//! Topology-preserving face offsets for bodies made of planes and cylinders (shells, offset
//! faces). Each face moves along its outward normal by its own distance: a plane is translated,
//! a cylinder changes radius about its axis. Every vertex and edge goes to where its faces'
//! moved surfaces meet; edges are lines or circular arcs. This is exact when the moved
//! surfaces meet the way the old ones did (prismatic parts, bosses, fillets between walls).
//! A corner where more than three planes meet, whose moved planes miss a common point (a
//! floor falling to a drain, a pyramid's apex), becomes several corners joined by short edges.
//! Anything else is refused, so callers can fall back.

use std::collections::HashMap;

use solidcraft_geom::Vec3;
use truck_modeling::{self as mt, builder};

use crate::body::{Body, Solid, from_p3, p3};
use crate::{KernelError, Result, guard};

#[derive(Clone, Copy, Debug)]
pub(crate) enum Surf {
    /// Outward normal and offset: n·x = d.
    Plane { n: Vec3, d: f64 },
    /// Axis point and direction, radius; `convex` when the outward normal points away from
    /// the axis.
    Cylinder { o: Vec3, a: Vec3, r: f64, convex: bool },
    /// Centre and radius; `convex` when the outward normal points away from the centre.
    Sphere { c: Vec3, r: f64, convex: bool },
    /// Apex, unit axis (into the cone, away from the apex) and half-angle; `convex` when the
    /// outward normal points away from the axis.
    Cone { v: Vec3, a: Vec3, half: f64, convex: bool },
    /// Axis point and unit axis, the tube's centre circle (radius `big` at height `h` along the
    /// axis) and the tube radius; `convex` when the outward normal points away from the tube's
    /// centre.
    Torus { o: Vec3, a: Vec3, big: f64, h: f64, r: f64, convex: bool },
}

/// The centre of a torus tube nearest `p`.
fn tube_centre(o: Vec3, a: Vec3, big: f64, h: f64, p: Vec3) -> Option<Vec3> {
    let w = p - o;
    let u = (w - a * w.dot(a)).normalized()?;
    Some(o + a * h + u * big)
}

impl Surf {
    /// Outward unit normal of the surface at `p` (on or near it).
    pub(crate) fn normal(&self, p: Vec3) -> Option<Vec3> {
        match *self {
            Surf::Plane { n, .. } => Some(n),
            Surf::Cylinder { o, a, convex, .. } => {
                let v = p - o;
                let radial = (v - a * v.dot(a)).normalized()?;
                Some(if convex { radial } else { -radial })
            }
            Surf::Sphere { c, convex, .. } => {
                let radial = (p - c).normalized()?;
                Some(if convex { radial } else { -radial })
            }
            Surf::Cone { v, a, half, convex } => {
                let w = p - v;
                let q = (w - a * w.dot(a)).normalized()?;
                let n = q * half.cos() - a * half.sin();
                Some(if convex { n } else { -n })
            }
            Surf::Torus { o, a, big, h, convex, .. } => {
                let n = (p - tube_centre(o, a, big, h, p)?).normalized()?;
                Some(if convex { n } else { -n })
            }
        }
    }
    /// The gradient of [`Surf::dist`] (the normal before orientation).
    fn grad(&self, p: Vec3) -> Option<Vec3> {
        match *self {
            Surf::Plane { n, .. } => Some(n),
            Surf::Cylinder { convex, .. } | Surf::Sphere { convex, .. } | Surf::Cone { convex, .. } | Surf::Torus { convex, .. } => {
                self.normal(p).map(|n| if convex { n } else { -n })
            }
        }
    }
    fn moved(&self, s: f64) -> Option<Surf> {
        match *self {
            Surf::Plane { n, d } => Some(Surf::Plane { n, d: d + s }),
            Surf::Cylinder { o, a, r, convex } => {
                let r2 = if convex { r + s } else { r - s };
                (r2 > 1e-9).then_some(Surf::Cylinder { o, a, r: r2, convex })
            }
            Surf::Sphere { c, r, convex } => {
                let r2 = if convex { r + s } else { r - s };
                (r2 > 1e-9).then_some(Surf::Sphere { c, r: r2, convex })
            }
            // Moving a cone along its normal moves its apex along the axis.
            Surf::Cone { v, a, half, convex } => {
                let s = if convex { s } else { -s };
                let sh = half.sin();
                (sh > 1e-9).then_some(Surf::Cone { v: v - a * (s / sh), a, half, convex })
            }
            Surf::Torus { o, a, big, h, r, convex } => {
                let r2 = if convex { r + s } else { r - s };
                (r2 > 1e-9).then_some(Surf::Torus { o, a, big, h, r: r2, convex })
            }
        }
    }
    fn dist(&self, p: Vec3) -> f64 {
        match *self {
            Surf::Plane { n, d } => n.dot(p) - d,
            Surf::Cylinder { o, a, r, .. } => {
                let v = p - o;
                (v - a * v.dot(a)).len() - r
            }
            Surf::Sphere { c, r, .. } => p.dist(c) - r,
            Surf::Cone { v, a, half, .. } => {
                let w = p - v;
                let h = w.dot(a);
                (w - a * h).len() * half.cos() - h * half.sin()
            }
            Surf::Torus { o, a, big, h, r, .. } => tube_centre(o, a, big, h, p).map(|c| p.dist(c) - r).unwrap_or(f64::INFINITY),
        }
    }
    /// The affine map taking this surface onto `to` (same kind, same axis).
    fn map_to(&self, to: &Surf) -> Option<mt::Matrix4> {
        match (*self, *to) {
            (Surf::Sphere { c, r, .. }, Surf::Sphere { r: r2, .. }) => {
                let k = r2 / r;
                let t = c * (1.0 - k);
                Some(mt::Matrix4::from_translation(mt::Vector3::new(t.x, t.y, t.z)) * mt::Matrix4::from_scale(k))
            }
            (Surf::Cone { v, .. }, Surf::Cone { v: v2, .. }) => {
                let t = v2 - v;
                Some(mt::Matrix4::from_translation(mt::Vector3::new(t.x, t.y, t.z)))
            }
            (Surf::Plane { n, d }, Surf::Plane { d: d2, .. }) => {
                let t = n * (d2 - d);
                Some(mt::Matrix4::from_translation(mt::Vector3::new(t.x, t.y, t.z)))
            }
            (Surf::Cylinder { o, a, r, .. }, Surf::Cylinder { r: r2, .. }) => {
                // Scale by k across the axis: x ↦ o + k (x − o) + (1 − k) a a·(x − o).
                let k = r2 / r;
                let m = |i: usize, j: usize| {
                    let (ai, aj) = ([a.x, a.y, a.z][i], [a.x, a.y, a.z][j]);
                    (if i == j { k } else { 0.0 }) + (1.0 - k) * ai * aj
                };
                let lin = [[m(0, 0), m(0, 1), m(0, 2)], [m(1, 0), m(1, 1), m(1, 2)], [m(2, 0), m(2, 1), m(2, 2)]];
                let ov = [o.x, o.y, o.z];
                let t: Vec<f64> = (0..3).map(|i| ov[i] - (0..3).map(|j| lin[i][j] * ov[j]).sum::<f64>()).collect();
                // Column-major.
                Some(mt::Matrix4::new(
                    lin[0][0],
                    lin[1][0],
                    lin[2][0],
                    0.0,
                    lin[0][1],
                    lin[1][1],
                    lin[2][1],
                    0.0,
                    lin[0][2],
                    lin[1][2],
                    lin[2][2],
                    0.0,
                    *t.first()?,
                    *t.get(1)?,
                    *t.get(2)?,
                    1.0,
                ))
            }
            _ => None,
        }
    }
}

fn fail(m: &str) -> KernelError {
    KernelError::Failed(format!("offset: {m}"))
}

/// Points and outward normals sampled over a face (boundary and a grid inside its
/// parameter box).
fn samples(f: &mt::Face) -> Vec<(Vec3, Vec3)> {
    use mt::{BoundedCurve, ParametricCurve, ParametricSurface, ParametricSurface3D, SearchNearestParameter};
    let surf = f.oriented_surface();
    let mut uv: Vec<(f64, f64)> = Vec::new();
    for w in f.boundaries() {
        for e in w.edge_iter() {
            let c = e.oriented_curve();
            let (t0, t1) = c.range_tuple();
            for k in 0..5 {
                // The previous point's parameters as a hint, unless that lands elsewhere (a
                // singular point such as a pole holds the search).
                let q = c.subs(t0 + (t1 - t0) * k as f64 / 5.0);
                let near = |x: (f64, f64)| from_p3(surf.subs(x.0, x.1)).dist(from_p3(q)) < 1e-6 * (1.0 + from_p3(q).len());
                let found = surf
                    .search_nearest_parameter(q, uv.last().copied(), 50)
                    .filter(|x| near(*x))
                    .or_else(|| surf.search_nearest_parameter(q, None, 100).filter(|x| near(*x)));
                if let Some(x) = found {
                    uv.push(x);
                }
            }
        }
    }
    let Some(&(u0, v0)) = uv.first() else { return Vec::new() };
    let (mut lo, mut hi) = ((u0, v0), (u0, v0));
    for (u, v) in &uv {
        lo = (lo.0.min(*u), lo.1.min(*v));
        hi = (hi.0.max(*u), hi.1.max(*v));
    }
    for i in 0..=4 {
        for j in 0..=4 {
            uv.push((lo.0 + (hi.0 - lo.0) * i as f64 / 4.0, lo.1 + (hi.1 - lo.1) * j as f64 / 4.0));
        }
    }
    // Parameter steps for checking that the normal is steady (not at a cone's apex).
    let (du, dv) = (((hi.0 - lo.0) * 1e-6).max(1e-9), ((hi.1 - lo.1) * 1e-6).max(1e-9));
    let nrm = |u: f64, v: f64| {
        let n = surf.normal(u, v);
        Vec3::new(n.x, n.y, n.z).normalized()
    };
    uv.iter()
        .filter_map(|(u, v)| {
            let n = nrm(*u, *v)?;
            let steady =
                [nrm(*u + du, *v), nrm(*u - du, *v), nrm(*u, *v + dv), nrm(*u, *v - dv)].iter().all(|m| m.is_some_and(|m| m.dot(n) > 1.0 - 1e-6));
            steady.then(|| (from_p3(surf.subs(*u, *v)), n))
        })
        .collect()
}

/// What surface a face lies on (planes and cylinders only).
pub(crate) fn surf_of(f: &mt::Face, tol: f64) -> Option<Surf> {
    if let mt::Surface::Plane(pl) = f.oriented_surface() {
        let n = pl.normal();
        let n = Vec3::new(n.x, n.y, n.z).normalized()?;
        return Some(Surf::Plane { n, d: n.dot(from_p3(pl.origin())) });
    }
    let s = samples(f);
    if let Some(t) = torus_of(f, &s, tol) {
        return Some(t);
    }
    let (p0, n0) = *s.first()?;
    // Flat though not built as a plane (a radial line swept round an axis).
    if s.iter().all(|(p, n)| n.dot(n0) > 1.0 - 1e-9 && (*p - p0).dot(n0).abs() < tol) {
        return Some(Surf::Plane { n: n0, d: n0.dot(p0) });
    }
    // The normal most square to the first (a half cylinder's two ends are opposite).
    let far = s.iter().map(|x| x.1).min_by(|a, b| a.dot(n0).abs().total_cmp(&b.dot(n0).abs()))?;
    let Some(a) = n0.cross(far).normalized() else { return sphere_or_cone(&s, tol) };
    if s.iter().any(|(_, n)| n.dot(a).abs() > 1e-6) {
        return sphere_or_cone(&s, tol);
    }
    // The axis: where the normal lines through two samples meet (seen along a).
    let (p1, n1) = *s.iter().min_by(|x, y| x.1.dot(n0).abs().total_cmp(&y.1.dot(n0).abs()))?;
    // p0 + t0 n0 = p1 + t1 n1 (projected): solve in the plane ⟂ a.
    let w = p1 - p0;
    let c = n0.dot(n1);
    let den = 1.0 - c * c;
    if den.abs() < 1e-9 {
        return None;
    }
    let t0 = (w.dot(n0) - c * w.dot(n1)) / den;
    let o = p0 + n0 * t0;
    let o = o - a * (o - p0).dot(a);
    let r = t0.abs();
    // Outward normal away from the axis: p = o + r n (t0 < 0 means o = p − r n).
    let convex = t0 < 0.0;
    let cyl = Surf::Cylinder { o, a, r, convex };
    let ok = s.iter().all(|(p, n)| cyl.dist(*p).abs() < tol && cyl.normal(*p).is_some_and(|m| m.dot(*n) > 1.0 - 1e-6));
    ok.then_some(cyl)
}

/// A torus: a surface of revolution whose normals (in the meridian plane) all point from one
/// circle, the tube's centre, at one distance.
fn torus_of(f: &mt::Face, s: &[(Vec3, Vec3)], tol: f64) -> Option<Surf> {
    let mt::Surface::RevolutedCurve(rc) = f.surface() else { return None };
    if *rc.transform() != <mt::Matrix4 as mt::One>::one() {
        return None;
    }
    let (o, ax) = (from_p3(rc.entity().origin()), rc.entity().axis());
    let a = Vec3::new(ax.x, ax.y, ax.z).normalized()?;
    // Meridian coordinates: ρ − r n_ρ = R and h − r n_h = h0 for every sample, least squares
    // in (R, h0, r).
    let mut m = [[0.0f64; 3]; 3];
    let mut rhs = [0.0f64; 3];
    for (p, n) in s {
        let w = *p - o;
        let hh = w.dot(a);
        let radial = w - a * hh;
        let u = radial.normalized()?;
        for (row, val) in [([1.0, 0.0, n.dot(u)], radial.len()), ([0.0, 1.0, n.dot(a)], hh)] {
            for i in 0..3 {
                for j in 0..3 {
                    m[i][j] += row[i] * row[j];
                }
                rhs[i] += row[i] * val;
            }
        }
    }
    let x = crate::polyhedron::solve3_pub(m, rhs)?;
    let (big, h, r) = (x[0], x[1], x[2]);
    // (A tube wider than its circle, a spindle torus, is fine: blends use its outer part, where
    // each point's tube centre is the one on its own side of the axis.)
    if !(big > 1e-9 && r.abs() > 1e-9) {
        return None;
    }
    let t = Surf::Torus { o, a, big, h, r: r.abs(), convex: r > 0.0 };
    s.iter().all(|(p, n)| t.dist(*p).abs() < tol && t.normal(*p).is_some_and(|m| m.dot(*n) > 1.0 - 1e-6)).then_some(t)
}

/// A sphere (normal lines all through one point) or a cone (tangent planes all through one
/// point, normals at one angle to an axis) through the samples.
fn sphere_or_cone(s: &[(Vec3, Vec3)], tol: f64) -> Option<Surf> {
    // Least squares point nearest all the normal lines: Σ (I − n nᵀ) x = Σ (I − n nᵀ) p.
    let mut m = [[0.0f64; 3]; 3];
    let mut r = [0.0f64; 3];
    for (p, n) in s {
        let nv = [n.x, n.y, n.z];
        let pv = [p.x, p.y, p.z];
        for i in 0..3 {
            for j in 0..3 {
                let a = (if i == j { 1.0 } else { 0.0 }) - nv[i] * nv[j];
                m[i][j] += a;
                r[i] += a * pv[j];
            }
        }
    }
    if let Some(x) = crate::polyhedron::solve3_pub(m, r) {
        let c = Vec3::new(x[0], x[1], x[2]);
        let (p0, n0) = *s.first()?;
        let rad = p0.dist(c);
        let convex = (p0 - c).dot(n0) > 0.0;
        let sp = Surf::Sphere { c, r: rad, convex };
        if rad > tol && s.iter().all(|(p, n)| sp.dist(*p).abs() < tol && sp.normal(*p).is_some_and(|q| q.dot(*n) > 1.0 - 1e-6)) {
            return Some(sp);
        }
    }
    // Cone: the apex lies on every tangent plane, n·(v − p) = 0.
    let mut m = [[0.0f64; 3]; 3];
    let mut r = [0.0f64; 3];
    for (p, n) in s {
        let nv = [n.x, n.y, n.z];
        let k = n.dot(*p);
        for i in 0..3 {
            for j in 0..3 {
                m[i][j] += nv[i] * nv[j];
            }
            r[i] += nv[i] * k;
        }
    }
    let x = crate::polyhedron::solve3_pub(m, r)?;
    let v = Vec3::new(x[0], x[1], x[2]);
    // Axis: from the apex toward the samples' centroid (the cone opens that way).
    let cen = s.iter().fold(Vec3::ZERO, |acc, (p, _)| acc + *p) * (1.0 / s.len() as f64);
    // Normals make a constant angle with the axis: fit it from three spread normals.
    let n0 = s.first()?.1;
    let n1 = s.iter().map(|x| x.1).max_by(|a, b| (*a - n0).len().total_cmp(&(*b - n0).len()))?;
    let n2 = s.iter().map(|x| x.1).max_by(|a, b| (*a - n0).cross(n1 - n0).len().total_cmp(&(*b - n0).cross(n1 - n0).len()))?;
    let mut a = (n1 - n0).cross(n2 - n0).normalized()?;
    if a.dot(cen - v) < 0.0 {
        a = -a;
    }
    let half = n0.dot(a).abs().clamp(0.0, 1.0).asin();
    if !(half > 1e-3 && half < std::f64::consts::FRAC_PI_2 - 1e-3) {
        return None;
    }
    let (p0, n0) = *s.first()?;
    let q = {
        let w = p0 - v;
        (w - a * w.dot(a)).normalized()?
    };
    let convex = n0.dot(q) > 0.0;
    let cone = Surf::Cone { v, a, half, convex };
    s.iter().all(|(p, n)| cone.dist(*p).abs() < tol && cone.normal(*p).is_some_and(|m| m.dot(*n) > 1.0 - 1e-6)).then_some(cone)
}

/// Move `p` (on all of `old`) onto the moved surfaces `new`: the smallest displacement δ with
/// n_i · δ = s_i for each face's normal n_i at p, then checked against the moved surfaces.
fn move_point(p: Vec3, old: &[Surf], shifts: &[f64], new: &[Surf], tol: f64) -> Option<Vec3> {
    let ns: Vec<Vec3> = old.iter().map(|sf| sf.normal(p)).collect::<Option<_>>()?;
    let mut q = p + span_step(&ns, shifts)?;
    // The step is exact for planes; a curved face meeting another at an angle needs a few
    // Newton steps onto the moved surfaces.
    for _ in 0..30 {
        let d: Vec<f64> = new.iter().map(|sf| sf.dist(q)).collect();
        if d.iter().all(|x| x.abs() < tol) {
            return Some(q);
        }
        let gs: Vec<Vec3> = new.iter().map(|sf| sf.grad(q)).collect::<Option<_>>()?;
        let rhs: Vec<f64> = d.iter().map(|x| -x).collect();
        q += span_step(&gs, &rhs)?;
    }
    new.iter().all(|sf| sf.dist(q).abs() < tol).then_some(q)
}

/// The smallest move `v` (in the span of `ns`) with `n_i · v = s_i` (least squares).
fn span_step(ns: &[Vec3], s: &[f64]) -> Option<Vec3> {
    // Orthonormal basis of the normals' span (tangent faces share a normal).
    let mut basis: Vec<Vec3> = Vec::new();
    for n in ns {
        let mut v = *n;
        for e in &basis {
            v = v - *e * v.dot(*e);
        }
        if let Some(u) = v.normalized().filter(|_| v.len() > 1e-6) {
            basis.push(u);
        }
    }
    // Least squares in the span: rows n_i · e_k, right-hand sides s_i.
    let r = basis.len();
    let mut m = [[0.0f64; 3]; 3];
    let mut rhs = [0.0f64; 3];
    for (n, si) in ns.iter().zip(s) {
        let row: Vec<f64> = basis.iter().map(|e| n.dot(*e)).collect();
        for i in 0..r {
            for j in 0..r {
                m[i][j] += row[i] * row[j];
            }
            rhs[i] += row[i] * si;
        }
    }
    // Pad unused dimensions with the identity.
    for (i, row) in m.iter_mut().enumerate().skip(r) {
        row[i] = 1.0;
    }
    let c = crate::polyhedron::solve3_pub(m, rhs)?;
    let mut v = Vec3::ZERO;
    for (k, e) in basis.iter().enumerate() {
        v += *e * c[k];
    }
    Some(v)
}

/// Most faces round one corner that [`split_corner`] takes on.
const MAX_SPLIT_FACES: usize = 48;

/// The corners in place of a corner where more than three planar faces meet and whose moved
/// planes miss a common point (a hip floor's corner, a pyramid's apex): each new corner with
/// the faces through it. Corners sharing two faces are joined by a new edge on those faces.
/// Only corners whose edges are all convex or all concave: the faces there bound a convex
/// region (the body's side or the outside's), and after the move its corners are the points on
/// three moved planes inside all the others.
fn split_corner(
    v: &mt::Vertex,
    at: &[usize],
    faces: &[mt::Face],
    efaces: &HashMap<mt::EdgeID, Vec<usize>>,
    old: &[Surf],
    new: &[Surf],
    size: f64,
) -> Option<Vec<(Vec3, Vec<usize>)>> {
    if at.len() < 4 || at.len() > MAX_SPLIT_FACES {
        return None;
    }
    let p = from_p3(v.point());
    // (face, outward normal, old offset, moved offset).
    let planes: Vec<(usize, Vec3, f64, f64)> = at
        .iter()
        .map(|i| match (old.get(*i)?, new.get(*i)?) {
            (Surf::Plane { n, d }, Surf::Plane { d: d2, .. }) => Some((*i, *n, *d, *d2)),
            _ => None,
        })
        .collect::<Option<_>>()?;
    let normal = |fi: usize| planes.iter().find(|x| x.0 == fi).map(|x| x.1);
    // Edge kinds round the corner: a face's loop runs counter-clockwise about its outward
    // normal, so its inside is to the left; the edge is convex when the face beyond it falls
    // away from there.
    let mut sign = 0.0f64;
    let mut mixed = false;
    let mut pairs: Vec<(usize, usize)> = Vec::new();
    for &(fi, na, ..) in &planes {
        for w in faces.get(fi)?.boundaries() {
            for e in w.edge_iter() {
                if e.front().id() != v.id() && e.back().id() != v.id() {
                    continue;
                }
                let fb = efaces.get(&e.id())?.iter().copied().find(|j| *j != fi)?;
                pairs.push((fi, fb));
                let dir = (from_p3(e.back().point()) - from_p3(e.front().point())).normalized()?;
                let s = normal(fb)?.dot(na.cross(dir));
                if s.abs() < 1e-9 {
                    continue;
                }
                if sign * s < 0.0 {
                    mixed = true;
                }
                sign = s.signum();
            }
        }
    }
    if sign == 0.0 {
        return None;
    }
    // Which triples of faces may meet at a new corner. All convex or all concave: any triple
    // whose point is inside all the other moved planes. Mixed (a saddle, such as a box on a
    // box that overhangs its side): the faces round the corner in order, and each new corner
    // on three faces in a row (the two new edges on the middle face meet there). A saddle's
    // faces bound no convex region, so a right corner can be outside another face's moved
    // plane, and the inside-all-the-others test does not apply. What keeps a corner in place
    // is that it is where its middle face's two new edges meet; a corner past the far end of
    // one of the corner's edges turns that edge round, and a face whose loops cross after the
    // move is refused (both checked in `offset_body`).
    let triples: Vec<[usize; 3]> = if mixed {
        let ring = face_ring(&pairs, planes.len())?;
        let n = ring.len();
        (0..n).map(|i| [ring[(i + n - 1) % n], ring[i], ring[(i + 1) % n]]).collect()
    } else {
        let mut t = Vec::new();
        for ia in 0..planes.len() {
            for ib in ia + 1..planes.len() {
                for ic in ib + 1..planes.len() {
                    t.push([planes[ia].0, planes[ib].0, planes[ic].0]);
                }
            }
        }
        t
    };
    let plane_of = |fi: usize| planes.iter().find(|x| x.0 == fi);
    // Convex: the body's side, n·x ≤ d; concave: the outside's, n·x ≥ d.
    let side = -sign;
    let merge = size * 1e-6;
    let mut nodes: Vec<(Vec3, Vec<usize>)> = Vec::new();
    for [fa, fb, fc] in triples {
        let (a, b, c) = (plane_of(fa)?, plane_of(fb)?, plane_of(fc)?);
        let den = a.1.dot(b.1.cross(c.1));
        if den.abs() < 1e-9 {
            continue;
        }
        let x = (b.1.cross(c.1) * a.3 + c.1.cross(a.1) * b.3 + a.1.cross(b.1) * c.3) * (1.0 / den);
        if !x.is_finite() || x.dist(p) > size || (!mixed && planes.iter().any(|q| side * (q.1.dot(x) - q.3) > merge)) {
            continue;
        }
        let fs = [fa, fb, fc];
        match nodes.iter_mut().find(|(q, _)| q.dist(x) < merge) {
            Some((_, nf)) => {
                for f in fs {
                    if !nf.contains(&f) {
                        nf.push(f);
                    }
                }
            }
            None => nodes.push((x, fs.to_vec())),
        }
    }
    // Every face keeps a corner here, and every edge an end.
    if planes.iter().any(|q| !nodes.iter().any(|(_, nf)| nf.contains(&q.0)))
        || pairs.iter().any(|(x, y)| !nodes.iter().any(|(_, nf)| nf.contains(x) && nf.contains(y)))
    {
        return None;
    }
    // Each corner where its planes meet best (least squares, for planes meeting at one point
    // to within the merge distance).
    for (q, nf) in nodes.iter_mut() {
        let (ns, ss): (Vec<Vec3>, Vec<f64>) = nf.iter().filter_map(|f| planes.iter().find(|x| x.0 == *f)).map(|x| (x.1, x.3 - x.2)).unzip();
        *q = p + span_step(&ns, &ss)?;
    }
    Some(nodes)
}

/// The faces round a corner in order, from the pairs of faces on each of its edges: `None`
/// unless they form one simple ring of `count` faces (each face meets the corner once, between
/// exactly two neighbours).
fn face_ring(pairs: &[(usize, usize)], count: usize) -> Option<Vec<usize>> {
    let mut next: HashMap<usize, Vec<usize>> = HashMap::new();
    for &(a, b) in pairs {
        for (x, y) in [(a, b), (b, a)] {
            let ns = next.entry(x).or_default();
            if !ns.contains(&y) {
                ns.push(y);
            }
        }
    }
    if next.len() != count || next.values().any(|ns| ns.len() != 2) {
        return None;
    }
    let first = *next.keys().min()?;
    let mut ring = vec![first];
    let mut prev = first;
    let mut cur = *next.get(&first)?.first()?;
    while cur != first {
        if ring.len() >= count {
            return None;
        }
        ring.push(cur);
        let ns = next.get(&cur)?;
        let to = if ns[0] == prev { ns[1] } else { ns[0] };
        prev = cur;
        cur = to;
    }
    (ring.len() == count).then_some(ring)
}

/// The corners of a split corner (see [`split_corner`]) on face `face`, in order from corner
/// `from` to corner `to`, along the new edges between them.
fn split_path(nodes: &[(Vec3, Vec<usize>)], face: usize, from: usize, to: usize) -> Option<Vec<usize>> {
    let joined = |x: usize, y: usize| {
        let (Some((_, a)), Some((_, b))) = (nodes.get(x), nodes.get(y)) else { return false };
        a.contains(&face) && b.contains(&face) && a.iter().filter(|f| b.contains(f)).count() >= 2
    };
    // Breadth first; `prev` leads back to `from`.
    let mut prev: Vec<Option<usize>> = vec![None; nodes.len()];
    let mut queue = std::collections::VecDeque::from([from]);
    let mut seen = vec![false; nodes.len()];
    *seen.get_mut(from)? = true;
    while let Some(x) = queue.pop_front() {
        if x == to {
            let mut path = vec![to];
            let mut cur = to;
            while let Some(p) = prev.get(cur).copied().flatten() {
                path.push(p);
                cur = p;
            }
            path.reverse();
            return Some(path);
        }
        for y in 0..nodes.len() {
            if !seen.get(y).copied().unwrap_or(true) && joined(x, y) {
                *seen.get_mut(y)? = true;
                *prev.get_mut(y)? = Some(x);
                queue.push_back(y);
            }
        }
    }
    None
}

/// A line or circular arc through an edge (by samples): `None` for other curves.
/// Maximum straight edges per face checked pairwise for crossings.
const MAX_CROSSING_EDGES: usize = 2000;

/// Where the moved straight edges of a planar face cross each other (or a loop of straight
/// edges turns round), if they do. Faces next to a collapsed round are left to the other checks.
fn crossed_loops(
    faces: &[mt::Face],
    old: &[Surf],
    pos: &dyn Fn(&mt::Vertex, &mt::Edge) -> Option<Vec3>,
    edge_collapsed: &dyn Fn(&mt::Edge) -> bool,
    size: f64,
) -> Option<Vec3> {
    let eps = size * 1e-9;
    for (f, sf) in faces.iter().zip(old) {
        let Surf::Plane { n, .. } = *sf else { continue };
        if f.edge_iter().any(|e| edge_collapsed(&e)) {
            continue;
        }
        let (u, w) = (n.any_perp(), n.cross(n.any_perp()));
        let flat = |p: Vec3| (p.dot(u), p.dot(w));
        // Straight edges as (vertex ids, new ends in the plane, the plane's new offset).
        let mut segs: Vec<([mt::VertexID; 2], [(f64, f64); 2], f64)> = Vec::new();
        for wire in f.absolute_boundaries() {
            let (mut area_old, mut area_new, mut straight) = (0.0, 0.0, true);
            for e in wire.edge_iter() {
                let (v0, v1) = (e.front(), e.back());
                let (Some(q0), Some(q1)) = (pos(v0, e), pos(v1, e)) else { return None };
                let (o0, o1, n0, n1) = (flat(from_p3(v0.point())), flat(from_p3(v1.point())), flat(q0), flat(q1));
                area_old += o0.0 * o1.1 - o1.0 * o0.1;
                area_new += n0.0 * n1.1 - n1.0 * n0.1;
                if arc_mid(e, size * 1e-6) == Some(None) {
                    segs.push(([v0.id(), v1.id()], [n0, n1], n.dot(q0)));
                } else {
                    straight = false;
                }
            }
            if straight && area_old.abs() > eps && area_old * area_new <= 0.0 {
                return wire.edge_iter().next().and_then(|e| pos(e.front(), e));
            }
        }
        if segs.len() > MAX_CROSSING_EDGES {
            continue;
        }
        let side = |a: (f64, f64), c: (f64, f64), p: (f64, f64)| (c.0 - a.0) * (p.1 - a.1) - (c.1 - a.1) * (p.0 - a.0);
        for (i, (ia, [a0, a1], d)) in segs.iter().enumerate() {
            for (ib, [b0, b1], _) in segs.iter().skip(i + 1) {
                if ia.iter().any(|v| ib.contains(v)) {
                    continue;
                }
                let (s0, s1) = (side(*a0, *a1, *b0), side(*a0, *a1, *b1));
                let (t0, t1) = (side(*b0, *b1, *a0), side(*b0, *b1, *a1));
                let tol = eps * size;
                if ((s0 > tol && s1 < -tol) || (s0 < -tol && s1 > tol)) && ((t0 > tol && t1 < -tol) || (t0 < -tol && t1 > tol)) {
                    let k = t0 / (t0 - t1);
                    let (x, y) = (a0.0 + (a1.0 - a0.0) * k, a0.1 + (a1.1 - a0.1) * k);
                    return Some(u * x + w * y + n * *d);
                }
            }
        }
    }
    None
}

pub(crate) fn arc_mid(e: &mt::Edge, tol: f64) -> Option<Option<Vec3>> {
    use mt::{BoundedCurve, ParametricCurve};
    let c = e.curve();
    let (t0, t1) = c.range_tuple();
    let pts: Vec<Vec3> = (0..=16).map(|k| from_p3(c.subs(t0 + (t1 - t0) * k as f64 / 16.0))).collect();
    let (a, b, m) = (*pts.first()?, *pts.last()?, *pts.get(8)?);
    if a.dist(b) < tol {
        return None;
    }
    if pts.iter().all(|p| p.dist_to_segment(a, b) < tol) {
        return Some(None);
    }
    let (u, v) = (b - a, m - a);
    let w = u.cross(v);
    let d = 2.0 * w.len2();
    if d < 1e-300 {
        return None;
    }
    let centre = a + (v.cross(w) * u.len2() + w.cross(u) * v.len2()) * (1.0 / d);
    let r = centre.dist(a);
    pts.iter().all(|p| (p.dist(centre) - r).abs() < tol && (*p - centre).dot(w.normalized().unwrap_or(Vec3::Z)).abs() < tol).then_some(Some(m))
}

/// The body with each face moved along its outward normal by `shift(face index, normal at a
/// face point)`, keeping the topology. Faces must be planes or cylinders. The result's history
/// is over `[b]`: each face is the face it was moved from, each edge the edge it was moved from;
/// faces moved to nothing are gone, and their neighbours meet along new edges.
pub(crate) fn offset_body(b: &Body, shift: impl Fn(usize, Vec3) -> f64) -> Result<Body> {
    let faces: Vec<mt::Face> = b.solid.face_iter().cloned().collect();
    let size = b.size();
    let tol = (size * 1e-7).max(1e-9);
    let mut old = Vec::with_capacity(faces.len());
    let mut new = Vec::with_capacity(faces.len());
    let mut shifts = Vec::with_capacity(faces.len());
    // Faces moved to nothing (a round thinner than the move): they go, their neighbours meet.
    let mut collapsed: Vec<bool> = Vec::with_capacity(faces.len());
    for (i, f) in faces.iter().enumerate() {
        let sf = surf_of(f, tol * 100.0).ok_or_else(|| {
            let at = samples(f).first().map(|(p, _)| format!(" (a face at [{:.2}, {:.2}, {:.2}])", p.x, p.y, p.z)).unwrap_or_default();
            fail(&format!("only bodies with planes, cylinders, cones, spheres and tori{at}"))
        })?;
        let n = match sf {
            Surf::Plane { n, .. } => n,
            Surf::Cylinder { a, .. } | Surf::Cone { a, .. } | Surf::Torus { a, .. } => a.any_perp(),
            Surf::Sphere { .. } => Vec3::Z,
        };
        let s = shift(i, n);
        old.push(sf);
        match sf.moved(s) {
            Some(x) => {
                new.push(x);
                collapsed.push(false);
            }
            None => {
                new.push(sf);
                collapsed.push(true);
            }
        }
        shifts.push(s);
    }
    let pick = |ids: &[usize]| -> (Vec<Surf>, Vec<f64>, Vec<Surf>) {
        (
            ids.iter().filter_map(|i| old.get(*i).copied()).collect(),
            ids.iter().filter_map(|i| shifts.get(*i).copied()).collect(),
            ids.iter().filter_map(|i| new.get(*i).copied()).collect(),
        )
    };
    // Vertex → its faces; edge → its faces.
    let mut vfaces: HashMap<mt::VertexID, Vec<usize>> = HashMap::new();
    let mut efaces: HashMap<mt::EdgeID, Vec<usize>> = HashMap::new();
    for (i, f) in faces.iter().enumerate() {
        for v in f.vertex_iter() {
            let e = vfaces.entry(v.id()).or_default();
            if !e.contains(&i) {
                e.push(i);
            }
        }
        for ed in f.edge_iter() {
            let e = efaces.entry(ed.id()).or_default();
            if !e.contains(&i) {
                e.push(i);
            }
        }
    }
    let is_collapsed = |i: usize| collapsed.get(i).copied().unwrap_or(false);
    let any_collapsed = collapsed.iter().any(|c| *c);
    // A collapsed face's place is taken by the faces around it (through other collapsed ones).
    let mut nbrs: HashMap<usize, Vec<usize>> = HashMap::new();
    for fs in efaces.values() {
        for a in fs {
            for b2 in fs {
                if a != b2 {
                    let e = nbrs.entry(*a).or_default();
                    if !e.contains(b2) {
                        e.push(*b2);
                    }
                }
            }
        }
    }
    let expand = |fs: &[usize]| -> Vec<usize> {
        let mut out: Vec<usize> = fs.iter().copied().filter(|i| !is_collapsed(*i)).collect();
        let mut queue: Vec<usize> = fs.iter().copied().filter(|i| is_collapsed(*i)).collect();
        let mut seen = queue.clone();
        while let Some(c) = queue.pop() {
            for nb in nbrs.get(&c).into_iter().flatten() {
                if is_collapsed(*nb) {
                    if !seen.contains(nb) {
                        seen.push(*nb);
                        queue.push(*nb);
                    }
                } else if !out.contains(nb) {
                    out.push(*nb);
                }
            }
        }
        out
    };
    // Each edge's ends and middle, and the faces either side.
    let edge_pts: Vec<(Vec<usize>, [Vec3; 3])> = b
        .solid
        .edge_iter()
        .filter_map(|e| {
            use mt::{BoundedCurve, ParametricCurve};
            let c = e.curve();
            let (t0, t1) = c.range_tuple();
            Some((
                efaces.get(&e.id())?.clone(),
                [from_p3(e.absolute_front().point()), from_p3(e.absolute_back().point()), from_p3(c.subs((t0 + t1) / 2.0))],
            ))
        })
        .collect();
    // Across the collapsed rounds next to edge `e` (its faces `fs`): a round is a strip, crossed
    // to its side that shares no vertex with the edge, through further collapsed rounds.
    let opposite_side = |e: &mt::Edge, fs: &[usize]| -> Vec<usize> {
        let mut out: Vec<usize> = Vec::new();
        let mut seen: Vec<usize> = Vec::new();
        let mut queue: Vec<(usize, [mt::VertexID; 2])> =
            fs.iter().copied().filter(|i| is_collapsed(*i)).map(|i| (i, [e.front().id(), e.back().id()])).collect();
        while let Some((cf, avoid)) = queue.pop() {
            if seen.contains(&cf) {
                continue;
            }
            seen.push(cf);
            let Some(face) = faces.get(cf) else { continue };
            for side in face.edge_iter() {
                if avoid.contains(&side.front().id()) || avoid.contains(&side.back().id()) {
                    continue;
                }
                for nb in efaces.get(&side.id()).into_iter().flatten().copied() {
                    if nb == cf || fs.contains(&nb) {
                        continue;
                    }
                    if is_collapsed(nb) {
                        queue.push((nb, [side.front().id(), side.back().id()]));
                    } else if !out.contains(&nb) {
                        out.push(nb);
                    }
                }
            }
        }
        out
    };
    // For a point on a collapsed round touching the faces `at`: the faces it keeps (`at`'s
    // own, not collapsed), and the faces along the rounds it is on, nearest `p` first.
    let across = |at: &[usize], p: Vec3| -> (Vec<usize>, Vec<usize>) {
        let own: Vec<usize> = at.iter().copied().filter(|i| !is_collapsed(*i)).collect();
        let mut cand: Vec<(f64, usize)> = expand(at)
            .into_iter()
            .filter(|nb| !own.contains(nb))
            .map(|nb| {
                let d = edge_pts
                    .iter()
                    .filter(|(fs, _)| fs.contains(&nb) && fs.iter().any(|i| is_collapsed(*i)))
                    .flat_map(|(_, q)| q.iter().map(|x| x.dist(p)))
                    .fold(f64::INFINITY, f64::min);
                (d, nb)
            })
            .collect();
        cand.sort_by(|x, y| x.0.total_cmp(&y.0));
        (own, cand.into_iter().map(|(_, nb)| nb).collect())
    };
    let mut newpos: HashMap<mt::VertexID, Vec3> = HashMap::new();
    // Corners whose moved faces meet at several points (see `split_corner`).
    let mut splits: HashMap<mt::VertexID, Vec<(Vec3, Vec<usize>)>> = HashMap::new();
    for v in b.solid.vertex_iter() {
        let at = vfaces.get(&v.id()).ok_or_else(|| fail("vertex"))?;
        let p = from_p3(v.point());
        let fs: Vec<usize> = if at.iter().any(|i| is_collapsed(*i)) {
            // On a collapsed round: its own faces, then the faces along the rounds it is on,
            // nearest first, until they fix a point.
            let (mut own, cand) = across(at, p);
            let rank = |ids: &[usize]| -> usize {
                let mut basis: Vec<Vec3> = Vec::new();
                for i in ids {
                    let Some(nv) = old.get(*i).and_then(|sf| sf.normal(p)) else { continue };
                    let mut w = nv;
                    for e in &basis {
                        w = w - *e * w.dot(*e);
                    }
                    if let Some(u) = w.normalized().filter(|_| w.len() > 1e-6) {
                        basis.push(u);
                    }
                }
                basis.len()
            };
            for c in cand {
                if rank(&own) >= 3 {
                    break;
                }
                own.push(c);
            }
            own
        } else {
            at.clone()
        };
        let (o, s, n) = pick(&fs);
        // A cone's apex goes with the cone (its normal is not defined there).
        let apex = o.iter().zip(&n).find_map(|(a, b)| match (a, b) {
            (Surf::Cone { v: va, .. }, Surf::Cone { v: vb, .. }) if va.dist(p) < size * 1e-6 => Some(*vb),
            _ => None,
        });
        let q = match apex {
            Some(q) if n.iter().all(|sf| sf.dist(q).abs() < size * 1e-6) => q,
            _ => match move_point(p, &o, &s, &n, size * 1e-6) {
                Some(q) => q,
                None => {
                    let nodes = (fs == *at)
                        .then(|| split_corner(&v, at, &faces, &efaces, &old, &new, size))
                        .flatten()
                        .ok_or_else(|| fail("a corner whose faces don't meet the same way after the move"))?;
                    let q = nodes.first().map(|x| x.0).ok_or_else(|| fail("corner"))?;
                    splits.insert(v.id(), nodes);
                    q
                }
            },
        };
        newpos.insert(v.id(), q);
    }
    // Which of a split corner's new corners an edge ends at.
    let node_of = |v: mt::VertexID, e: mt::EdgeID| -> Option<usize> {
        let fs = efaces.get(&e)?;
        splits.get(&v)?.iter().position(|(_, nf)| fs.iter().all(|i| nf.contains(i)))
    };
    // Where an edge's end goes.
    let end_pos = |v: &mt::Vertex, e: &mt::Edge| -> Option<Vec3> {
        match splits.get(&v.id()) {
            Some(nodes) => nodes.get(node_of(v.id(), e.id())?).map(|x| x.0),
            None => newpos.get(&v.id()).copied(),
        }
    };
    // An edge turned round: faces crossed over (a wall thinner than the move), which would
    // leave the body inside out.
    let edge_collapsed = |e: &mt::Edge| efaces.get(&e.id()).is_some_and(|fs| fs.iter().any(|i| is_collapsed(*i)));
    for e in b.solid.edge_iter() {
        if edge_collapsed(&e) {
            continue;
        }
        let (f0, f1) = (e.absolute_front(), e.absolute_back());
        let (Some(q0), Some(q1)) = (end_pos(f0, &e), end_pos(f1, &e)) else { continue };
        let (old_v, new_v) = (from_p3(f1.point()) - from_p3(f0.point()), q1 - q0);
        if old_v.len() > size * 1e-6 && old_v.dot(new_v) < 0.0 {
            return Err(fail("not supported yet: the move is larger than a wall or step is thick (faces would cross)"));
        }
    }
    // A plane's loops crossing after the move: a rim or ledge narrower than the move (the hole
    // in a top face sliding past its outer edge) turns part of the face inside out, though no
    // edge turns round.
    if let Some(at) = crossed_loops(&faces, &old, &end_pos, &edge_collapsed, size) {
        return Err(fail(&format!(
            "not supported yet: the move is larger than a rim or ledge near [{:.1}, {:.1}, {:.1}] is wide (a face would turn inside out)",
            at.x, at.y, at.z
        )));
    }
    guard("offset", || {
        // Vertices of collapsed faces that land together become one.
        let on_collapsed: std::collections::HashSet<mt::VertexID> = faces
            .iter()
            .enumerate()
            .filter(|(i, _)| is_collapsed(*i))
            .flat_map(|(_, f)| f.vertex_iter().map(|v| v.id()).collect::<Vec<_>>())
            .collect();
        let mut verts: HashMap<mt::VertexID, mt::Vertex> = HashMap::new();
        let mut welded: Vec<(Vec3, mt::Vertex)> = Vec::new();
        for v in b.solid.vertex_iter() {
            let Some(p) = newpos.get(&v.id()).copied() else { continue };
            let nv = if any_collapsed && on_collapsed.contains(&v.id()) {
                match welded.iter().find(|(q, _)| q.dist(p) < size * 1e-6) {
                    Some((_, w)) => w.clone(),
                    None => {
                        let w = builder::vertex(p3(p));
                        welded.push((p, w.clone()));
                        w
                    }
                }
            } else {
                builder::vertex(p3(p))
            };
            verts.insert(v.id(), nv);
        }
        let node_verts: HashMap<mt::VertexID, Vec<mt::Vertex>> =
            splits.iter().map(|(id, nodes)| (*id, nodes.iter().map(|(q, _)| builder::vertex(p3(*q))).collect())).collect();
        let vert_at = |v: &mt::Vertex, e: &mt::Edge| -> Option<&mt::Vertex> {
            match node_verts.get(&v.id()) {
                Some(nv) => nv.get(node_of(v.id(), e.id())?),
                None => verts.get(&v.id()),
            }
        };
        // New edges between a split corner's corners (stored from the lower index).
        let mut links: HashMap<(mt::VertexID, usize, usize), mt::Edge> = HashMap::new();
        // Old edge → new edge, and whether it runs the other way (merged with its twin).
        let mut edges: HashMap<mt::EdgeID, (mt::Edge, bool)> = HashMap::new();
        // Moved faces and edges; an edge where neighbours of a collapsed face meet is new.
        let mut rec = crate::record::Rebuild::default();
        // New edges along collapsed rounds: ends and points along them, for their twins.
        let mut merged: Vec<(mt::VertexID, mt::VertexID, Vec<Vec3>, mt::Edge)> = Vec::new();
        for e in b.solid.edge_iter() {
            if edges.contains_key(&e.id()) {
                continue;
            }
            let (Some(a), Some(c)) = (vert_at(e.absolute_front(), &e), vert_at(e.absolute_back(), &e)) else { return Err(fail("edge")) };
            let touches = edge_collapsed(&e);
            if a == c || from_p3(a.point()).dist(from_p3(c.point())) < size * 1e-9 {
                if touches {
                    continue;
                }
                return Err(fail("an edge vanishes (the offset is too large)"));
            }
            if efaces.get(&e.id()).is_some_and(|fs| fs.iter().all(|i| is_collapsed(*i))) {
                continue;
            }
            if touches {
                // Along a collapsed round: where the edge's own face meets the face across the
                // round, between the edge's moved ends, through the edge's middle moved onto
                // both: a line (planes, or a plane along a cylinder's axis) or an arc (a plane
                // square to a coaxial cylinder); faces that meet in another curve are refused.
                // Shared with its twin from the round's other side: the new edge with the same
                // ends through the same middle (two arcs of one circle have the same ends).
                use mt::{BoundedCurve, ParametricCurve};
                let fs = efaces.get(&e.id()).ok_or_else(|| fail("edge faces"))?;
                let cv = e.curve();
                let (t0, t1) = cv.range_tuple();
                let mid = from_p3(cv.subs((t0 + t1) / 2.0));
                // The face across: the strip's far side (`opposite_side`), the nearest of them
                // when several; the nearest along the rounds when the strip has none.
                let (mut meet, cand) = across(fs, mid);
                let far = opposite_side(&e, fs);
                meet.extend(cand.iter().find(|c| far.contains(c)).or(cand.first()));
                if meet.len() < 2 {
                    return Err(fail("an edge along a collapsed round with no face across it"));
                }
                let (o, s, n) = pick(&meet);
                let m = move_point(mid, &o, &s, &n, size * 1e-6).ok_or_else(|| fail("faces that don't meet along a collapsed round"))?;
                let near = size * 1e-5;
                let twin = merged.iter().find(|(x, y, pts, _)| {
                    ((*x == a.id() && *y == c.id()) || (*x == c.id() && *y == a.id()))
                        && pts.windows(2).any(|w| w.first().zip(w.get(1)).is_some_and(|(p, q)| m.dist_to_segment(*p, *q) < near))
                });
                let ne = match twin {
                    Some((_, _, _, x)) => x.clone(),
                    None => {
                        let (pa, pc) = (from_p3(a.point()), from_p3(c.point()));
                        let x = if m.dist_to_segment(pa, pc) < near { builder::line(a, c) } else { builder::circle_arc(a, c, p3(m)) };
                        let xc = x.curve();
                        let (u0, u1) = xc.range_tuple();
                        let pts: Vec<Vec3> = (0..=16).map(|j| from_p3(xc.subs(u0 + (u1 - u0) * j as f64 / 16.0))).collect();
                        if !pts.iter().all(|p| n.iter().all(|sf| sf.dist(*p).abs() < near)) {
                            return Err(fail("not supported yet: faces meeting along a collapsed round in a curve other than a line or circle"));
                        }
                        merged.push((a.id(), c.id(), pts, x.clone()));
                        x
                    }
                };
                let flipped = ne.front() != a;
                edges.insert(e.id(), (ne, flipped));
                continue;
            }
            let Some(kind) = arc_mid(&e, size * 1e-6) else {
                // Another curve (where two cylinders meet): its points moved onto the moved
                // faces, through a smooth curve.
                use mt::{BoundedCurve, ParametricCurve};
                let fs = efaces.get(&e.id()).ok_or_else(|| fail("edge faces"))?;
                let (o, s, n) = pick(fs);
                let cv = e.curve();
                let (t0, t1) = cv.range_tuple();
                let k = 16;
                let mut pts = vec![from_p3(a.point())];
                for j in 1..k {
                    let p = from_p3(cv.subs(t0 + (t1 - t0) * j as f64 / k as f64));
                    pts.push(
                        move_point(p, &o, &s, &n, size * 1e-6).ok_or_else(|| fail("an edge whose faces don't meet the same way after the move"))?,
                    );
                }
                pts.push(from_p3(c.point()));
                let curve = crate::build::interpolate_cubic(&pts).ok_or_else(|| fail("edge curve"))?;
                let ne = mt::Edge::new(a, c, mt::Curve::BSplineCurve(curve));
                rec.piece(&ne, &e);
                edges.insert(e.id(), (ne, false));
                continue;
            };
            let ne = match kind {
                None => builder::line(a, c),
                Some(m) => {
                    let fs = efaces.get(&e.id()).ok_or_else(|| fail("edge faces"))?;
                    let (o, s, n) = pick(fs);
                    let m2 =
                        move_point(m, &o, &s, &n, size * 1e-6).ok_or_else(|| fail("an edge whose faces don't meet the same way after the move"))?;
                    builder::circle_arc(a, c, p3(m2))
                }
            };
            rec.piece(&ne, &e);
            edges.insert(e.id(), (ne, false));
        }
        let mut out = Vec::new();
        for (i, f) in faces.iter().enumerate() {
            if is_collapsed(i) {
                continue;
            }
            let mut wires: Vec<mt::Wire> = Vec::new();
            for w in f.absolute_boundaries() {
                let old_edges: Vec<mt::Edge> = w.edge_iter().cloned().collect();
                let mut wire: Vec<mt::Edge> = Vec::new();
                for (j, e) in old_edges.iter().enumerate() {
                    if let Some((ne, flip)) = edges.get(&e.id()) {
                        wire.push(if (e.front() == e.absolute_front()) != *flip { ne.clone() } else { ne.inverse() });
                    }
                    // At a split corner, along the new edges from this edge's end to the next's.
                    let v = e.back();
                    let (Some(nodes), Some(next)) = (splits.get(&v.id()), old_edges.get((j + 1) % old_edges.len())) else { continue };
                    let (Some(from), Some(to)) = (node_of(v.id(), e.id()), node_of(v.id(), next.id())) else { return Err(fail("split corner")) };
                    let path = split_path(nodes, i, from, to).ok_or_else(|| fail("split corner"))?;
                    for pair in path.windows(2) {
                        let (Some(&x), Some(&y)) = (pair.first(), pair.get(1)) else { continue };
                        let key = (v.id(), x.min(y), x.max(y));
                        let link = match links.get(&key) {
                            Some(l) => l.clone(),
                            None => {
                                let nv = node_verts.get(&v.id()).ok_or_else(|| fail("split corner"))?;
                                let (Some(va), Some(vb)) = (nv.get(key.1), nv.get(key.2)) else { return Err(fail("split corner")) };
                                let l = builder::line(va, vb);
                                links.insert(key, l.clone());
                                l
                            }
                        };
                        wire.push(if x < y { link } else { link.inverse() });
                    }
                }
                wires.push(wire.into());
            }
            let (Some(o), Some(n)) = (old.get(i), new.get(i)) else { return Err(fail("face")) };
            let surface = match (o.map_to(n), *o, *n) {
                (Some(mat), _, _) => mt::Transformed::transformed(&f.surface(), mat),
                // A torus: its tube's circle scaled about the tube's centre, revolved again.
                (None, Surf::Torus { o: to, a, big, h, r, .. }, Surf::Torus { r: r2, .. }) => {
                    use mt::{BoundedCurve, ParametricCurve};
                    let mt::Surface::RevolutedCurve(rc) = f.surface() else { return Err(fail("torus surface")) };
                    let curve = rc.entity().entity_curve().clone();
                    let (t0, t1) = curve.range_tuple();
                    let q = from_p3(curve.subs((t0 + t1) / 2.0));
                    let c = tube_centre(to, a, big, h, q).ok_or_else(|| fail("torus"))?;
                    let k = r2 / r;
                    let t = c * (1.0 - k);
                    let mat = mt::Matrix4::from_translation(mt::Vector3::new(t.x, t.y, t.z)) * mt::Matrix4::from_scale(k);
                    let moved = mt::Transformed::transformed(&curve, mat);
                    let mut p = mt::Processor::new(mt::RevolutedCurve::by_revolution(moved, rc.entity().origin(), rc.entity().axis()));
                    if !rc.orientation() {
                        mt::Invertible::invert(&mut p);
                    }
                    mt::Surface::RevolutedCurve(p)
                }
                _ => return Err(fail("surface")),
            };
            let mut nf = mt::Face::try_new(wires, surface).map_err(|e| fail(&e.to_string()))?;
            if !f.orientation() {
                nf.invert();
            }
            rec.face(&nf, f);
            out.push(nf);
        }
        // One shell per shell of the body (pieces, voids), less the collapsed faces.
        let mut shells: Vec<mt::Shell> = Vec::new();
        let mut it = out.into_iter();
        let mut k = 0;
        for sh in b.solid.boundaries() {
            let keep = (k..k + sh.len()).filter(|i| !is_collapsed(*i)).count();
            k += sh.len();
            shells.push(it.by_ref().take(keep).collect::<Vec<_>>().into());
        }
        let solid = Solid::try_new(shells).map_err(|e| fail(&e.to_string()))?;
        crate::record::rebuilt(solid, &[&b.solid], &rec)
    })
}

/// Draft walls parallel to `pull` (planes, and cylinders about axes along `pull`) by `angle`
/// about their line on the neutral plane: at height h along `pull` a wall moves along its
/// outward normal by −h·tan(angle), so planes tilt and cylinders become cones. The faces that
/// share a corner with a drafted wall must be caps square to `pull` or walls; the rest of the
/// body (a rounded edge elsewhere, say) is left as it is. The topology is kept, and the
/// result's history is over `[b]` (each face and edge the one it was moved from).
pub(crate) fn draft_walls(b: &Body, chosen: &[usize], neutral: &solidcraft_geom::Plane, pull: Vec3, angle: f64) -> Result<Body> {
    let faces: Vec<mt::Face> = b.solid.face_iter().cloned().collect();
    let size = b.size();
    let tol = (size * 1e-7).max(1e-9);
    let nn = neutral.normal();
    let den = pull.dot(nn);
    if den.abs() < 1e-9 {
        return Err(fail("the neutral plane is parallel to the pull direction"));
    }
    let height = |p: Vec3| (p - neutral.origin).dot(nn) / den;
    let tan = angle.tan();
    let mut vfaces: HashMap<mt::VertexID, Vec<usize>> = HashMap::new();
    let mut efaces: HashMap<mt::EdgeID, Vec<usize>> = HashMap::new();
    for (i, f) in faces.iter().enumerate() {
        for v in f.vertex_iter() {
            let e = vfaces.entry(v.id()).or_default();
            if !e.contains(&i) {
                e.push(i);
            }
        }
        for ed in f.edge_iter() {
            let e = efaces.entry(ed.id()).or_default();
            if !e.contains(&i) {
                e.push(i);
            }
        }
    }
    // A corner moves when one of its faces is drafted; the faces round a moving corner must be
    // walls or caps for the corner to follow them. Faces away from the drafted walls may be
    // anything: they keep their surface and their edges.
    let moves = |vid: &mt::VertexID| vfaces.get(vid).is_some_and(|fs| fs.iter().any(|i| chosen.contains(i)));
    let touched: Vec<bool> = faces.iter().enumerate().map(|(i, f)| chosen.contains(&i) || f.vertex_iter().any(|v| moves(&v.id()))).collect();
    let mut old: Vec<Option<Surf>> = Vec::with_capacity(faces.len());
    for (i, f) in faces.iter().enumerate() {
        if !touched.get(i).copied().unwrap_or(false) {
            old.push(None);
            continue;
        }
        let sf = surf_of(f, tol * 100.0).ok_or_else(|| fail("draft: the faces next to the drafted walls must be planes or cylinders"))?;
        let wall = match sf {
            Surf::Plane { n, .. } => n.dot(pull).abs() < 1e-9,
            Surf::Cylinder { a, .. } => a.dot(pull).abs() > 1.0 - 1e-9,
            Surf::Sphere { .. } | Surf::Cone { .. } | Surf::Torus { .. } => false,
        };
        let cap = matches!(sf, Surf::Plane { n, .. } if n.dot(pull).abs() > 1.0 - 1e-9);
        if chosen.contains(&i) && !wall {
            return Err(fail("draft: a chosen face isn't parallel to the pull direction"));
        }
        if !chosen.contains(&i) && !cap && !wall {
            return Err(fail("draft: faces next to the drafted walls must be square to the pull direction"));
        }
        old.push(Some(sf));
    }
    // Shifts and moved surfaces at a height.
    let at = |ids: &[usize], h: f64| -> Option<(Vec<Surf>, Vec<f64>, Vec<Surf>)> {
        let mut o = Vec::new();
        let mut s = Vec::new();
        let mut n = Vec::new();
        for i in ids {
            let sf = (*old.get(*i)?)?;
            let sh = if chosen.contains(i) { -h * tan } else { 0.0 };
            o.push(sf);
            s.push(sh);
            n.push(sf.moved(sh)?);
        }
        Some((o, s, n))
    };
    let mut newpos: HashMap<mt::VertexID, Vec3> = HashMap::new();
    for v in b.solid.vertex_iter() {
        let p = from_p3(v.point());
        if !moves(&v.id()) {
            newpos.insert(v.id(), p);
            continue;
        }
        let fs = vfaces.get(&v.id()).ok_or_else(|| fail("vertex"))?;
        let (o, s, n) = at(fs, height(p)).ok_or_else(|| fail("draft: a wall shrinks to nothing"))?;
        let q = move_point(p, &o, &s, &n, size * 1e-6).ok_or_else(|| fail("draft: a corner whose faces don't meet the same way after the draft"))?;
        newpos.insert(v.id(), q);
    }
    // New surfaces for the drafted walls: planes through their hinge line, cylinders → cones.
    let mut surfaces: Vec<Option<mt::Surface>> = vec![None; faces.len()];
    for (i, sf) in old.iter().enumerate() {
        let (true, Some(sf)) = (chosen.contains(&i), *sf) else { continue };
        let surface = match sf {
            Surf::Plane { n, d } => {
                let n2 = (n * angle.cos() + pull * angle.sin()).normalized().ok_or_else(|| fail("draft"))?;
                // A point of the old plane on the neutral plane: the hinge.
                let o = neutral.intersect_ray(n * d, pull).ok_or_else(|| fail("draft: hinge"))?;
                let u = n2.cross(pull).normalized().ok_or_else(|| fail("draft"))?;
                let w = n2.cross(u);
                mt::Surface::Plane(mt::Plane::new(p3(o), p3(o + u), p3(o + w)))
            }
            Surf::Cylinder { o, a, r, convex } => {
                // Generator: the wall at two heights along one radial direction.
                let u = a.any_perp();
                // Over the face's heights (and a little beyond), short of the apex.
                let hs: Vec<f64> = faces.get(i).map(|f| f.vertex_iter().map(|v| height(from_p3(v.point()))).collect()).unwrap_or_default();
                let (lo, hi) = hs.iter().fold((f64::INFINITY, f64::NEG_INFINITY), |m, h| (m.0.min(*h), m.1.max(*h)));
                if !(lo.is_finite() && hi.is_finite()) {
                    return Err(fail("draft: a wall without corners"));
                }
                let pad = (hi - lo).max(size * 1e-3) * 0.25;
                let (mut h0, mut h1) = (lo - pad, hi + pad);
                // Keep the radius positive along the generator.
                let r_at = |h: f64| if convex { r - h * tan } else { r + h * tan };
                for h in [&mut h0, &mut h1] {
                    if r_at(*h) <= r * 1e-3 {
                        *h = if (*h - lo).abs() < (*h - hi).abs() { lo } else { hi };
                    }
                }
                let base = |h: f64| {
                    let rr = if convex { r - h * tan } else { r + h * tan };
                    o + pull * (h - height(o)) + u * rr
                };
                let (pa, pb) = (base(h0), base(h1));
                let line = mt::Curve::Line(mt::Line(p3(pa), p3(pb)));
                let rc = mt::RevolutedCurve::by_revolution(line, p3(o), mt::Vector3::new(a.x, a.y, a.z));
                mt::Surface::RevolutedCurve(mt::Processor::new(rc))
            }
            Surf::Sphere { .. } | Surf::Cone { .. } | Surf::Torus { .. } => {
                return Err(fail("draft: only planes and cylinders along the pull can be drafted"));
            }
        };
        if let Some(slot) = surfaces.get_mut(i) {
            *slot = Some(surface);
        }
    }
    guard("draft", || {
        let verts: HashMap<mt::VertexID, mt::Vertex> = newpos.iter().map(|(k, p)| (*k, builder::vertex(p3(*p)))).collect();
        let mut edges: HashMap<mt::EdgeID, mt::Edge> = HashMap::new();
        let mut rec = crate::record::Rebuild::default();
        for e in b.solid.edge_iter() {
            if edges.contains_key(&e.id()) {
                continue;
            }
            let (Some(a), Some(c)) = (verts.get(&e.absolute_front().id()), verts.get(&e.absolute_back().id())) else { return Err(fail("edge")) };
            // An edge whose ends stay keeps its curve (it may be anything: a blend's spline).
            if !moves(&e.absolute_front().id()) && !moves(&e.absolute_back().id()) {
                let ne = mt::Edge::try_new(a, c, e.curve().clone()).map_err(|e| fail(&e.to_string()))?;
                rec.piece(&ne, &e);
                edges.insert(e.id(), ne);
                continue;
            }
            let kind = arc_mid(&e, size * 1e-6).ok_or_else(|| fail("draft: edges next to the drafted walls must be lines or arcs"))?;
            let ne = match kind {
                None => builder::line(a, c),
                Some(m) => {
                    let fs = efaces.get(&e.id()).ok_or_else(|| fail("edge faces"))?;
                    let (o, s, n) = at(fs, height(m)).ok_or_else(|| fail("draft"))?;
                    let m2 = move_point(m, &o, &s, &n, size * 1e-6).ok_or_else(|| fail("draft: an arc that can't follow its faces"))?;
                    builder::circle_arc(a, c, p3(m2))
                }
            };
            rec.piece(&ne, &e);
            edges.insert(e.id(), ne);
        }
        let mut out = Vec::new();
        for (i, f) in faces.iter().enumerate() {
            let wires: Vec<mt::Wire> = f
                .boundaries()
                .iter()
                .map(|w| {
                    w.edge_iter()
                        .filter_map(|e| edges.get(&e.id()).map(|ne| if e.front() == e.absolute_front() { ne.clone() } else { ne.inverse() }))
                        .collect::<Vec<_>>()
                        .into()
                })
                .collect();
            let nf = match surfaces.get(i).cloned().flatten() {
                Some(mut surface) => {
                    // Outward like the old face (checked at a boundary point).
                    let probe = f.boundaries().first().and_then(|w| w.vertex_iter().next()).map(|v| from_p3(v.point()));
                    let want = probe.and_then(|p| old.get(i).copied().flatten().and_then(|s| s.normal(p)));
                    if let (Some(p), Some(want)) = (probe, want) {
                        use mt::{ParametricSurface3D, SearchNearestParameter};
                        if let Some((u, v)) = surface.search_nearest_parameter(p3(p), None, 100) {
                            let nn = surface.normal(u, v);
                            if Vec3::new(nn.x, nn.y, nn.z).dot(want) < 0.0 {
                                surface = mt::Invertible::inverse(&surface);
                            }
                        }
                    }
                    mt::Face::try_new(wires, surface).map_err(|e| fail(&e.to_string()))?
                }
                None => {
                    // Unchanged surface: rebuild on the oriented boundary.
                    let mut nf = mt::Face::try_new(
                        f.absolute_boundaries()
                            .iter()
                            .map(|w| {
                                w.edge_iter()
                                    .filter_map(|e| {
                                        edges.get(&e.id()).map(|ne| if e.front() == e.absolute_front() { ne.clone() } else { ne.inverse() })
                                    })
                                    .collect::<Vec<_>>()
                                    .into()
                            })
                            .collect(),
                        f.surface(),
                    )
                    .map_err(|e| fail(&e.to_string()))?;
                    if !f.orientation() {
                        nf.invert();
                    }
                    nf
                }
            };
            rec.face(&nf, f);
            out.push(nf);
        }
        let shell: mt::Shell = out.into();
        let solid = Solid::try_new(vec![shell]).map_err(|e| fail(&e.to_string()))?;
        crate::record::rebuilt(solid, &[&b.solid], &rec)
    })
}

/// The chosen faces and every face on the same surface joined to them (the halves of a hole's
/// wall), as face indices. Pieces of a face split on purpose (Split Face) stay apart.
pub fn with_same_surface(b: &Body, chosen: &[usize]) -> Vec<usize> {
    let faces: Vec<mt::Face> = b.solid.face_iter().cloned().collect();
    let size = b.size();
    let tol = (size * 1e-7).max(1e-9) * 100.0;
    let surfs: Vec<Option<Surf>> = faces.iter().map(|f| surf_of(f, tol)).collect();
    let same = |x: &Surf, y: &Surf| match (*x, *y) {
        (Surf::Plane { n, d }, Surf::Plane { n: m, d: e }) => n.dot(m) > 1.0 - 1e-9 && (d - e).abs() < tol,
        (Surf::Cylinder { o, a, r, convex }, Surf::Cylinder { o: o2, a: a2, r: r2, convex: c2 }) => {
            a.dot(a2).abs() > 1.0 - 1e-9 && (r - r2).abs() < tol && convex == c2 && { (o2 - o - a * (o2 - o).dot(a)).len() < tol }
        }
        (Surf::Sphere { c, r, convex }, Surf::Sphere { c: c2, r: r2, convex: k2 }) => c.dist(c2) < tol && (r - r2).abs() < tol && convex == k2,
        (Surf::Cone { v, a, half, convex }, Surf::Cone { v: v2, a: a2, half: h2, convex: k2 }) => {
            v.dist(v2) < tol && a.dot(a2) > 1.0 - 1e-9 && (half - h2).abs() < 1e-9 && convex == k2
        }
        (Surf::Torus { o, a, big, h, r, convex }, Surf::Torus { o: o2, a: a2, big: b2, h: h2, r: r2, convex: k2 }) => {
            (o + a * h).dist(o2 + a2 * h2) < tol && a.dot(a2).abs() > 1.0 - 1e-9 && (big - b2).abs() < tol && (r - r2).abs() < tol && convex == k2
        }
        _ => false,
    };
    let keep = b.split_keep();
    let mut out: Vec<usize> = chosen.to_vec();
    let mut grew = true;
    while grew {
        grew = false;
        for (i, f) in faces.iter().enumerate() {
            if out.contains(&i) {
                continue;
            }
            let Some(Some(si)) = surfs.get(i) else { continue };
            let joins = out.iter().any(|j| {
                surfs.get(*j).and_then(|x| x.as_ref()).is_some_and(|sj| same(si, sj))
                    && faces.get(*j).is_some_and(|g| {
                        g.edge_iter().any(|e| {
                            f.edge_iter().any(|x| x.id() == e.id())
                                && (keep.is_empty() || !crate::heal::on_split_line(&crate::heal::edge_points(&e), keep))
                        })
                    })
            });
            if joins {
                out.push(i);
                grew = true;
            }
        }
    }
    out
}