<'a, V, R> {
    /// The tile this piece belongs to.
    #[must_use]
    pub fn tile(&self) -> TileId {
        self.tile
    }

    /// The piece's rings, exterior first, each closed and in the tile's local frame.
    pub fn rings(&self) -> impl Iterator<Item = Local<'a, V>> + use<'a, V, R> {
        let rings = self.rings;
        let n = match rings {
            PieceRings::Built { ends, .. } => ends.len(),
            PieceRings::Whole { rings, .. } => rings.len(),
        };
        (0..n).map(move |i| match rings {
            PieceRings::Built {
                verts,
                ends,
                origin,
            } => {
                let start = i.checked_sub(1).map_or(0, |p| ends[p] as usize);
                Local::ring(&verts[start..ends[i] as usize], origin, true)
            }
            PieceRings::Whole {
                rings,
                input,
                copies,
                origin,
            } => Local::ring(rings[i].pts(input, copies), origin, false),
        })
    }
}

/// Where one kept ring's distinct vertices are.
#[derive(Debug, Clone, Copy)]
enum Src {
    /// The first `len` vertices of the input ring.
    Input,
    /// `copies[start..start + len]`, the ring with its duplicate vertices dropped.
    Copy(usize),
}

/// One kept ring (at least three distinct vertices) of the polygon being sliced: ring 0 is its
/// exterior. Edge `i` runs from vertex `i` to vertex `(i + 1) % len`, and is edge `edge0 + i` of the
/// polygon.
#[derive(Debug, Clone, Copy)]
struct RingInfo {
    input: u32,
    src: Src,
    len: u32,
    edge0: u32,
}

impl RingInfo {
    fn pts<'a, V, R: AsRef<[V]>>(&self, input: &'a [R], copies: &'a [V]) -> &'a [V] {
        let len = self.len as usize;
        match self.src {
            Src::Input => &input[self.input as usize].as_ref()[..len],
            Src::Copy(start) => &copies[start..start + len],
        }
    }
}

/// [`CHUNK`] consecutive edges of one ring, and the tile rows their buffered `y`-range reaches.
#[derive(Debug, Clone, Copy)]
struct Chunk {
    ring: u32,
    first: u32,
    count: u32,
    rows: (i64, i64),
}

/// A ring edge touching tile `tx` of the current row. An edge touching fewer than [`LONG_EDGE`]
/// tiles there has one hit per tile, so a tile's hits are one contiguous run, used as is.
#[derive(Debug, Clone, Copy)]
struct Hit {
    tx: i32,
    edge: u32,
    ring: u32,
}

/// A long edge touching tiles `first..=last` of the current row: one entry for the row, so a row
/// costs memory per edge, never per tile, however far an edge runs along it.
#[derive(Debug, Clone, Copy)]
struct Span {
    first: i32,
    last: i32,
    edge: u32,
    ring: u32,
}

/// A ring edge crossing the current row's center line, the line of every winding ray in that row
/// (half-open in `y`, so each crossing is counted once). The crossing lies right of tile `tx`'s
/// buffered box — on its winding ray — iff `tx <= right_of`.
#[derive(Debug, Clone, Copy)]
struct Crossing {
    edge: u32,
    ring: u32,
    up: bool,
    right_of: i64,
}

impl Crossing {
    fn sign(self) -> i64 {
        if self.up { 1 } else { -1 }
    }
}

/// Row `ty`'s center line: the midpoint of its buffered box `[ty·e − b, ty·e + e − 1 + b]`, rounded
/// toward zero, as the clip computes it with `midpoint` — in closed form: the buffer cancels, and the
/// sum `2·ty·e + e − 1` is negative exactly when `ty < 0`, where rounding toward zero rounds up.
fn row_center(ty: i64, extent: i64) -> i64 {
    ty * extent + if ty < 0 { extent / 2 } else { (extent - 1) / 2 }
}

/// Rebuild `tree` as a Fenwick (binary indexed) tree over `values`, in `O(n)`.
fn fenwick_build(tree: &mut Vec<i64>, values: impl Iterator<Item = i64>) {
    tree.clear();
    tree.extend(values);
    for i in 0..tree.len() {
        let parent = i | (i + 1);
        if parent < tree.len() {
            tree[parent] += tree[i];
        }
    }
}

/// Add `delta` at position `i`.
fn fenwick_add(tree: &mut [i64], mut i: usize, delta: i64) {
    while i < tree.len() {
        tree[i] += delta;
        i |= i + 1;
    }
}

/// Sum of the first `end` positions.
fn fenwick_prefix(tree: &[i64], mut end: usize) -> i64 {
    let mut sum = 0;
    while end > 0 {
        sum += tree[end - 1];
        end &= end - 1;
    }
    sum
}

/// Sort one row's `hits` by `(tx, edge)`. They arrive in ascending edge order, so a stable counting
/// sort by column finishes the job in `O(n + columns)`; a column range much wider than the hits are
/// many falls back to a comparison sort. `tmp` and `counts` are reused scratch.
fn sort_hits(hits: &mut [Hit], tmp: &mut Vec<Hit>, counts: &mut Vec<usize>) {
    let Some(head) = hits.first() else {
        return;
    };
    let (lo, hi) = hits.iter().fold((head.tx, head.tx), |(lo, hi), h| {
        (lo.min(h.tx), hi.max(h.tx))
    });
    let columns = hi.abs_diff(lo) as usize + 1;
    if columns == 1 {
        // One column: already in edge order.
    } else if hits.len() < 16 || columns > hits.len().saturating_mul(4) {
        hits.sort_unstable_by_key(|h| (h.tx, h.edge));
    } else {
        counts.clear();
        counts.resize(columns, 0);
        for h in hits.iter() {
            counts[h.tx.abs_diff(lo) as usize] += 1;
        }
        // Exclusive prefix sums: where each column starts.
        let mut start = 0;
        for c in counts.iter_mut() {
            (*c, start) = (start, start + *c);
        }
        tmp.clear();
        tmp.extend_from_slice(hits);
        for h in tmp.iter() {
            let at = &mut counts[h.tx.abs_diff(lo) as usize];
            hits[*at] = *h;
            *at += 1;
        }
    }
    debug_assert!(hits.is_sorted_by_key(|h| (h.tx, h.edge)));
}

/// Check that freshly written vertices fit the tile-local frame of `origin`, as [`Local`] lends them.
fn check_local<V: Vertex>(verts: &[V], origin: Coord<i32>) -> Result<(), TileError> {
    for &v in verts {
        to_local(v, origin)?;
    }
    Ok(())
}

/// Restore edge order to `spans`, whose first `kept` are in edge order and the rest were appended (in
/// first-column order, so in edge order too when they all start in the same column): sort the
/// appended part only if needed, then merge the two in one pass through `tmp`.
fn merge_by_edge<T: Copy>(
    spans: &mut Vec<T>,
    kept: usize,
    tmp: &mut Vec<T>,
    edge: impl Fn(&T) -> u32,
) {
    let added = &mut spans[kept..];
    if !added.is_sorted_by_key(&edge) {
        added.sort_unstable_by_key(&edge);
    }
    if kept == 0 || kept == spans.len() || edge(&spans[kept - 1]) < edge(&spans[kept]) {
        return;
    }
    tmp.clear();
    let (old, new) = spans.split_at(kept);
    let (mut i, mut j) = (0, 0);
    while i < old.len() && j < new.len() {
        if edge(&old[i]) < edge(&new[j]) {
            tmp.push(old[i]);
            i += 1;
        } else {
            tmp.push(new[j]);
            j += 1;
        }
    }
    tmp.extend_from_slice(&old[i..]);
    tmp.extend_from_slice(&new[j..]);
    core::mem::swap(spans, tmp);
}

/// `hits` (the spans covering one tile, in edge order, so each ring's are contiguous) split into runs
/// of the same ring.
fn ring_groups(mut hits: &[Hit]) -> impl Iterator<Item = &[Hit]> {
    iter::from_fn(move || {
        let ring = hits.first()?.ring;
        let (group, rest) = hits.split_at(hits.partition_point(|x| x.ring == ring));
        hits = rest;
        Some(group)
    })
}

/// Signed crossings, right of the current tile, of `ring`'s `count` edges from local edge `first`
/// (cyclically): the row's still-active crossings in that edge range, summed by the Fenwick tree.
fn excursion_winding(
    cs: &[Crossing],
    fenwick: &[i64],
    ring: RingInfo,
    first: usize,
    count: usize,
) -> i64 {
    let (start, len) = (ring.edge0 as usize, ring.len as usize);
    let sum = |lo: usize, hi: usize| {
        let i = cs.partition_point(|c| (c.edge as usize) < lo);
        let j = cs.partition_point(|c| (c.edge as usize) < hi);
        fenwick_prefix(fenwick, j) - fenwick_prefix(fenwick, i)
    };
    if first + count <= len {
        sum(start + first, start + first + count)
    } else {
        sum(start + first, start + len) + sum(start, start + first + count - len)
    }
}

/// Reusable working memory for polygon slicing, kept in the [`Slicer`](crate::Slicer) between
/// features (see [`Scratch`]): only its capacity survives a call.
pub(crate) struct PolyScratch<V> {
    /// The current polygon's kept rings, copies of rings that had duplicate vertices, and chunks.
    rings: Vec<RingInfo>,
    copies: Vec<V>,
    chunks: Vec<Chunk>,
    /// Chunk ids by first row, and the chunks reaching the current row (ascending, so edge order).
    order: Vec<u32>,
    active: Vec<u32>,
    /// The current row: its per-tile spans of short edges (by column, then edge), the spans of its
    /// long edges (by first column, then edge), the long ones covering the sweep's current tile (by
    /// edge), and a tile's merged spans when a long edge crosses it. Then its crossings (by edge), the
    /// crossings by `right_of`, and the Fenwick tree of the signs of those still right of the sweep.
    hits: Vec<Hit>,
    long: Vec<Span>,
    long_cover: Vec<Span>,
    covering: Vec<Hit>,
    crossings: Vec<Crossing>,
    by_x: Vec<u32>,
    /// Sorting scratch for the row's spans.
    hits_tmp: Vec<Hit>,
    spans_tmp: Vec<Span>,
    counts: Vec<usize>,
    fenwick: Vec<i64>,
    /// Per ring: odd crossings right of the current tile.
    inside: Vec<bool>,
    /// The piece being yielded.
    piece: PieceBuf<V>,
    arcs: Vec<(usize, usize)>,
    clip: ClipScratch<V>,
    /// Working memory of the piece's planarity check.
    segs: PlanarScratch,
    /// A repair's frame: the outside points it is built around, and their convex hull.
    probe: Probe,
    frame_hull: Vec<Coord<i32>>,
    /// How many vertices a frame keeps past each tip of each bridge, and each input ring's first
    /// bridge (see [`RingPlan::depths`]).
    depths: Vec<(u32, u32)>,
    ring_bases: Vec<usize>,
    /// Each input ring's arcs for the tile, scanned once for the rounds of a frame's probe.
    probe_arcs: Vec<(usize, usize)>,
    probe_rings: Vec<Option<(usize, Range<usize>)>>,
}

impl<V> PolyScratch<V> {
    pub(crate) const fn new() -> Self {
        Self {
            rings: Vec::new(),
            copies: Vec::new(),
            chunks: Vec::new(),
            order: Vec::new(),
            active: Vec::new(),
            hits: Vec::new(),
            long: Vec::new(),
            long_cover: Vec::new(),
            covering: Vec::new(),
            crossings: Vec::new(),
            by_x: Vec::new(),
            hits_tmp: Vec::new(),
            spans_tmp: Vec::new(),
            counts: Vec::new(),
            fenwick: Vec::new(),
            inside: Vec::new(),
            piece: PieceBuf::new(),
            arcs: Vec::new(),
            clip: ClipScratch::new(),
            segs: PlanarScratch::new(),
            probe: Probe::new(),
            frame_hull: Vec::new(),
            depths: Vec::new(),
            ring_bases: Vec::new(),
            probe_arcs: Vec::new(),
            probe_rings: Vec::new(),
        }
    }
}

/// The [`Slicer`](crate::Slicer)'s polygon working memory, allocated by its first polygon, so a
/// slicer used only for lines and points never builds it. It holds nothing between calls that affects
/// results: it compares equal, clones empty, and only its capacity survives a call.
pub(crate) struct Scratch<V>(Option<Box<PolyState<V>>>);

/// The working memory, and the state of the current polygon's sweep (kept out of the iterator, so the
/// iterator stays small to move).
pub(crate) struct PolyState<V> {
    pub(crate) scratch: PolyScratch<V>,
    sweep: Option<Sweep>,
    /// The walk over the few tiles of a small polygon.
    small: Option<SmallWalk>,
}

impl<V> Scratch<V> {
    pub(crate) const fn new() -> Self {
        Self(None)
    }

    /// The working memory, allocated on first use.
    #[inline]
    pub(crate) fn get(&mut self) -> &mut PolyState<V> {
        #[cold]
        fn alloc<V>() -> Box<PolyState<V>> {
            Box::new(PolyState {
                scratch: PolyScratch::new(),
                sweep: None,
                small: None,
            })
        }
        self.0.get_or_insert_with(alloc)
    }
}

impl<V> Clone for Scratch<V> {
    fn clone(&self) -> Self {
        Self::new()
    }
}

impl<V> PartialEq for Scratch<V> {
    fn eq(&self, _: &Self) -> bool {
        true
    }
}

impl<V> Eq for Scratch<V> {}

impl<V> core::fmt::Debug for Scratch<V> {
    fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {
        f.debug_struct("PolyScratch").finish_non_exhaustive()
    }
}

/// The fill ring of `run` on `grid`: tile (0, 0)'s buffered box `[-buffer, extent − 1 + buffer]`
/// grown by one unit, wound like the run's polygon — the same in every tile's local frame. It fits
/// `i32` because [`Grid::new`] keeps `extent + buffer` within it.
pub(crate) fn fill_ring<V: PolyVertex>(grid: Grid, run: &FillRun) -> [V; 5] {
    let (extent, buffer) = (grid.extent_i32(), grid.buffer_i32());
    let sw = Coord {
        x: -buffer - 1,
        y: -buffer - 1,
    };
    let ne = Coord {
        x: extent + buffer,
        y: extent + buffer,
    };
    fill_box(run.orient, sw, ne).map(V::synthetic_at)
}

/// What the next slice is; the borrowed view is built from it.
enum Event {
    /// The whole current polygon, for `tile`: it lies in the tile's inner box, or is unclipped.
    Whole {
        tile: TileId,
        origin: Coord<i32>,
    },
    /// A piece built into the piece buffer, for the tile at `origin`.
    Built {
        tile: TileId,
        origin: Coord<i32>,
    },
    Run(FillRun),
}

/// The row sweep of the current polygon.
struct Sweep {
    /// The rows not yet loaded.
    rows: RangeInclusive<i64>,
    /// Chunks not yet active, as an index into the by-first-row order.
    next_chunk: usize,
    /// Whether a row is loaded, and the sweep's place in it: the next crossing (by `right_of`), the
    /// next short-edge span, the next long-edge span, and the first tile not yet yielded or filled.
    in_row: bool,
    ty: i32,
    k: usize,
    h: usize,
    lh: usize,
    next: i64,
    /// The spans of the current tile: its short-edge spans as they are (`Some`, the usual case: no
    /// copy), or merged with the long edges crossing it into the scratch's `covering` (`None`).
    cover: Option<Range<usize>>,
    /// The last column a long edge covering the sweep reaches (`i64::MIN` when none does).
    long_last: i64,
    /// How many holes have odd parity right of the sweep.
    holes_in: u32,
    /// A covered span of the current row not yet yielded (it may still grow), as `start..end`.
    pending: Option<(i64, i64)>,
    /// A run to yield before anything else, and an edge tile to build after it (from the spans
    /// covering it).
    ready: Option<FillRun>,
    tile_ready: Option<i32>,
    /// The exterior's winding, once a fill needs it.
    orient: Option<Ordering>,
    /// The columns yielded: see [`PolygonSlices::within`].
    cols: RangeInclusive<i32>,
}

/// Lazy iterator over the slices of a polygon in every tile it reaches; see
/// [`Slicer::polygon`](crate::Slicer::polygon).
///
/// A **lending** iterator: a [`PolygonPiece`] borrows the iterator's piece buffer, so it is valid
/// until the next call to [`next`](Self::next). Loop with `while let Some(slice) = slices.next()`.
pub struct PolygonSlices<'s, V, R> {
    grid: Grid,
    /// [`ClipMode::Unclipped`]: every tile gets the whole polygon.
    whole: bool,
    rings: &'s [R],
    s: &'s mut PolyScratch<V>,
    /// How the polygon starts, until the first step; then its sweep, if it has one.
    start: Option<Start>,
    sweep: &'s mut Option<Sweep>,
    small: &'s mut Option<SmallWalk>,
    done: bool,
    /// The columns and rows yielded: see [`Self::within`].
    cols: RangeInclusive<i32>,
    rows: RangeInclusive<i32>,
}

/// How a loaded polygon starts: small, so the iterator can be built in place, its sweep only on the
/// first step.
enum Start {
    /// It lies wholly in `tile`'s inner box.
    Whole { tile: TileId, origin: Coord<i32> },
    /// It is swept over `rows`.
    Sweep { rows: RangeInclusive<i64> },
    /// It is smaller than a tile, so no tile is covered: each of the tiles `x0..=x1` by `y0..=y1`
    /// its buffered box reaches gets what [`clip_polygon_to_tile`] gives it, with no sweep.
    Small { x0: i32, x1: i32, y0: i32, y1: i32 },
}

/// The walk over a small polygon's tiles: the next tile, and the columns and last row to go.
#[derive(Clone, Copy)]
struct SmallWalk {
    x: i32,
    y: i32,
    x0: i32,
    x1: i32,
    y1: i32,
}

/// The most vertices of a polygon walked tile by tile (each tile scans all of them) rather than swept.
const SMALL_VERTICES: u32 = 128;

impl<'s, V: PolyVertex, R: AsRef<[V]>> PolygonSlices<'s, V, R> {
    /// Load the polygon `rings`, bounding and indexing it.
    #[cfg_attr(feature = "hotpath", hotpath::measure)]
    #[inline]
    pub(crate) fn new(
        grid: Grid,
        mode: ClipMode,
        rings: &'s [R],
        state: &'s mut PolyState<V>,
    ) -> Result<Self, TileError> {
        let PolyState {
            scratch: s,
            sweep,
            small,
        } = state;
        *sweep = None;
        *small = None;
        let whole = mode == ClipMode::Unclipped;
        let start = prepare(grid, rings, s, !whole)?;
        Ok(Self {
            grid,
            whole,
            rings,
            s,
            start,
            sweep,
            small,
            done: false,
            cols: i32::MIN..=i32::MAX,
            rows: i32::MIN..=i32::MAX,
        })
    }

    /// Yield only the slices in tiles `min..=max` (columns `min.x..=max.x`, rows `min.y..=max.y`).
    /// Needs to be called before [`next`](Self::next).
    #[must_use]
    pub fn within(mut self, min: TileId, max: TileId) -> Self {
        debug_assert!(self.sweep.is_none(), "within() before the first next()");
        self.cols = min.x..=max.x;
        self.rows = min.y..=max.y;
        self
    }

    /// The next slice, or `None` when the polygon is done. An error ends the iteration: only
    /// [`TileError::GeometryTooLarge`], for a piece of more than `u32::MAX` vertices, since
    /// [`Slicer::polygon`](crate::Slicer::polygon) settled every overflow up front.
    #[expect(
        clippy::should_implement_trait,
        reason = "a lending iterator: each item borrows the iterator"
    )]
    pub fn next(&mut self) -> Option<Result<PolygonSlice<'_, V, R>, TileError>> {
        if self.done {
            return None;
        }
        // A polygon inside one tile's inner box (most of them) is that tile's only slice.
        if let Some(&Start::Whole { tile, origin }) = self.start.as_ref() {
            self.start = None;
            self.done = true;
            let inside = self.cols.contains(&tile.x) && self.rows.contains(&tile.y);
            return inside.then(|| Ok(self.view(Event::Whole { tile, origin })));
        }
        match self.step() {
            Ok(Some(event)) => Some(Ok(self.view(event))),
            Ok(None) => {
                self.done = true;
                None
            }
            Err(e) => {
                self.done = true;
                Some(Err(e))
            }
        }
    }

    /// The ring [`Slicer::tile_polygon`](crate::Slicer::tile_polygon) yields for `run`'s polygon in
    /// each tile of the run: the tile's buffered box grown by one unit, tile-local and closed (its
    /// first vertex repeated), all synthetic vertices, wound like the polygon's exterior. It is the
    /// polygon's only ring there. The same as [`Slicer::fill_ring`](crate::Slicer::fill_ring), for use
    /// while the iterator borrows the slicer.
    #[must_use]
    pub fn fill_ring(&self, run: &FillRun) -> [V; 5] {
        fill_ring(self.grid, run)
    }

    fn view(&self, event: Event) -> PolygonSlice<'_, V, R> {
        let (tile, rings) = match event {
            Event::Run(run) => return PolygonSlice::Run(run),
            Event::Built { tile, origin } => (
                tile,
                PieceRings::Built {
                    verts: self.s.piece.verts(),
                    ends: self.s.piece.ring_ends(),
                    origin,
                },
            ),
            Event::Whole { tile, origin } => (
                tile,
                PieceRings::Whole {
                    rings: &self.s.rings,
                    input: self.rings,
                    copies: &self.s.copies,
                    origin,
                },
            ),
        };
        PolygonSlice::Piece(PolygonPiece { tile, rings })
    }

    /// The next tile of a small polygon's walk ([`Start::Small`]).
    #[inline(always)]
    #[expect(
        clippy::inline_always,
        reason = "the walk of most polygons that cross a tile edge; out of line it costs them ~2%"
    )]
    fn step_small(&mut self) -> Result<Option<Event>, TileError> {
        let Some(walk) = self.small.as_mut() else {
            return Ok(None);
        };
        while walk.y <= walk.y1 {
            let tile = TileId::new(walk.x, walk.y);
            if walk.x < walk.x1 {
                walk.x += 1;
            } else {
                (walk.x, walk.y) = (walk.x0, walk.y + 1);
            }
            // (`prepare` proved the box of every tile the polygon reaches fits `i32`.)
            let (min, max) = self.grid.tile_box(tile);
            // A lone ring inside the tile's buffered box (a building near a tile edge) is its
            // piece there as it is: lent, not clipped.
            if let [info] = self.s.rings[..]
                // (The box's side may pass `i32::MAX`, but fits `u32`.)
                && let span = max.x.wrapping_sub(min.x).cast_unsigned()
                && ring_within(info.pts(self.rings, &self.s.copies), min, span).is_some()
            {
                let origin = self.grid.origin_of(min);
                return Ok(Some(Event::Whole { tile, origin }));
            }
            let built = build_tile_piece(self.grid, min, max, self.rings, self.s)?;
            if let Some(origin) = built {
                return Ok(Some(Event::Built { tile, origin }));
            }
        }
        *self.small = None;
        Ok(None)
    }

    fn step(&mut self) -> Result<Option<Event>, TileError> {
        match self.start.take() {
            // (`next` yields a polygon that starts whole itself.)
            Some(Start::Whole { .. }) | None => {}
            Some(Start::Sweep { rows }) => {
                let first = (*rows.start()).max(i64::from(*self.rows.start()));
                let last = (*rows.end()).min(i64::from(*self.rows.end()));
                *self.sweep = Some(Sweep::new(first..=last, self.cols.clone()));
            }
            Some(Start::Small { x0, x1, y0, y1 }) => {
                let (x0, x1) = (x0.max(*self.cols.start()), x1.min(*self.cols.end()));
                let (y0, y1) = (y0.max(*self.rows.start()), y1.min(*self.rows.end()));
                if x0 <= x1 {
                    *self.small = Some(SmallWalk {
                        x: x0,
                        y: y0,
                        x0,
                        x1,
                        y1,
                    });
                }
            }
        }
        // (No sweep: a small polygon's walk, if any.)
        let Some(sw) = self.sweep.as_mut() else {
            return self.step_small();
        };
        let mut cx = Cx {
            grid: self.grid,
            s: &mut *self.s,
            rings: self.rings,
        };
        if self.whole {
            sw.step_whole(&mut cx)
        } else {
            sw.step::<false, V, R>(&mut cx)
        }
    }
}

/// The most vertices of a ring [`Grid::inner_tile_of_ring`] is asked about: a building, not a coastline.
const LONE_RING: usize = 16;

/// Load the polygon `rings` into `s`: how it starts, or `None` if there is nothing to slice (a
/// degenerate exterior). Every overflow the slicing could meet is settled here, from the polygon's
/// bounds.
// Once per polygon: inlined into `PolygonSlices::new`, it costs small polygons ~4%.
#[inline(never)]
fn prepare<V: PolyVertex, R: AsRef<[V]>>(
    grid: Grid,
    rings: &[R],
    s: &mut PolyScratch<V>,
    allow_small: bool,
) -> Result<Option<Start>, TileError> {
    let (extent, buffer) = (i64::from(grid.extent()), i64::from(grid.buffer()));
    s.rings.clear();
    s.copies.clear();
    // The usual polygon is one small ring with no repeated vertex (a building, most often four
    // corners) lying inside a tile's inner box: found in one pass, with none of the loop's bookkeeping.
    if let [ring] = rings
        && let ring = ring.as_ref()
        && (3..=LONE_RING).contains(&ring.len())
        && let Some((tile, origin, vertices)) = grid.inner_tile_of_ring(ring)
    {
        s.rings.push(RingInfo {
            input: 0,
            src: Src::Input,
            len: u32::try_from(vertices).unwrap_or(0),
            edge0: 0,
        });
        return Ok(Some(Start::Whole { tile, origin }));
    }
    u32::try_from(rings.len()).map_err(|_| TileError::GeometryTooLarge)?;
    let mut lo = Coord {
        x: i32::MAX,
        y: i32::MAX,
    };
    let mut hi = Coord {
        x: i32::MIN,
        y: i32::MIN,
    };
    let mut edges = 0_u32;
    for (index, ring) in (0_u32..).zip(rings) {
        let Some((info, rlo, rhi)) = load_ring(s, index, ring.as_ref(), edges)? else {
            if index == 0 {
                return Ok(None); // a degenerate exterior drops the polygon
            }
            continue;
        };
        edges = edges
            .checked_add(info.len)
            .ok_or(TileError::GeometryTooLarge)?;
        s.rings.push(info);
        (lo.x, lo.y) = (lo.x.min(rlo.x), lo.y.min(rlo.y));
        (hi.x, hi.y) = (hi.x.max(rhi.x), hi.y.max(rhi.y));
    }

    // A polygon inside one tile's inner box touches only that tile, whose box `inner_tile` proved
    // fits: it goes there whole, lent as it is, with nothing else to check.
    if let Some((tile, origin)) = grid.inner_tile(lo, hi) {
        return Ok(Some(Start::Whole { tile, origin }));
    }
    grid.check_polygon_bounds(lo, hi)?;

    // A polygon smaller than a tile, with few vertices (a building): no tile it reaches is covered,
    // so each gets its edge piece from the one-tile clip, which costs less than the sweep's setup.
    if allow_small
        && edges <= SMALL_VERTICES
        && i64::from(hi.x) - i64::from(lo.x) < extent
        && i64::from(hi.y) - i64::from(lo.y) < extent
    {
        let tile = |c: i32, d: i64| {
            // The tile index of `c + d`: it fits `i32` (`check_polygon_bounds` proved the boxes do).
            i32::try_from((i64::from(c) + d).div_euclid(extent)).unwrap_or(0)
        };
        return Ok(Some(Start::Small {
            x0: tile(lo.x, -buffer),
            x1: tile(hi.x, buffer),
            y0: tile(lo.y, -buffer),
            y1: tile(hi.y, buffer),
        }));
    }

    s.chunks.clear();
    index_chunks(s, rings, extent, buffer);
    s.order.clear();
    #[expect(
        clippy::cast_possible_truncation,
        reason = "at most one chunk per edge"
    )]
    s.order.extend(0..s.chunks.len() as u32);
    let chunks = &s.chunks;
    s.order.sort_unstable_by_key(|&c| chunks[c as usize].rows.0);
    s.active.clear();
    s.inside.clear();
    s.inside.resize(s.rings.len(), false);
    let first = (i64::from(lo.y) - buffer).div_euclid(extent);
    let last = (i64::from(hi.y) + buffer).div_euclid(extent);
    Ok(Some(Start::Sweep { rows: first..=last }))
}

/// What the sweep reads besides its own state: the grid, the scratch, and the polygon being swept.
struct Cx<'x, V, R> {
    grid: Grid,
    s: &'x mut PolyScratch<V>,
    rings: &'x [R],
}

impl Sweep {
    /// A sweep over `rows`, before its first row.
    fn new(rows: RangeInclusive<i64>, cols: RangeInclusive<i32>) -> Self {
        Self {
            rows,
            next_chunk: 0,
            in_row: false,
            ty: 0,
            k: 0,
            h: 0,
            next: i64::MIN,
            holes_in: 0,
            pending: None,
            ready: None,
            tile_ready: None,
            orient: None,
            lh: 0,
            cover: Some(0..0),
            long_last: i64::MIN,
            cols,
        }
    }

    /// [`Self::step`] unclipped.
    // Keeps this second copy of the sweep out of `PolygonSlices::next`, so the clipped one inlines as
    // before (any extra code there costs the clipped sweep 2-5%).
    #[inline(never)]
    fn step_whole<V: PolyVertex, R: AsRef<[V]>>(
        &mut self,
        cx: &mut Cx<'_, V, R>,
    ) -> Result<Option<Event>, TileError> {
        self.step::<true, V, R>(cx)
    }

    /// Advance the sweep to its next slice; `None` when the polygon is done. `WHOLE`
    /// ([`ClipMode::Unclipped`]): give every tile the whole polygon, fill runs tile by tile.
    fn step<const WHOLE: bool, V: PolyVertex, R: AsRef<[V]>>(
        &mut self,
        cx: &mut Cx<'_, V, R>,
    ) -> Result<Option<Event>, TileError> {
        loop {
            if let Some(run) = self.ready.take() {
                return Ok(Some(if WHOLE {
                    self.whole_run_tile(cx.grid, run)
                } else {
                    Event::Run(run)
                }));
            }
            if let Some(tx) = self.tile_ready.take() {
                let tile = TileId::new(tx, self.ty);
                if WHOLE {
                    if let Some(event) = self.whole_tile(cx.grid, cx.s, tile) {
                        return Ok(Some(event));
                    }
                } else if let Some(origin) = self.build_tile(cx, tile)? {
                    return Ok(Some(Event::Built { tile, origin }));
                }
                continue;
            }
            if !self.in_row {
                let Some(row) = self.rows.next() else {
                    return Ok(None);
                };
                self.load_row(cx, row)?;
                self.ty = i32::try_from(row).map_err(|_| TileError::Overflow)?;
                (self.in_row, self.k, self.h, self.lh, self.next) = (true, 0, 0, 0, i64::MIN);
                (self.cover, self.long_last) = (Some(0..0), i64::MIN);
                continue;
            }
            if self.next > i64::from(*self.cols.end()) {
                // Past the window: nothing more in this row, whose parity starts the next one even.
                self.ready = self.take_pending();
                cx.s.inside.fill(false);
                self.holes_in = 0;
                self.in_row = false;
                continue;
            }
            // Tiles `next..=bound` all see the same crossings on their winding rays.
            let s = &*cx.s;
            let bound = s
                .by_x
                .get(self.k)
                .map_or(i64::MAX, |&i| s.crossings[i as usize].right_of);
            // The next edge tile, and its spans. Rows without long edges (most) take the short path.
            let tile = if s.long.is_empty() {
                s.hits.get(self.h).map(|c| i64::from(c.tx))
            } else {
                // The sweep's next tile if a long edge still covers it, else where the next short or
                // long span starts.
                let next = self.next;
                // (`long_last` is `i64::MIN` when no long edge covers the sweep, as at a row start.)
                let covered =
                    (self.long_last != i64::MIN && self.long_last >= next).then_some(next);
                let short = s.hits.get(self.h).map(|c| i64::from(c.tx));
                let long = s.long.get(self.lh).map(|c| i64::from(c.first).max(next));
                [covered, short, long].into_iter().flatten().min()
            };
            if let Some(tx) = tile
                && tx <= bound
            {
                let tx = i32::try_from(tx).map_err(|_| TileError::Overflow)?;
                let s = &mut *cx.s;
                // The tile's short-edge spans: one contiguous run.
                let start = self.h;
                self.h += s.hits[start..].partition_point(|c| c.tx == tx);
                self.cover = Some(start..self.h);
                if !s.long.is_empty() {
                    self.cover_long(s, tx, start);
                }
                self.fill(cx, self.next, i64::from(tx))?;
                // The edge tile ends any covered span before it.
                self.ready = self.take_pending();
                self.tile_ready = self.cols.contains(&tx).then_some(tx);
                self.next = i64::from(tx) + 1;
                continue;
            }
            if self.k == s.by_x.len() {
                // Past every crossing each ring's parity is even again: nothing covers.
                debug_assert!(cx.s.inside.iter().all(|&i| !i), "parity ends even");
                self.ready = self.take_pending();
                self.in_row = false;
                continue;
            }
            // `bound` is a crossing's `right_of` (|·| ≤ 2^32), so `+ 1` cannot overflow.
            self.fill(cx, self.next, bound + 1)?;
            self.next = bound + 1;
            // Tiles past `bound` have these crossings on their left.
            let s = &mut *cx.s;
            while let Some(&i) = s.by_x.get(self.k)
                && s.crossings[i as usize].right_of == bound
            {
                let crossing = s.crossings[i as usize];
                let r = crossing.ring as usize;
                s.inside[r] = !s.inside[r];
                if r != 0 {
                    if s.inside[r] {
                        self.holes_in += 1;
                    } else {
                        // Parity alternates from an empty row start, so a hole leaves only after
                        // entering.
                        self.holes_in = self.holes_in.saturating_sub(1);
                    }
                }
                fenwick_add(&mut s.fenwick, i as usize, -crossing.sign());
                self.k += 1;
            }
            // Coverage that ends here ends the covered span: it cannot grow past this crossing.
            if !(s.inside[0] && self.holes_in == 0) {
                self.ready = self.take_pending();
            }
        }
    }

    /// Unclipped, each covered tile gets the whole polygon: the first tile of `run`, keeping the rest
    /// of it for the next steps.
    fn whole_run_tile(&mut self, grid: Grid, mut run: FillRun) -> Event {
        let tile = TileId::new(run.x.start, run.y);
        run.x.start += 1;
        if !run.x.is_empty() {
            self.ready = Some(run);
        }
        Event::Whole {
            tile,
            origin: grid.tile_origin(tile),
        }
    }

    /// Add the long edges covering tile `tx` to its spans (the short ones are `s.hits[start..h]`):
    /// drop those ending before it, add those starting by it, and merge both in edge order.
    fn cover_long<V>(&mut self, s: &mut PolyScratch<V>, tx: i32, start: usize) {
        if self.long_last < i64::from(tx) {
            s.long_cover.clear();
        } else {
            s.long_cover.retain(|c| c.last >= tx);
        }
        let kept = s.long_cover.len();
        while let Some(&span) = s.long.get(self.lh)
            && span.first <= tx
        {
            s.long_cover.push(span);
            self.lh += 1;
        }
        if s.long_cover.is_empty() {
            self.long_last = i64::MIN;
            return;
        }
        merge_by_edge(&mut s.long_cover, kept, &mut s.spans_tmp, |c| c.edge);
        self.long_last = s
            .long_cover
            .iter()
            .map(|c| i64::from(c.last))
            .max()
            .unwrap_or(i64::MIN);
        s.covering.clear();
        s.covering.extend(s.long_cover.iter().map(|c| Hit {
            tx,
            edge: c.edge,
            ring: c.ring,
        }));
        let kept = s.covering.len();
        s.covering.extend_from_slice(&s.hits[start..self.h]);
        merge_by_edge(&mut s.covering, kept, &mut s.hits_tmp, |c| c.edge);
        self.cover = None;
    }

    /// The covered span not yet yielded, as a run (ending it).
    fn take_pending(&mut self) -> Option<FillRun> {
        let (start, end) = self.pending.take()?;
        #[expect(
            clippy::cast_possible_truncation,
            reason = "both ends were checked to fit i32 when the span was recorded"
        )]
        Some(FillRun {
            y: self.ty,
            x: start as i32..end as i32,
            orient: self.orient.unwrap_or(Ordering::Equal),
        })
    }

    /// Record tiles `start..end` of the current row as covered, if the polygon covers them,
    /// extending the span not yet yielded when they abut.
    fn fill<V: PolyVertex, R: AsRef<[V]>>(
        &mut self,
        cx: &mut Cx<'_, V, R>,
        start: i64,
        end: i64,
    ) -> Result<(), TileError> {
        let covered = cx.s.inside.first().copied().unwrap_or(false) && self.holes_in == 0;
        if start >= end || !covered {
            return Ok(());
        }
        i32::try_from(start).map_err(|_| TileError::Overflow)?;
        i32::try_from(end).map_err(|_| TileError::Overflow)?;
        let start = start.max(i64::from(*self.cols.start()));
        let end = end.min(i64::from(*self.cols.end()) + 1);
        if start >= end {
            return Ok(());
        }
        if self.orient.is_none() {
            let ext = cx.s.rings[0].pts(cx.rings, &cx.s.copies);
            self.orient = Some(ring_orientation(ext));
        }
        match &mut self.pending {
            Some((_, e)) if *e == start => *e = end,
            pending => {
                debug_assert!(
                    pending.is_none(),
                    "an edge tile or row end yields a span first"
                );
                *pending = Some((start, end));
            }
        }
        Ok(())
    }

    /// Gather row `ty`'s hits and crossings from the chunks reaching it.
    #[cfg_attr(feature = "hotpath", hotpath::measure)]
    // (The `hotpath` wrapper hides the body from this lint.)
    #[cfg_attr(
        not(feature = "hotpath"),
        expect(
            clippy::many_single_char_names,
            reason = "edge endpoints and the scratch"
        )
    )]
    fn load_row<V: PolyVertex, R: AsRef<[V]>>(
        &mut self,
        cx: &mut Cx<'_, V, R>,
        ty: i64,
    ) -> Result<(), TileError> {
        let (extent, buffer) = (i64::from(cx.grid.extent()), i64::from(cx.grid.buffer()));
        let s = &mut *cx.s;
        let rings = cx.rings;
        while let Some(&c) = s.order.get(self.next_chunk)
            && s.chunks[c as usize].rows.0 <= ty
        {
            let at = s.active.partition_point(|&a| a < c);
            s.active.insert(at, c);
            self.next_chunk += 1;
        }
        let chunks = &s.chunks;
        s.active.retain(|&c| chunks[c as usize].rows.1 >= ty);

        s.hits.clear();
        s.long.clear();
        s.long_cover.clear();
        s.covering.clear();
        s.crossings.clear();
        let (slab_lo, slab_hi) = (ty * extent - buffer, ty * extent + extent - 1 + buffer);
        let cy = row_center(ty, extent);
        for &c in &s.active {
            let chunk = s.chunks[c as usize];
            let info = s.rings[chunk.ring as usize];
            let pts = info.pts(rings, &s.copies);
            let m = pts.len();
            for i in chunk.first as usize..(chunk.first + chunk.count) as usize {
                let (a, b) = (pts[i].position(), pts[(i + 1) % m].position());
                let (lo, hi) = if a.y <= b.y { (a, b) } else { (b, a) };
                if i64::from(hi.y) < slab_lo || i64::from(lo.y) > slab_hi {
                    continue;
                }
                #[expect(clippy::cast_possible_truncation, reason = "edge ids fit u32")]
                let edge = info.edge0 + i as u32;
                let wide = |p: Coord<i32>| Coord {
                    x: i64::from(p.x),
                    y: i64::from(p.y),
                };
                let (first, last) = Seg::new(wide(lo), wide(hi)).row_span(ty, extent, buffer);
                let fits = |v: i64| i32::try_from(v).map_err(|_| TileError::Overflow);
                if last - first + 1 >= LONG_EDGE {
                    s.long.push(Span {
                        first: fits(first)?,
                        last: fits(last)?,
                        edge,
                        ring: chunk.ring,
                    });
                } else {
                    for tx in first..=last {
                        s.hits.push(Hit {
                            tx: fits(tx)?,
                            edge,
                            ring: chunk.ring,
                        });
                    }
                }
                // Crossings of the center line, half-open in `y`.
                if i64::from(lo.y) <= cy && cy < i64::from(hi.y) {
                    let x =
                        crossing_x_ceil(a, b, i32::try_from(cy).map_err(|_| TileError::Overflow)?);
                    s.crossings.push(Crossing {
                        edge,
                        ring: chunk.ring,
                        up: b.y > a.y,
                        // Right of tile `tx`'s box iff x > tx·extent + extent − 1 + buffer.
                        right_of: (x - extent - buffer).div_euclid(extent),
                    });
                }
            }
        }
        sort_hits(&mut s.hits, &mut s.hits_tmp, &mut s.counts);
        s.long.sort_unstable_by_key(|c| (c.first, c.edge));
        debug_assert!(s.crossings.is_sorted_by_key(|c| c.edge));
        s.by_x.clear();
        #[expect(clippy::cast_possible_truncation, reason = "one crossing per edge")]
        s.by_x.extend(0..s.crossings.len() as u32);
        let crossings = &s.crossings;
        s.by_x
            .sort_unstable_by_key(|&i| crossings[i as usize].right_of);
        fenwick_build(&mut s.fenwick, s.crossings.iter().map(|c| c.sign()));
        Ok(())
    }

    /// The spans covering the current edge tile, if the polygon leaves anything there. Only the
    /// touching rings are visited: the untouched holes are settled at once by how many of them
    /// contain the tile. Takes the scratch's row fields, `hits`, `covering` and `inside`, rather than
    /// the scratch, so the caller can still write the piece buffer.
    #[expect(
        clippy::inline_always,
        reason = "run per edge tile; out of line it costs the polygon benchmarks 2%"
    )]
    #[inline(always)]
    fn tile_hits<'x>(
        &self,
        hits: &'x [Hit],
        covering: &'x [Hit],
        inside: &[bool],
    ) -> Option<&'x [Hit]> {
        let hits = match &self.cover {
            Some(range) => &hits[range.clone()],
            None => covering,
        };
        // Hits come in edge order, so the exterior's (if any) come first, then each hole's.
        let exterior_touched = hits.first().is_some_and(|x| x.ring == 0);
        if !exterior_touched && !inside[0] {
            return None; // the exterior misses the tile
        }
        // `holes_in` counts every hole whose ray parity is odd, but a touched hole's parity says
        // nothing about the tile: what remains are untouched holes containing it.
        let holes_in = self.holes_in as usize;
        if holes_in > 0
            && holes_in
                > ring_groups(hits)
                    .filter(|g| g[0].ring != 0 && inside[g[0].ring as usize])
                    .count()
        {
            return None;
        }
        Some(hits)
    }

    /// Unclipped, edge tile `tile` gets the whole polygon if it would get a piece.
    fn whole_tile<V>(&self, grid: Grid, s: &PolyScratch<V>, tile: TileId) -> Option<Event> {
        self.tile_hits(&s.hits, &s.covering, &s.inside)?;
        let origin = grid.tile_origin(tile);
        Some(Event::Whole { tile, origin })
    }

    /// Build the polygon's piece in edge tile `tile` from [the spans covering it](Self::tile_hits)
    /// into the piece buffer, returning the tile origin; `None` if it leaves nothing there. Every
    /// vertex fits the tile's local frame: the polygon's bounds were checked when it was loaded.
    #[cfg_attr(feature = "hotpath", hotpath::measure)]
    fn build_tile<V: PolyVertex, R: AsRef<[V]>>(
        &mut self,
        cx: &mut Cx<'_, V, R>,
        tile: TileId,
    ) -> Result<Option<Coord<i32>>, TileError> {
        let (min, max) = cx.grid.tile_buffered_bounds(tile)?;
        let origin = cx.grid.origin_of(min);
        let s = &mut *cx.s;
        let rings = cx.rings;
        let Some(hits) = self.tile_hits(&s.hits, &s.covering, &s.inside) else {
            return Ok(None);
        };
        let exterior_touched = hits.first().is_some_and(|x| x.ring == 0);
        // The plain clip: what [`Self::piece`] builds for [`Attempt::PLAIN`], without its bookkeeping.
        s.piece.clear();
        if !exterior_touched {
            // A tile the exterior covers gets a fill box, wound like the exterior.
            let orient = *self
                .orient
                .get_or_insert_with(|| ring_orientation(s.rings[0].pts(rings, &s.copies)));
            s.piece.push_fill_box(orient, min, max)?;
        }
        for ring_hits in ring_groups(hits) {
            let info = s.rings[ring_hits[0].ring as usize];
            s.arcs.clear();
            for x in ring_hits {
                keep_edge(&mut s.arcs, (x.edge - info.edge0) as usize);
            }
            let (cs, fenwick) = (&s.crossings, &s.fenwick);
            let winding =
                |first: usize, count: usize| excursion_winding(cs, fenwick, info, first, count);
            let pts = info.pts(rings, &s.copies);
            let (arcs, piece) = (&mut s.arcs, &mut s.piece);
            s.clip.close_ring(pts, arcs, (min, max), winding, piece)?;
        }
        if !self.make_valid(s, rings, (min, max), origin, exterior_touched)? {
            return Ok(None);
        }
        Ok(Some(origin))
    }

    /// [`Self::build_tile`]'s plain clip, just built, made planar outside the box if it is not:
    /// `false` if the polygon then leaves nothing in the tile. Out of line, to keep the plain clip's
    /// small.
    #[inline(never)]
    fn make_valid<V: PolyVertex, R: AsRef<[V]>>(
        &self,
        s: &mut PolyScratch<V>,
        rings: &[R],
        (min, max): (Coord<i32>, Coord<i32>),
        origin: Coord<i32>,
        exterior_touched: bool,
    ) -> Result<bool, TileError> {
        let Some(budget) = needs_repair(s, (min, max)) else {
            return Ok(true);
        };
        // (The plain clip of a covered tile just found the exterior's winding.)
        let fill = (!exterior_touched).then(|| self.orient.unwrap_or(Ordering::Equal));
        let whole_fits = fits_whole(rings);
        repair(
            s,
            (min, max),
            origin,
            rings.len(),
            whole_fits,
            budget,
            |s, a, b| self.piece(s, rings, (min, max), fill, a, b),
        )
    }

    /// Build the piece of the edge tile with box `(min, max)`, which [`Self::tile_hits`] found gets one,
    /// by `attempt` into the piece buffer, from the spans covering it and the winding index, so a repair costs the piece, not
    /// the ring. `fill` is the exterior's winding if it covers the tile, which then gets a fill box.
    #[inline]
    fn piece<V: PolyVertex, R: AsRef<[V]>>(
        &self,
        s: &mut PolyScratch<V>,
        rings: &[R],
        (min, max): (Coord<i32>, Coord<i32>),
        fill: Option<Ordering>,
        attempt: Attempt,
        budget: &mut usize,
    ) -> Result<Built, TileError> {
        // (The caller checked the tile gets a piece: these are the spans covering it.)
        let hits = match &self.cover {
            Some(range) => &s.hits[range.clone()],
            None => &s.covering[..],
        };
        let covered = fill.is_some();
        // Growing the fill box does nothing unless the exterior covers the tile and there are holes.
        if attempt == Attempt::ENCLOSE && (!covered || rings.len() == 1) {
            return Ok(Built::Failed);
        }
        s.piece.clear();
        s.clip.start_attempt();
        if attempt.frame {
            let probed = adapt_frame(
                |depths, probe, bases| {
                    bases.clear();
                    bases.resize(rings.len(), 0);
                    for ring_hits in ring_groups(hits) {
                        let info = s.rings[ring_hits[0].ring as usize];
                        s.arcs.clear();
                        for x in ring_hits {
                            keep_edge(&mut s.arcs, (x.edge - info.edge0) as usize);
                        }
                        let plan = RingPlan {
                            depths: Some(depths),
                            bridge_base: probe.bridges,
                            ..RingPlan::PLAIN
                        };
                        bases[info.input as usize] = probe.bridges;
                        let pts = info.pts(rings, &s.copies);
                        arcs_points(pts, &mut s.arcs, plan, (min, max), probe);
                    }
                    Ok(())
                },
                &mut s.depths,
                &mut s.probe,
                &mut s.frame_hull,
                &mut s.ring_bases,
                (min, max),
                budget,
            )?;
            if !probed {
                return Ok(Built::Failed);
            }
        }
        if let Some(orient) = fill {
            s.piece.push_fill_box(orient, min, max)?;
        }
        for ring_hits in ring_groups(hits) {
            let info = s.rings[ring_hits[0].ring as usize];
            s.arcs.clear();
            for x in ring_hits {
                keep_edge(&mut s.arcs, (x.edge - info.edge0) as usize);
            }
            let (cs, fenwick) = (&s.crossings, &s.fenwick);
            let winding =
                |first: usize, count: usize| excursion_winding(cs, fenwick, info, first, count);
            let pts = info.pts(rings, &s.copies);
            let plan = if attempt == Attempt::PLAIN {
                RingPlan::PLAIN
            } else {
                let frame = attempt.frame.then(|| {
                    let ring = info.input as usize;
                    (
                        s.frame_hull.as_slice(),
                        s.depths.as_slice(),
                        s.ring_bases[ring],
                    )
                });
                attempt.ring_plan(info.input as usize, rings.len(), frame)
            };
            let (arcs, piece) = (&mut s.arcs, &mut s.piece);
            let closed = s
                .clip
                .close_ring_planned(pts, arcs, (min, max), winding, &plan, piece)?;
            if closed == Closed::FrameMissed {
                return Ok(Built::Failed);
            }
        }
        if attempt.enclose && covered && s.piece.ring_ends().len() > 1 {
            s.piece.grow_fill_box(min, max)?;
        }
        Ok(Built::Piece)
    }
}

/// A kept ring and its bounding box `(lo, hi)`.
type LoadedRing = (RingInfo, Coord<i32>, Coord<i32>);

/// Load one ring of the polygon in `s` (its `index` among the input rings, its first edge `edge0`):
/// borrowed when it is distinct as given — no consecutive duplicates once its repeated closing
/// vertices are dropped — else its distinct vertices copied, with its bounding box. `None` for a
/// degenerate ring (fewer than 3 distinct vertices).
#[expect(
    clippy::inline_always,
    reason = "run per ring of every polygon; out of line it costs the small-polygon benchmarks 3-5%"
)]
#[inline(always)]
fn load_ring<V: Vertex>(
    s: &mut PolyScratch<V>,
    index: u32,
    ring: &[V],
    edge0: u32,
) -> Result<Option<LoadedRing>, TileError> {
    let Some(first) = ring.first().map(Vertex::position) else {
        return Ok(None);
    };
    let mut n = ring.len();
    while n >= 2 && ring[n - 1].position() == first {
        n -= 1;
    }
    // One pass: the bounding box, and whether the ring is distinct as given.
    let (mut lo, mut hi, mut prev, mut distinct) = (first, first, first, true);
    for v in &ring[1..n] {
        let p = v.position();
        (lo.x, lo.y, hi.x, hi.y) = (lo.x.min(p.x), lo.y.min(p.y), hi.x.max(p.x), hi.y.max(p.y));
        distinct &= p != prev;
        prev = p;
    }
    let (src, len) = if distinct {
        (Src::Input, n)
    } else {
        let start = s.copies.len();
        (Src::Copy(start), push_distinct(ring, &mut s.copies))
    };
    if len < 3 {
        if let Src::Copy(start) = src {
            s.copies.truncate(start);
        }
        return Ok(None);
    }
    let info = RingInfo {
        input: index,
        src,
        len: u32::try_from(len).map_err(|_| TileError::GeometryTooLarge)?,
        edge0,
    };
    Ok(Some((info, lo, hi)))
}

/// The row index of the polygon in `s.rings`: every run of [`CHUNK`] edges, with the rows its buffered
/// `y`-range reaches.
fn index_chunks<V: Vertex, R: AsRef<[V]>>(
    s: &mut PolyScratch<V>,
    rings: &[R],
    extent: i64,
    buffer: i64,
) {
    for (ring_id, info) in (0_u32..).zip(&s.rings) {
        let pts = info.pts(rings, &s.copies);
        for first in (0..pts.len()).step_by(CHUNK) {
            let count = CHUNK.min(pts.len() - first);
            let (mut y0, mut y1) = (i32::MAX, i32::MIN);
            for v in pts[first..first + count]
                .iter()
                .chain(iter::once(&pts[(first + count) % pts.len()]))
            {
                let y = v.position().y;
                (y0, y1) = (y0.min(y), y1.max(y));
            }
            #[expect(
                clippy::cast_possible_truncation,
                reason = "within the ring's u32 length"
            )]
            s.chunks.push(Chunk {
                ring: ring_id,
                first: first as u32,
                count: count as u32,
                rows: (
                    (i64::from(y0) - buffer).div_euclid(extent),
                    (i64::from(y1) + buffer).div_euclid(extent),
                ),
            });
        }
    }
}

/// How the rings of one repair attempt place their detours: all in the same lane, or each ring (and
/// each of its bridges) one lane further out — the exterior innermost, or outermost.
#[derive(Debug, Clone, Copy, PartialEq, Eq)]
enum Lanes {
    Shared,
    HolesOut,
    ExteriorOut,
}

/// A frame's data for one ring: the hull to go around, how deep each bridge's tips go, and the ring's
/// first bridge.
type Frame<'a> = (&'a [Coord<i32>], &'a [(u32, u32)], usize);

/// One attempt at building a piece: how deep each excursion keeps its original vertices, where the
/// detours run, and whether a covered tile's fill box grows to enclose the holes.
#[derive(Debug, Clone, Copy, PartialEq, Eq)]
struct Attempt {
    deepen: usize,
    lanes: Lanes,
    enclose: bool,
    /// Close excursions on a frame around the hull of everything kept, each tip keeping as many
    /// original vertices as it takes to be a vertex of that hull (see [`RingPlan::frame`]); `deepen`
    /// is not used.
    frame: bool,
}

impl Attempt {
    /// The plain clip, whose pieces are what the slicer yields whenever they are planar.
    const PLAIN: Self = Self {
        deepen: 0,
        lanes: Lanes::Shared,
        enclose: false,
        frame: false,
    };

    /// Only the fill box of a covered tile grows around the holes: the plain clip otherwise.
    const ENCLOSE: Self = Self {
        enclose: true,
        ..Self::PLAIN
    };

    fn ring_plan(self, ring: usize, rings: usize, frame: Option<Frame<'_>>) -> RingPlan<'_> {
        let (lane_base, lane_step) = match self.lanes {
            Lanes::Shared => (0, 0),
            Lanes::HolesOut => (i32::try_from(ring).unwrap_or(i32::MAX), 1),
            Lanes::ExteriorOut => (i32::try_from(rings - 1 - ring).unwrap_or(i32::MAX), 1),
        };
        RingPlan {
            deepen: self.deepen,
            lane_base,
            lane_step,
            frame: frame.map(|f| f.0),
            depths: frame.map(|f| f.1),
            bridge_base: frame.map_or(0, |f| f.2),
        }
    }
}

/// How many times a frame probe may deepen the tips that are not on the hull before giving up. A tip
/// goes one vertex deeper the first rounds, then more each round, so this reaches depths over 40.
const FRAME_ROUNDS: usize = 16;

/// What a probe round costs, per point probed, in segment pairs of the tile's budget (the hull sorts).
const ROUND_COST: usize = 8;

/// Find a frame for the piece: how many vertices each bridge's tips keep (into `depths`) so that every
/// tip is a vertex of the convex hull (into `hull`) of what the piece keeps outside the box, and the
/// corners of `B⁺`. `probe_all(depths, probe)` probes every ring of the piece at those depths, filling
/// `probe` and the first bridge of each input ring (`bases`). `false` if the tips do not get there.
fn adapt_frame(
    mut probe_all: impl FnMut(&[(u32, u32)], &mut Probe, &mut Vec<usize>) -> Result<(), TileError>,
    depths: &mut Vec<(u32, u32)>,
    probe: &mut Probe,
    hull: &mut Vec<Coord<i32>>,
    bases: &mut Vec<usize>,
    (min, max): (Coord<i32>, Coord<i32>),
    budget: &mut usize,
) -> Result<bool, TileError> {
    let (lo, hi) = lane_bounds(min, max, 1)?;
    depths.clear();
    for round in 0..FRAME_ROUNDS {
        probe.clear();
        probe_all(depths, probe, bases)?;
        let cost = probe.pts.len().saturating_mul(ROUND_COST);
        if cost > *budget {
            *budget = 0;
            return Ok(false);
        }
        *budget -= cost;
        depths.resize(probe.bridges, (0, 0));
        probe.pts.extend([
            lo,
            Coord { x: hi.x, y: lo.y },
            hi,
            Coord { x: lo.x, y: hi.y },
        ]);
        convex_hull(&mut probe.pts, hull);
        let mut deepened = false;
        for &(tip, bridge, side) in &probe.tips {
            if !hull.contains(&tip) {
                let d = &mut depths[bridge as usize];
                let step = 1 + u32::try_from(round / 4).unwrap_or(0);
                if side == 0 {
                    d.0 += step;
                } else {
                    d.1 += step;
                }
                deepened = true;
            }
        }
        if !deepened {
            return Ok(true);
        }
    }
    Ok(false)
}

/// The most vertices a repair may fall back to keeping whole, as its last resort, counted over all
/// the polygon's rings (as given): a bigger polygon keeps the plain clip (invalid outside the box, but
/// bounded) rather than costing a tile its whole ring.
const WHOLE_LIMIT: usize = 2048;

/// Whether the polygon `rings` is small enough for its excursions to be kept whole ([`WHOLE_LIMIT`]).
fn fits_whole<V, R: AsRef<[V]>>(rings: &[R]) -> bool {
    rings.iter().map(|r| r.as_ref().len()).sum::<usize>() <= WHOLE_LIMIT
}

/// How deep the repairs without a frame keep each excursion's original vertices, shallowest first.
const DEPTHS: [usize; 5] = [1, 2, 4, 8, 16];

/// The repairs tried after the plain clip, cheapest first: the fill box grown around the holes; the
/// frame (whose lanes nest the rings); the plain way with more vertices kept; and last every excursion
/// whole (`deepen == usize::MAX`), which is valid because the input is.
fn repair_attempts(rings: usize) -> impl Iterator<Item = Attempt> {
    let lanes = move |frame: bool, deepen: usize| {
        [Lanes::Shared, Lanes::HolesOut, Lanes::ExteriorOut]
            .into_iter()
            // One ring has no holes to put outside it, but its excursions still take lanes of their own.
            .filter(move |&l| rings > 1 || l != Lanes::ExteriorOut)
            .map(move |lanes| Attempt {
                deepen,
                lanes,
                enclose: true,
                frame,
            })
    };
    iter::once(Attempt::ENCLOSE)
        .chain(lanes(true, 0))
        .chain(DEPTHS.into_iter().flat_map(move |d| lanes(false, d)))
        .chain(iter::once(Attempt {
            deepen: usize::MAX,
            lanes: Lanes::Shared,
            enclose: true,
            frame: false,
        }))
}

/// Build `rings`' piece in the box `[min, max]` for one `attempt` into the scratch's piece buffer,
/// in the input's frame: what [`Sweep::piece`] builds for the same attempt, from the rings.
#[expect(
    clippy::too_many_lines,
    reason = "one cohesive build of an attempt's piece; splitting it would only scatter shared state"
)]
fn build_piece<V: PolyVertex, R: AsRef<[V]>>(
    rings: &[R],
    min: Coord<i32>,
    max: Coord<i32>,
    attempt: Attempt,
    s: &mut PolyScratch<V>,
    budget: &mut usize,
) -> Result<Built, TileError> {
    s.piece.clear();
    s.clip.start_attempt();
    let Some((exterior, holes)) = rings.split_first() else {
        return Ok(Built::Empty);
    };
    // A frame goes around the hull of everything the piece keeps outside the box: find how deep each
    // tip must go to be on it.
    if attempt.frame {
        // Scan every ring once: the rounds of the probe only change how deep the tips go.
        s.probe_arcs.clear();
        s.probe_rings.clear();
        for ring in rings {
            s.probe_rings
                .push(scan_arcs(ring.as_ref(), min, max, &mut s.probe_arcs));
        }
        let probed = adapt_frame(
            |depths, probe, bases| {
                bases.clear();
                for (ring, scan) in rings.iter().zip(&s.probe_rings) {
                    bases.push(probe.bridges);
                    let plan = RingPlan {
                        depths: Some(depths),
                        bridge_base: probe.bridges,
                        ..RingPlan::PLAIN
                    };
                    match scan {
                        Some((m, arcs)) => {
                            if !arcs.is_empty() {
                                let ring = &ring.as_ref()[..*m];
                                arcs_points(
                                    ring,
                                    &mut s.probe_arcs[arcs.clone()],
                                    plan,
                                    (min, max),
                                    probe,
                                );
                            }
                        }
                        // A ring with repeated vertices is scanned from a copy, each round.
                        None => {
                            clip_ring_planned(
                                ring.as_ref(),
                                min,
                                max,
                                &mut s.clip,
                                &plan,
                                Some(&mut *probe),
                                &mut s.piece,
                            )?;
                        }
                    }
                }
                Ok(())
            },
            &mut s.depths,
            &mut s.probe,
            &mut s.frame_hull,
            &mut s.ring_bases,
            (min, max),
            budget,
        )?;
        if !probed {
            return Ok(Built::Failed);
        }
    }
    let frame = |ring: usize| {
        attempt.frame.then(|| {
            (
                s.frame_hull.as_slice(),
                s.depths.as_slice(),
                s.ring_bases[ring],
            )
        })
    };
    let plan = attempt.ring_plan(0, rings.len(), frame(0));
    // If the exterior misses the tile entirely, the whole polygon is absent here.
    let Some(exterior_clip) = clip_ring_planned(
        exterior.as_ref(),
        min,
        max,
        &mut s.clip,
        &plan,
        None,
        &mut s.piece,
    )?
    else {
        return Ok(Built::Failed);
    };
    if exterior_clip == RingClip::Outside {
        return Ok(Built::Empty);
    }
    // Growing the fill box does nothing unless the exterior covers the tile and there are holes.
    if attempt == Attempt::ENCLOSE && (exterior_clip != RingClip::Covers || rings.len() == 1) {
        return Ok(Built::Failed);
    }
    for (i, hole) in (1..).zip(holes) {
        let plan = attempt.ring_plan(i, rings.len(), frame(i));
        match clip_ring_planned(
            hole.as_ref(),
            min,
            max,
            &mut s.clip,
            &plan,
            None,
            &mut s.piece,
        )? {
            None => return Ok(Built::Failed),
            // A hole covering the whole tile leaves nothing to draw here.
            Some(RingClip::Covers) => return Ok(Built::Empty),
            Some(RingClip::Clipped | RingClip::Outside) => {}
        }
    }
    if attempt.enclose && exterior_clip == RingClip::Covers && s.piece.ring_ends().len() > 1 {
        s.piece.grow_fill_box(min, max)?;
    }
    Ok(Built::Piece)
}

/// What building one attempt's piece came to.
#[derive(Debug, Clone, Copy, PartialEq, Eq)]
enum Built {
    /// The polygon leaves nothing in the tile.
    Empty,
    /// The attempt does not apply here, or can't be built: a fill box grown with no covered tile or
    /// no hole, a frame whose tips never reach its hull (within the budget) or can't close an
    /// excursion, or a lane past the `i32` limits.
    Failed,
    /// The piece is in the scratch's piece buffer.
    Piece,
}

/// For the plain clip just built into `s` for the box `[min, max]`: `None` if it is planar outside
/// the box, or nothing says it is not (nothing synthetic can cross anything, and a check that runs
/// out of budget passes it); else the budget the check left for repairs.
#[inline]
fn needs_repair<V: PolyVertex>(
    s: &mut PolyScratch<V>,
    (min, max): (Coord<i32>, Coord<i32>),
) -> Option<usize> {
    if s.piece.synthetic().is_empty() {
        return None;
    }
    let mut budget = PAIR_BUDGET;
    let verdict = is_planar(&s.piece, false, min, max, &mut s.segs, &mut budget);
    (verdict == Verdict::Crossing).then_some(budget)
}

/// Rebuild the piece in `[min, max]`, whose plain clip is not planar outside the box, by the cheapest
/// repair that is (see [`repair_attempts`]), into the scratch's piece buffer; if none is found within
/// `budget`, or the last resort would keep more than [`WHOLE_LIMIT`] vertices (`!whole_fits`), the
/// plain clip again. `attempt_piece(s, attempt, budget)` builds one attempt's piece: [`Sweep::piece`] for the
/// all-tiles slicer, [`build_piece`] for one tile, which build the same pieces, so this one search
/// gives a tile the same piece either way. `false` if the polygon leaves nothing in the tile (as its
/// plain clip says).
#[cold]
#[inline(never)]
fn repair<V: PolyVertex>(
    s: &mut PolyScratch<V>,
    (min, max): (Coord<i32>, Coord<i32>),
    origin: Coord<i32>,
    rings: usize,
    whole_fits: bool,
    mut budget: usize,
    mut attempt_piece: impl FnMut(&mut PolyScratch<V>, Attempt, &mut usize) -> Result<Built, TileError>,
) -> Result<bool, TileError> {
    for attempt in repair_attempts(rings) {
        let whole = attempt.deepen == usize::MAX;
        if whole && !whole_fits {
            break;
        }
        let built = match attempt_piece(s, attempt, &mut budget) {
            // The plain clip fits `i32`: only a repair's growth past it can overflow.
            Err(TileError::Overflow) => Built::Failed,
            built => built?,
        };
        match built {
            Built::Empty => return Ok(false),
            Built::Failed => {}
            Built::Piece => {
                // Every excursion whole is the input's own rings: as valid as it is.
                let verdict = if whole {
                    Verdict::Planar
                } else {
                    is_planar(&s.piece, attempt.frame, min, max, &mut s.segs, &mut budget)
                };
                match verdict {
                    // (A piece grown past the tile's local frame is only a failed attempt.)
                    Verdict::Planar if check_local(s.piece.verts(), origin).is_ok() => {
                        return Ok(true);
                    }
                    Verdict::Planar | Verdict::Crossing => {}
                    Verdict::OverBudget => break,
                }
            }
        }
        if budget == 0 {
            break;
        }
    }
    // Nothing better within the budget: the plain clip, as it was.
    let built = attempt_piece(s, Attempt::PLAIN, &mut budget)?;
    debug_assert_ne!(built, Built::Failed, "the plain clip was built before");
    Ok(built == Built::Piece)
}

/// Clip a polygon's `rings` to one `tile` into the scratch's piece buffer, and lend the piece; `None`
/// if the polygon leaves nothing in the tile. See [`Slicer::tile_polygon`](crate::Slicer::tile_polygon).
#[cfg_attr(feature = "hotpath", hotpath::measure)]
pub(crate) fn clip_polygon_to_tile<'s, V: PolyVertex, R: AsRef<[V]>>(
    grid: Grid,
    tile: TileId,
    rings: &'s [R],
    s: &'s mut PolyScratch<V>,
) -> Result<Option<PolygonPiece<'s, V, R>>, TileError> {
    let (min, max) = grid.tile_buffered_bounds(tile)?;
    // A lone small ring inside the tile's buffered box (a building, most often) is its piece there as
    // it is: lent, as `polygon` lends it, not clipped.
    if let [ring] = rings
        && let ring = ring.as_ref()
        && ring.len() <= LONE_RING
        && let Some(n) = ring_within(ring, min, max.x.wrapping_sub(min.x).cast_unsigned())
    {
        s.rings.clear();
        s.rings.push(RingInfo {
            input: 0,
            src: Src::Input,
            len: u32::try_from(n).unwrap_or(0),
            edge0: 0,
        });
        return Ok(Some(PolygonPiece {
            tile,
            rings: PieceRings::Whole {
                rings: &s.rings,
                input: rings,
                copies: &s.copies,
                origin: grid.origin_of(min),
            },
        }));
    }
    let Some(origin) = build_tile_piece(grid, min, max, rings, s)? else {
        return Ok(None);
    };
    Ok(Some(PolygonPiece {
        tile,
        rings: PieceRings::Built {
            verts: s.piece.verts(),
            ends: s.piece.ring_ends(),
            origin,
        },
    }))
}

/// Clip a polygon's `rings` to the tile with buffered box `(min, max)` into the scratch's piece buffer
/// (tile-global frame), and return the tile's origin; `None` if the polygon leaves nothing in it.
/// A piece not planar outside the box is repaired.
#[inline(always)]
#[expect(
    clippy::inline_always,
    reason = "out of line, the one-tile call costs the polygon_one benchmark about 4%"
)]
fn build_tile_piece<V: PolyVertex, R: AsRef<[V]>>(
    grid: Grid,
    min: Coord<i32>,
    max: Coord<i32>,
    rings: &[R],
    s: &mut PolyScratch<V>,
) -> Result<Option<Coord<i32>>, TileError> {
    let origin = grid.origin_of(min);
    s.piece.clear();
    let Some((exterior, holes)) = rings.split_first() else {
        return Ok(None);
    };
    // If the exterior misses the tile entirely, the whole polygon is absent here.
    if clip_ring(exterior.as_ref(), min, max, &mut s.clip, &mut s.piece)? == RingClip::Outside {
        return Ok(None);
    }
    check_local(s.piece.verts(), origin)?;
    for hole in holes {
        let start = s.piece.len();
        match clip_ring(hole.as_ref(), min, max, &mut s.clip, &mut s.piece)? {
            // A hole covering the whole tile leaves nothing to draw here.
            RingClip::Covers => return Ok(None),
            RingClip::Clipped => check_local(&s.piece.verts()[start..], origin)?,
            RingClip::Outside => {}
        }
    }
    if !make_valid(rings, (min, max), origin, s)? {
        return Ok(None);
    }
    Ok(Some(origin))
}

/// [`build_tile_piece`]'s plain clip, just built, made planar outside the box if it is not: `false`
/// if the polygon then leaves nothing in the tile. Out of line, to keep the plain clip's code small.
#[inline(never)]
fn make_valid<V: PolyVertex, R: AsRef<[V]>>(
    rings: &[R],
    (min, max): (Coord<i32>, Coord<i32>),
    origin: Coord<i32>,
    s: &mut PolyScratch<V>,
) -> Result<bool, TileError> {
    let Some(budget) = needs_repair(s, (min, max)) else {
        return Ok(true);
    };
    let whole_fits = fits_whole(rings);
    repair(
        s,
        (min, max),
        origin,
        rings.len(),
        whole_fits,
        budget,
        |s, a, b| build_piece(rings, min, max, a, s, b),
    )
}

/// A polygon's `rings` whole, lent in one `tile`'s local frame; `None` if the polygon leaves nothing
/// in the tile — the test [`clip_polygon_to_tile`] makes, without building the piece. See
/// [`Slicer::tile_polygon`](crate::Slicer::tile_polygon) in [`ClipMode::Unclipped`].
pub(crate) fn whole_polygon_in_tile<'s, V: PolyVertex, R: AsRef<[V]>>(
    grid: Grid,
    tile: TileId,
    rings: &'s [R],
    s: &'s mut PolyScratch<V>,
) -> Result<Option<PolygonPiece<'s, V, R>>, TileError> {
    let (min, max) = grid.tile_buffered_bounds(tile)?;
    let origin = grid.origin_of(min);
    s.rings.clear();
    s.copies.clear();
    for (index, ring) in (0_u32..).zip(rings) {
        // (Edge ids only index the sweep's rows.)
        let Some((info, lo, hi)) = load_ring(s, index, ring.as_ref(), 0)? else {
            if index == 0 {
                return Ok(None); // a degenerate exterior drops the polygon
            }
            continue;
        };
        match ring_meets_box(info.pts(rings, &s.copies), min, max) {
            // An exterior missing the tile, or a hole covering it, leaves nothing to draw here.
            RingClip::Outside if index == 0 => return Ok(None),
            RingClip::Covers if index > 0 => return Ok(None),
            _ => {}
        }
        check_local(&[lo, hi], origin)?;
        s.rings.push(info);
    }
    if s.rings.is_empty() {
        return Ok(None); // no rings at all
    }
    Ok(Some(PolygonPiece {
        tile,
        rings: PieceRings::Whole {
            rings: &s.rings,
            input: rings,
            copies: &s.copies,
            origin,
        },
    }))
}