diff --git a/crates/engine/src/cmd/joints.rs b/crates/engine/src/cmd/joints.rs
index 6e94d472..09a90321 100644
--- a/crates/engine/src/cmd/joints.rs
+++ b/crates/engine/src/cmd/joints.rs
@@ -73,7 +73,7 @@ pub static COMMANDS: &[CommandSpec] = &[
     ),
     CommandSpec::new("joint.solve", "Solve Joints", solve_cmd).icon("compute").params("re-place the occurrences from the joints → conflicts, errors, dof"),
     CommandSpec::new("inspect.interference", "Interference", interference).at("SOLID", "INSPECT").icon("interference").noundo().params(
-        "bodies?: [names] (default: all, in world placement) → interferences: pairs that overlap with the overlap volume, area, bounding box and centre; unchecked: pairs that could not be checked, with the reason. Bodies are pruned by bounding box first, so any number of bodies is fine; at most 20000 pairs whose boxes overlap are intersected per call (truncated: true and skipped_pairs: n say when more were left out)",
+        "bodies?: [names] (default: all, in world placement) → interferences: pairs that overlap with the overlap volume, area, bounding box and centre (a pin a hair off the axis of a hole of its size also has near_coaxial: {offset_mm, length_mm}, its volume the crescent the walls share); unchecked: pairs that could not be checked, with the reason. Bodies are pruned by bounding box first, so any number of bodies is fine; at most 20000 pairs whose boxes overlap are intersected per call (truncated: true and skipped_pairs: n say when more were left out)",
     ),
 ];
 
@@ -1156,6 +1156,16 @@ fn interference(s: &mut Session, p: &Value) -> Result<Value> {
     let (mut hits, mut unchecked) = (Vec::new(), Vec::new());
     for (i, j) in pairs {
         let (Some(a), Some(b)) = (bodies.get(i), bodies.get(j)) else { continue };
+        let threshold = 1e-9 * a.body.size().max(1.0).powi(3);
+        // A pin a hair off the axis of its hole: the crescent the walls share, which the
+        // boolean can't intersect.
+        if let Some(o) = solidcraft_kernel::coaxial_overlap(&a.body, &b.body) {
+            if o.volume > threshold {
+                hits.push(json!({"a": a.name, "b": b.name, "volume_mm3": o.volume, "area_mm2": o.area, "center": o.centroid,
+                    "bbox": {"min": o.bbox.min, "max": o.bbox.max}, "near_coaxial": {"offset_mm": o.offset, "length_mm": o.length}}));
+            }
+            continue;
+        }
         // A pair whose shared part can't be worked out is reported, not passed as clear.
         let (shared, report) =
             solidcraft_kernel::capture_boolean_diagnostics(|| solidcraft_kernel::boolean(&a.body, &b.body, solidcraft_kernel::BoolOp::Intersect));
@@ -1178,7 +1188,7 @@ fn interference(s: &mut Session, p: &Value) -> Result<Value> {
                 continue;
             }
         };
-        if m.volume > 1e-9 * a.body.size().max(1.0).powi(3) {
+        if m.volume > threshold {
             hits.push(json!({"a": a.name, "b": b.name, "volume_mm3": m.volume, "area_mm2": m.area, "center": m.centroid, "bbox": {"min": m.bbox.min, "max": m.bbox.max}}));
         }
     }
diff --git a/crates/engine/src/cmd/joints_tests.rs b/crates/engine/src/cmd/joints_tests.rs
index f0b42b4e..4c5e4a6c 100644
--- a/crates/engine/src/cmd/joints_tests.rs
+++ b/crates/engine/src/cmd/joints_tests.rs
@@ -1452,3 +1452,31 @@ mod cam;
 // Plots and interference of motion studies, on the four-bar and slider-crank above.
 #[path = "motion_analysis_tests.rs"]
 mod motion_analysis;
+
+/// A pin through a hole of its own radius whose axis is a hair off the hole's (parts placed from
+/// rounded coordinates): the walls are all but tangent along two lines, which the boolean can't
+/// intersect. Each offset gets a checked answer: clear on the axis and within the kernel's
+/// same-surface tolerance, and otherwise the crescent the pin shares with the plate, against
+/// the textbook lens of two circles. The pin is chamfered, so it is not a plain prism.
+#[test]
+fn interference_of_a_pin_a_hair_off_its_hole() {
+    let crescent = |r: f64, e: f64| std::f64::consts::PI * r * r - (2.0 * r * r * (e / (2.0 * r)).acos() - e / 2.0 * (4.0 * r * r - e * e).sqrt());
+    for offset in [0.0, 1e-6, 5.9e-4, 1e-2] {
+        let mut s = Session::default();
+        run(&mut s, "solid.box", json!({"length": 20, "width": 20, "height": 5, "corner": [-10, -10, 0], "body_name": "Plate"}));
+        run(&mut s, "solid.cylinder", json!({"radius": 3, "height": 9, "base": [0, 0, -2], "operation": "cut", "targets": ["Plate"]}));
+        run(&mut s, "solid.cylinder", json!({"radius": 3, "height": 9, "base": [offset, 0, -2], "operation": "new", "body_name": "Pin"}));
+        run(&mut s, "solid.chamfer", json!({"edges": [[offset + 3.0, 0, 7]], "distance": 0.5, "body": "Pin"}));
+        let i = run(&mut s, "inspect.interference", json!({"bodies": ["Plate", "Pin"]}));
+        assert_eq!(i["unchecked"].as_array().map(Vec::len), Some(0), "{offset}: {i}");
+        if offset < 1e-5 {
+            assert_eq!(i["count"], 0, "{offset}: {i}");
+            continue;
+        }
+        assert_eq!(i["count"], 1, "{offset}: {i}");
+        let want = 5.0 * crescent(3.0, offset);
+        let got = i["interferences"][0]["volume_mm3"].as_f64().unwrap_or(0.0);
+        assert!((got - want).abs() < 1e-6 * want, "{offset}: {got} against {want}: {i}");
+        assert!((i["interferences"][0]["near_coaxial"]["offset_mm"].as_f64().unwrap_or(0.0) - offset).abs() < 1e-9, "{i}");
+    }
+}
diff --git a/crates/kernel/src/coaxial_overlap.rs b/crates/kernel/src/coaxial_overlap.rs
new file mode 100644
index 00000000..378a4fbe
--- /dev/null
+++ b/crates/kernel/src/coaxial_overlap.rs
@@ -0,0 +1,424 @@
+//! The shared volume of a pin in a hole of (nearly) its own size whose axis is a hair off the
+//! hole's (parts placed from rounded coordinates).
+//!
+//! The pin's wall and the hole's wall then meet along two lines where they are all but tangent,
+//! which the boolean cannot intersect, and the facets of the two walls pass through each other
+//! by less than the boolean's sampled shortcuts can see. So the overlap is worked out in two
+//! parts instead. The pin's wall is moved off the hole's by a clear step (`offset_body`) and the
+//! bodies are intersected as they then are, which the boolean can do: narrowed, that gives the
+//! overlap away from the walls (the core). In the band the move swept, what the two walls
+//! share is a crescent in every section, whose area follows from the two circles alone. The
+//! band is assumed to hold nothing but the two walls; that is checked by intersecting with the
+//! pin widened by half the step and by the whole step, whose volumes must come out as the core
+//! plus the crescents those wider circles give. Where they don't (a cross hole through the
+//! band, a hole that is not round all the way, a pin end inside the hole), the overlap is not
+//! answered here and the caller takes the general boolean.
+
+use solidcraft_geom::{Aabb3, Vec3};
+
+use crate::body::Body;
+use crate::{BoolOp, FaceSurface};
+
+/// The overlap of two bodies with a pin of one in a hole of the other, worked out as above.
+#[derive(Clone, Debug)]
+pub struct CoaxialOverlap {
+    pub volume: f64,
+    /// Approximate: the core's area plus the crescents' walls and ends.
+    pub area: f64,
+    pub centroid: Vec3,
+    pub bbox: Aabb3,
+    /// The largest offset between a pin's axis and its hole's, and the length they share.
+    pub offset: f64,
+    pub length: f64,
+}
+
+/// Largest axis offset plus radius difference handled, as a share of the radius.
+const NEAR: f64 = 0.01;
+/// The step the pin's wall is moved by, as a share of its radius.
+const STEP: f64 = 0.1;
+
+/// The faces of one body on one cylinder, the side its material is on and their extent along
+/// the axis.
+#[derive(Clone, Debug)]
+struct Wall {
+    faces: Vec<usize>,
+    origin: Vec3,
+    axis: Vec3,
+    radius: f64,
+    /// Material inside the cylinder (a pin), not outside it (a hole).
+    boss: bool,
+    /// Intervals along `axis` from `origin`, one per face.
+    spans: Vec<(f64, f64)>,
+}
+
+/// A pin of one body (`pin_in_a`) in a hole of the other, and the length they share.
+#[derive(Clone, Debug)]
+struct Pair {
+    pin: Wall,
+    pin_in_a: bool,
+    hole: Wall,
+    /// The hole's axis seen from the pin's, across it, and how far.
+    toward: Vec3,
+    offset: f64,
+    /// Where they share length along the pin's axis.
+    shared: Vec<(f64, f64)>,
+    step: f64,
+}
+
+/// The overlap of `a` and `b` when a pin of one sits in a hole of the other on nearly the same
+/// cylinder (axes parallel, offset plus radius difference within 1 % of the radius, but not on
+/// the same surface), worked out as the module says. `None` when there is no such pin, or the
+/// bodies hold more than the pin and the hole where the work assumes they don't: the caller
+/// then takes the general boolean.
+pub fn coaxial_overlap(a: &Body, b: &Body) -> Option<CoaxialOverlap> {
+    if a.is_mesh() || b.is_mesh() {
+        return None;
+    }
+    crate::guard("near-coaxial overlap", || Ok(overlap(a, b))).ok().flatten()
+}
+
+fn overlap(a: &Body, b: &Body) -> Option<CoaxialOverlap> {
+    let size = a.size().max(b.size());
+    if !(size.is_finite() && size > 0.0) {
+        return None;
+    }
+    // The same surface as the kernel takes it elsewhere: walls that close are one wall.
+    let same = (size * 1e-7).max(1e-9) * 10.0;
+    // Meshed only when two cylinders are close enough to be asked about.
+    let (ca, cb) = (cylinders(a), cylinders(b));
+    let close = |x: &Wall, y: &Wall| x.axis.dot(y.axis).abs() > 1.0 - 1e-9 && near(x, y, same).is_some();
+    if !ca.iter().any(|x| cb.iter().any(|y| close(x, y))) {
+        return None;
+    }
+    let (wa, wb) = (walls(a, ca, same)?, walls(b, cb, same)?);
+    let mut pairs = Vec::new();
+    for x in &wa {
+        for y in &wb {
+            if x.axis.dot(y.axis).abs() <= 1.0 - 1e-9 {
+                continue;
+            }
+            let (pin, hole, pin_in_a) = match (x.boss, y.boss) {
+                (true, false) => (x, y, true),
+                (false, true) => (y, x, false),
+                // Two pins or two holes on nearly one cylinder: not this case.
+                _ if near(x, y, same).is_some() => return None,
+                _ => continue,
+            };
+            let Some((toward, offset)) = near(pin, hole, same) else { continue };
+            let shared = intersect(&union(&pin.spans), &union(&project(hole, pin)));
+            if total(&shared) <= same {
+                continue;
+            }
+            let step = (pin.radius * STEP).min(size * 0.02);
+            pairs.push(Pair { pin: pin.clone(), pin_in_a, hole: hole.clone(), toward, offset, shared, step });
+        }
+    }
+    if pairs.is_empty() {
+        return None;
+    }
+    // A pin's faces move together: one step for each of them (the pin's walls do not repeat
+    // across pairs unless one pin is near two holes, which the check below then answers).
+    let shared_at = |k: f64| -> Option<(f64, Option<Body>)> {
+        let moved = |body: &Body, in_a: bool| -> Option<Body> {
+            let mine: Vec<&Pair> = pairs.iter().filter(|p| p.pin_in_a == in_a).collect();
+            if mine.is_empty() {
+                return Some(body.clone());
+            }
+            let shift = |i: usize, _| mine.iter().find(|p| p.pin.faces.contains(&i)).map(|p| p.step * k).unwrap_or(0.0);
+            let m = crate::offset::offset_body(body, shift).ok()?;
+            m.validity().is_empty().then_some(m)
+        };
+        let (ma, mb) = (moved(a, true)?, moved(b, false)?);
+        let (r, report) = crate::capture_boolean_diagnostics(|| crate::ops::boolean(&ma, &mb, BoolOp::Intersect));
+        if report.unverified().is_some() || report.warning().is_some() {
+            return None;
+        }
+        match r.ok()? {
+            None => Some((0.0, None)),
+            Some(body) => Some((crate::measure(&body).ok()?.volume, Some(body))),
+        }
+    };
+    // The crescents' volume with every pin grown by `k` steps.
+    let crescents =
+        |k: f64| -> f64 { pairs.iter().map(|p| total(&p.shared) * crescent(p.pin.radius + p.step * k, p.hole.radius, p.offset).area).sum() };
+    let (core, core_body) = shared_at(-1.0)?;
+    for k in [0.5, 1.0] {
+        let (v, _) = shared_at(k)?;
+        let gain = crescents(k) - crescents(-1.0);
+        if !((v - core - gain).abs() <= 0.02 * gain + 1e-4 * v.abs() + 1e-12 * size.powi(3)) {
+            return None;
+        }
+    }
+    // The band held the two walls only: the overlap is the core plus the crescents.
+    let mut volume = core;
+    let mut moment = Vec3::ZERO;
+    let mut area = 0.0;
+    let mut bbox = Aabb3::EMPTY;
+    if let Some(body) = &core_body
+        && let Ok(m) = crate::measure(body)
+    {
+        moment = m.centroid * m.volume;
+        area = m.area;
+        bbox = m.bbox;
+        volume = m.volume;
+    }
+    for p in &pairs {
+        let c = crescent(p.pin.radius, p.hole.radius, p.offset);
+        let length = total(&p.shared);
+        // Less what the narrowed pin already reached (nothing, unless the hole is the smaller).
+        let v = length * c.area - length * crescent(p.pin.radius - p.step, p.hole.radius, p.offset).area;
+        let mid = p.shared.iter().map(|(s, t)| (s + t) * 0.5 * (t - s)).sum::<f64>() / length;
+        let side = p.toward.cross(p.pin.axis);
+        moment += (p.pin.origin + p.pin.axis * mid + p.toward * c.centroid) * v;
+        volume += v;
+        area += p.shared.len() as f64 * 2.0 * c.area + length * c.perimeter;
+        for (s, t) in &p.shared {
+            for q in crescent_outline(p.pin.radius, p.hole.radius, p.offset) {
+                let at = p.pin.origin + p.toward * q.0 + side * q.1;
+                bbox.add(at + p.pin.axis * *s);
+                bbox.add(at + p.pin.axis * *t);
+            }
+        }
+    }
+    let centroid = if volume > 0.0 { moment / volume } else { bbox.center() };
+    let offset = pairs.iter().map(|p| p.offset).fold(0.0, f64::max);
+    let length = pairs.iter().map(|p| total(&p.shared)).sum();
+    Some(CoaxialOverlap { volume: volume.max(0.0), area, centroid, bbox, offset, length })
+}
+
+/// The pin's offset to the hole, when the two are nearly on one cylinder but not on the same
+/// surface: the unit direction from the pin's axis to the hole's (any, for one axis) and the
+/// distance.
+fn near(pin: &Wall, hole: &Wall, same: f64) -> Option<(Vec3, f64)> {
+    let d = hole.origin - pin.origin;
+    let across = d - pin.axis * d.dot(pin.axis);
+    let offset = across.len();
+    let gap = (pin.radius - hole.radius).abs();
+    if offset <= same && gap <= same {
+        return None;
+    }
+    if !(offset + gap <= NEAR * pin.radius.min(hole.radius)) {
+        return None;
+    }
+    let toward = across.normalized().unwrap_or_else(|| pin.axis.any_perp());
+    Some((toward, offset))
+}
+
+/// The body's cylindrical faces, each as a wall of its own with no side or extent yet.
+fn cylinders(b: &Body) -> Vec<Wall> {
+    (0..b.face_count())
+        .filter_map(|i| match crate::face_surface(b, i)? {
+            FaceSurface::Cylinder { origin, axis, radius } if origin.is_finite() && radius.is_finite() && radius > 0.0 => {
+                Some(Wall { faces: vec![i], origin, axis: axis.normalized()?, radius, boss: false, spans: Vec::new() })
+            }
+            _ => None,
+        })
+        .collect()
+}
+
+/// The cylindrical walls of a body from its `cylinders`: its faces on each cylinder, which side
+/// its material is on and where they run along the axis. `None` when the body can't be meshed.
+fn walls(b: &Body, cylinders: Vec<Wall>, same: f64) -> Option<Vec<Wall>> {
+    let mut out: Vec<Wall> = Vec::new();
+    let mesh = b.tessellate_with((b.size() * 1e-3).max(1e-4), true).ok()?;
+    for Wall { faces, origin, axis, radius, .. } in cylinders {
+        let Some(&face) = faces.first() else { continue };
+        // Which way the face looks: out of the axis for a pin, toward it for a hole.
+        let (mut facing, mut lo, mut hi) = (0.0, f64::INFINITY, f64::NEG_INFINITY);
+        for (t, f) in mesh.triangles.iter().zip(&mesh.tri_face) {
+            if *f as usize != face {
+                continue;
+            }
+            let Some([p, q, r]) = mesh.tri(t) else { continue };
+            let c = (p + q + r) / 3.0;
+            let radial = (c - origin) - axis * (c - origin).dot(axis);
+            facing += (q - p).cross(r - p).dot(radial);
+            for x in [p, q, r] {
+                let s = (x - origin).dot(axis);
+                lo = lo.min(s);
+                hi = hi.max(s);
+            }
+        }
+        if !(lo < hi) || facing == 0.0 {
+            continue;
+        }
+        let boss = facing > 0.0;
+        let wall = Wall { faces: vec![face], origin, axis, radius, boss, spans: vec![(lo, hi)] };
+        match out.iter_mut().find(|w| w.boss == boss && on_one_cylinder(w, &wall, same)) {
+            Some(w) => {
+                let s = (origin - w.origin).dot(w.axis) * if axis.dot(w.axis) < 0.0 { -1.0 } else { 1.0 };
+                let (a, b) = if axis.dot(w.axis) < 0.0 { (-hi, -lo) } else { (lo, hi) };
+                w.faces.push(face);
+                w.spans.push((a + s, b + s));
+            }
+            None => out.push(wall),
+        }
+    }
+    Some(out)
+}
+
+fn on_one_cylinder(x: &Wall, y: &Wall, same: f64) -> bool {
+    let d = y.origin - x.origin;
+    x.axis.dot(y.axis).abs() > 1.0 - 1e-9 && (x.radius - y.radius).abs() < same && (d - x.axis * d.dot(x.axis)).len() < same
+}
+
+/// `wall`'s spans along `onto`'s axis.
+fn project(wall: &Wall, onto: &Wall) -> Vec<(f64, f64)> {
+    let flip = wall.axis.dot(onto.axis) < 0.0;
+    let s = (wall.origin - onto.origin).dot(onto.axis);
+    wall.spans.iter().map(|&(lo, hi)| if flip { (s - hi, s - lo) } else { (s + lo, s + hi) }).collect()
+}
+
+/// Sorted, disjoint intervals covering `spans`.
+fn union(spans: &[(f64, f64)]) -> Vec<(f64, f64)> {
+    let mut v: Vec<(f64, f64)> = spans.iter().copied().filter(|(a, b)| a < b).collect();
+    v.sort_by(|x, y| x.0.total_cmp(&y.0));
+    let mut out: Vec<(f64, f64)> = Vec::with_capacity(v.len());
+    for (a, b) in v {
+        match out.last_mut() {
+            Some(last) if a <= last.1 => last.1 = last.1.max(b),
+            _ => out.push((a, b)),
+        }
+    }
+    out
+}
+
+fn intersect(x: &[(f64, f64)], y: &[(f64, f64)]) -> Vec<(f64, f64)> {
+    let mut out = Vec::new();
+    for &(a, b) in x {
+        for &(c, d) in y {
+            let (lo, hi) = (a.max(c), b.min(d));
+            if lo < hi {
+                out.push((lo, hi));
+            }
+        }
+    }
+    out
+}
+
+fn total(spans: &[(f64, f64)]) -> f64 {
+    spans.iter().map(|(a, b)| b - a).sum()
+}
+
+/// The part of a disc of radius `pin` outside a disc of radius `hole` whose centre is `offset`
+/// away along +x: its area, its centroid's x (y is 0) and its perimeter.
+#[derive(Clone, Copy, Debug, PartialEq)]
+pub(crate) struct Crescent {
+    pub(crate) area: f64,
+    pub(crate) centroid: f64,
+    pub(crate) perimeter: f64,
+}
+
+pub(crate) fn crescent(pin: f64, hole: f64, offset: f64) -> Crescent {
+    use std::f64::consts::PI;
+    let (r, q, e) = (pin.max(0.0), hole.max(0.0), offset.abs());
+    let disc = PI * r * r;
+    if e >= r + q {
+        return Crescent { area: disc, centroid: 0.0, perimeter: 2.0 * PI * r };
+    }
+    if e <= (r - q).abs() {
+        return if q >= r {
+            Crescent { area: 0.0, centroid: 0.0, perimeter: 0.0 }
+        } else {
+            let area = disc - PI * q * q;
+            Crescent { area, centroid: -PI * q * q * e / area, perimeter: 2.0 * PI * (r + q) }
+        };
+    }
+    // Half-angles of the lens's two segments, seen from each centre.
+    let alpha = ((e * e + r * r - q * q) / (2.0 * e * r)).clamp(-1.0, 1.0).acos();
+    let beta = ((e * e + q * q - r * r) / (2.0 * e * q)).clamp(-1.0, 1.0).acos();
+    let segment = |radius: f64, half: f64| {
+        let area = radius * radius * (half - half.sin() * half.cos());
+        let w = 2.0 * half - (2.0 * half).sin();
+        let from_centre = if w > 1e-12 { 4.0 * radius * half.sin().powi(3) / (3.0 * w) } else { radius };
+        (area, from_centre)
+    };
+    let (sa, da) = segment(r, alpha);
+    let (sb, db) = segment(q, beta);
+    let lens = sa + sb;
+    let area = (disc - lens).max(0.0);
+    let moment = -(sa * da + sb * (e - db));
+    let centroid = if area > 0.0 { moment / area } else { 0.0 };
+    Crescent { area, centroid, perimeter: r * (2.0 * PI - 2.0 * alpha) + q * 2.0 * beta }
+}
+
+/// Points around the crescent's outline (x toward the hole's centre, y across), for its box.
+fn crescent_outline(pin: f64, hole: f64, offset: f64) -> Vec<(f64, f64)> {
+    let (r, q, e) = (pin, hole, offset);
+    let alpha = if e > 0.0 { ((e * e + r * r - q * q) / (2.0 * e * r)).clamp(-1.0, 1.0).acos() } else { 0.0 };
+    let n = 32;
+    (0..=n)
+        .map(|i| {
+            let t = alpha + (2.0 * std::f64::consts::PI - 2.0 * alpha) * f64::from(i) / f64::from(n);
+            (r * t.cos(), r * t.sin())
+        })
+        .collect()
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+
+    /// The crescent against the textbook lens of two equal circles, and its limits.
+    #[test]
+    fn crescent_matches_the_lens_formula() {
+        let (r, e) = (3.0f64, 0.01f64);
+        let lens = 2.0 * r * r * (e / (2.0 * r)).acos() - e / 2.0 * (4.0 * r * r - e * e).sqrt();
+        let c = crescent(r, r, e);
+        assert!((c.area - (std::f64::consts::PI * r * r - lens)).abs() < 1e-12, "{c:?}");
+        // Thin: twice the radius times the offset, its centroid π/4 of the radius away from
+        // the hole's centre, its outline nearly two half circles.
+        assert!((c.area - 2.0 * r * e).abs() < 1e-6, "{c:?}");
+        assert!((c.centroid + std::f64::consts::FRAC_PI_4 * r).abs() < 1e-2, "{c:?}");
+        assert!((c.perimeter - 2.0 * std::f64::consts::PI * r).abs() < 1e-3, "{c:?}");
+        assert_eq!(crescent(r, r, 0.0).area, 0.0);
+        // A pin with clearance in its hole shares nothing; a pin wider than its hole a ring.
+        assert_eq!(crescent(2.99, 3.0, 0.005).area, 0.0);
+        let ring = crescent(3.01, 3.0, 0.0);
+        assert!((ring.area - std::f64::consts::PI * (3.01f64.powi(2) - 9.0)).abs() < 1e-12, "{ring:?}");
+    }
+
+    /// A 20 x 20 x 5 plate with a hole of radius 3 through it, and a pin of radius `pin` through
+    /// the hole, `offset` off its axis along X.
+    fn plate_and_pin(pin: f64, offset: f64) -> (Body, Body) {
+        let block = crate::box_solid(Vec3::new(-10.0, -10.0, 0.0), Vec3::new(10.0, 10.0, 5.0)).unwrap();
+        let drill = crate::cylinder(Vec3::new(0.0, 0.0, -2.0), Vec3::Z, 3.0, 9.0).unwrap();
+        let plate = crate::boolean(&block, &drill, BoolOp::Cut).unwrap().unwrap();
+        (plate, crate::cylinder(Vec3::new(offset, 0.0, -2.0), Vec3::Z, pin, 9.0).unwrap())
+    }
+
+    /// The pin shares a crescent with the plate over the plate's thickness, in either order.
+    #[test]
+    fn a_pin_off_its_hole_shares_a_crescent() {
+        let (plate, pin) = plate_and_pin(3.0, 0.01);
+        let want = 5.0 * crescent(3.0, 3.0, 0.01).area;
+        for (x, y) in [(&plate, &pin), (&pin, &plate)] {
+            let o = coaxial_overlap(x, y).unwrap();
+            assert!((o.volume - want).abs() < 1e-6 * want, "{o:?} {want}");
+            assert!((o.offset - 0.01).abs() < 1e-9 && (o.length - 5.0).abs() < 1e-9, "{o:?}");
+            // The crescent is on the pin's side away from the hole's axis (+X).
+            assert!(o.centroid.x > 2.0 && o.bbox.min.z > -1e-9 && o.bbox.max.z < 5.0 + 1e-9, "{o:?}");
+        }
+    }
+
+    /// A pin with clearance all round, though off the axis, shares nothing.
+    #[test]
+    fn a_pin_with_clearance_shares_nothing() {
+        let (plate, pin) = plate_and_pin(2.995, 0.003);
+        let o = coaxial_overlap(&plate, &pin).unwrap();
+        assert!(o.volume < 1e-9, "{o:?}");
+    }
+
+    /// A slot through the hole's wall where the crescent would be: the band holds more than the
+    /// two walls, the widened pin says so, and the pair is left to the boolean.
+    #[test]
+    fn a_slot_through_the_band_is_left_to_the_boolean() {
+        let (plate, pin) = plate_and_pin(3.0, 0.01);
+        let slot = crate::box_solid(Vec3::new(2.5, -1.0, -1.0), Vec3::new(4.0, 1.0, 6.0)).unwrap();
+        let slotted = crate::boolean(&plate, &slot, BoolOp::Cut).unwrap().unwrap();
+        assert!(coaxial_overlap(&slotted, &pin).is_none());
+        // Pins on the very axis, or far off it, aren't asked about.
+        assert!(coaxial_overlap(&plate, &plate_and_pin(3.0, 0.0).1).is_none());
+        assert!(coaxial_overlap(&plate, &plate_and_pin(3.0, 0.5).1).is_none());
+    }
+}
diff --git a/crates/kernel/src/lib.rs b/crates/kernel/src/lib.rs
index 814d935a..7775d2b4 100644
--- a/crates/kernel/src/lib.rs
+++ b/crates/kernel/src/lib.rs
@@ -28,6 +28,7 @@ mod boundary;
 mod bridge;
 mod build;
 mod clearance;
+mod coaxial_overlap;
 mod continuity;
 mod coplanar;
 mod corner_g2;
@@ -139,6 +140,7 @@ pub use build::{
     PathSeg, box_solid, canonical_revolve_axis, cylinder, extrude, extrude_tapered, extrude_tapered_between, loft, path_polyline, revolve,
     revolve_whole, sphere, sweep, sweep_with, torus,
 };
+pub use coaxial_overlap::{CoaxialOverlap, coaxial_overlap};
 pub use continuity::{
     ContinuityReport, ContinuityTolerances, EdgePick, MAX_SAMPLES, MIN_SAMPLES, SeamReport, SeamVerdict, SeamWorst, pick_edges, seam_diagnostics,
 };