diff --git a/crates/kernel/src/blend_corner.rs b/crates/kernel/src/blend_corner.rs
index de1da03493..e1d7081ae8 100644
--- a/crates/kernel/src/blend_corner.rs
+++ b/crates/kernel/src/blend_corner.rs
@@ -404,8 +404,8 @@ mod tests {
             assert_eq!(result.unmeshed_faces(), 0);
             assert!(result.validity().is_empty(), "{:?}", result.validity());
             assert!((vol(&result) - vol(&reference)).abs() < 0.05);
-            assert!(!result.contains(Vec3::new(24.9, 0.1, 5.0)));
-            assert!(result.contains(Vec3::new(24.0, 1.0, 5.0)));
+            assert_eq!(result.classify_point(Vec3::new(24.9, 0.1, 5.0)).unwrap(), crate::PointClass::Outside);
+            assert_eq!(result.classify_point(Vec3::new(24.0, 1.0, 5.0)).unwrap(), crate::PointClass::Inside);
         }
     }
 
@@ -424,8 +424,8 @@ mod tests {
             assert!(result.validity().is_empty(), "{:?}", result.validity());
             assert!((vol(&result) - vol(&reference)).abs() < 0.05, "radius {radius}");
             // The old bottom radius is still 2, away from the newly rounded corner.
-            assert!(!result.contains(Vec3::new(12.5, 0.5, 0.5)));
-            assert!(result.contains(Vec3::new(12.5, 1.0, 1.0)));
+            assert_eq!(result.classify_point(Vec3::new(12.5, 0.5, 0.5)).unwrap(), crate::PointClass::Outside);
+            assert_eq!(result.classify_point(Vec3::new(12.5, 1.0, 1.0)).unwrap(), crate::PointClass::Inside);
         }
         // A tangent boundary of an earlier round is not a sharp edge, and cannot be
         // reinterpreted as the sharp edge exposed when that round is taken off.
@@ -470,8 +470,8 @@ mod tests {
                 let result = fillet(&rounded, &crate::tangent_chain(&rounded, selection).unwrap(), 2.0).unwrap();
                 assert!(result.validity().is_empty(), "{:?}", result.validity());
                 assert!((vol(&result) - want).abs() < 1e-6 * want, "{selection:?}: {} vs {want}", vol(&result));
-                assert!(!result.contains(Vec3::new(-4.9, 87.1, 10.0)));
-                assert!(result.contains(Vec3::new(-4.1, 86.3, 10.0)));
+                assert_eq!(result.classify_point(Vec3::new(-4.9, 87.1, 10.0)).unwrap(), crate::PointClass::Outside);
+                assert_eq!(result.classify_point(Vec3::new(-4.1, 86.3, 10.0)).unwrap(), crate::PointClass::Inside);
             }
         }
     }
diff --git a/crates/kernel/src/body.rs b/crates/kernel/src/body.rs
index 49b46bb0fa..44c42974ce 100644
--- a/crates/kernel/src/body.rs
+++ b/crates/kernel/src/body.rs
@@ -422,6 +422,17 @@ pub struct Refusal {
     pub why: String,
 }
 
+/// Where a point lies against a body ([`Body::classify_point`]).
+#[derive(Clone, Copy, Debug, PartialEq, Eq, Hash)]
+pub enum PointClass {
+    /// In the body's material.
+    Inside,
+    /// Outside the material, or in a cavity.
+    Outside,
+    /// Within the body's tolerance of a face, an edge or a vertex.
+    OnBoundary,
+}
+
 /// The kind of surface a face lies on, as a person modelling would name it.
 #[derive(Clone, Copy, Debug, PartialEq, Eq)]
 pub enum SurfaceKind {
@@ -1348,6 +1359,39 @@ impl Body {
         self.mesh(&MeshTolerance::chord(self.size() * 2e-3)).map(|m| m.contains(p)).unwrap_or(false)
     }
 
+    /// Where `p` lies against the body, decided exactly on its faces' surfaces and edges' curves
+    /// (SMLib's point classification at the body's own tolerance, no mesh): on the boundary when
+    /// within that tolerance of a face, an edge or a vertex, else inside its material or outside
+    /// (a cavity is outside). A mesh body's triangles are its exact shape: on its boundary
+    /// within the record's tolerance of a triangle, else inside when rays from the point cross
+    /// it an odd number of times.
+    pub fn classify_point(&self, p: Vec3) -> Result<PointClass> {
+        if !(p.x.is_finite() && p.y.is_finite() && p.z.is_finite()) {
+            return Err(KernelError::Invalid(format!("point classification: the point {p:?} is not finite")));
+        }
+        if let Some(m) = &self.mesh {
+            let mesh = m.to_mesh();
+            let near = mesh
+                .triangles
+                .iter()
+                .filter_map(|t| mesh.tri(t))
+                .any(|[a, b, c]| solidcraft_geom::compare::closest_on_triangle(p, a, b, c).dist(p) <= self.record.tolerance);
+            return Ok(if near {
+                PointClass::OnBoundary
+            } else if mesh.contains(p) {
+                PointClass::Inside
+            } else {
+                PointClass::Outside
+            });
+        }
+        let class = self.smlib()?.classify_point([p.x, p.y, p.z]).map_err(|e| KernelError::Failed(format!("SMLib point classification: {e}")))?;
+        Ok(match class {
+            solidcraft_smlib::PointClass::Inside => PointClass::Inside,
+            solidcraft_smlib::PointClass::Outside => PointClass::Outside,
+            solidcraft_smlib::PointClass::OnBoundary => PointClass::OnBoundary,
+        })
+    }
+
     /// B-rep faces (a mesh body is one face; [`crate::measure`] counts its triangles).
     pub fn face_count(&self) -> usize {
         if self.mesh.is_some() { 1 } else { self.record.faces.len() }
diff --git a/crates/kernel/src/body_lump_tests.rs b/crates/kernel/src/body_lump_tests.rs
index b8ac628bc0..95db09a81f 100644
--- a/crates/kernel/src/body_lump_tests.rs
+++ b/crates/kernel/src/body_lump_tests.rs
@@ -259,3 +259,52 @@ fn a_long_thin_cylinder_comes_out_outward() {
         assert!((volume - want).abs() < want * 0.02, "{lo} {seam}: {volume} vs {want}");
     }
 }
+
+/// Exact point classification: a box's inside, outside, faces, edges and corners; a cylinder's
+/// curved wall within its tolerance (on it), 5 µm inside it (inside: the old mesh test, at a
+/// chord of 2e-3 of the size, missed this band) and just beyond; a cavity is outside.
+#[test]
+fn points_classify_exactly_against_the_body() {
+    use crate::{PointClass::*, cylinder};
+    let b = box_solid(Vec3::ZERO, Vec3::new(10.0, 20.0, 30.0)).unwrap();
+    for (p, want) in [
+        (Vec3::new(5.0, 10.0, 15.0), Inside),
+        (Vec3::new(5.0, 10.0, 31.0), Outside),
+        (Vec3::new(5.0, 10.0, 30.0 + 1e-4), Outside),
+        (Vec3::new(5.0, 10.0, 30.0 - 1e-4), Inside),
+        (Vec3::new(5.0, 10.0, 30.0), OnBoundary),
+        (Vec3::new(10.0, 10.0, 30.0), OnBoundary),
+        (Vec3::new(10.0, 20.0, 30.0), OnBoundary),
+        (Vec3::ZERO, OnBoundary),
+    ] {
+        assert_eq!(b.classify_point(p).unwrap(), want, "{p:?}");
+    }
+    assert!(b.classify_point(Vec3::new(f64::NAN, 0.0, 0.0)).is_err());
+
+    let c = cylinder(Vec3::ZERO, Vec3::Z, 5.0, 10.0).unwrap();
+    let tol = c.record().tolerance;
+    let at = |r: f64| Vec3::new(r * std::f64::consts::FRAC_1_SQRT_2, r * std::f64::consts::FRAC_1_SQRT_2, 5.0);
+    assert_eq!(c.classify_point(at(5.0 + 0.5 * tol)).unwrap(), OnBoundary);
+    assert_eq!(c.classify_point(at(5.0 - 0.5 * tol)).unwrap(), OnBoundary);
+    assert_eq!(c.classify_point(at(5.0 - 5e-3)).unwrap(), Inside);
+    assert_eq!(c.classify_point(at(5.0 + 1e-4)).unwrap(), Outside);
+    assert_eq!(c.classify_point(Vec3::new(0.0, 0.0, 10.0)).unwrap(), OnBoundary);
+
+    let hollow = boolean(&cube(50.0), &cube(40.0), BoolOp::Cut).unwrap().unwrap();
+    assert_eq!(hollow.classify_point(Vec3::ZERO).unwrap(), Outside, "the cavity");
+    assert_eq!(hollow.classify_point(Vec3::X * 45.0).unwrap(), Inside);
+    assert_eq!(hollow.classify_point(Vec3::X * 40.0).unwrap(), OnBoundary, "the cavity's wall");
+}
+
+/// A mesh body's triangles are its shape: on them within the record's tolerance, else inside or
+/// outside by the rays' crossings.
+#[test]
+fn points_classify_against_a_mesh_body() {
+    use crate::PointClass::*;
+    let p = [Vec3::ZERO, Vec3::X * 10.0, Vec3::Y * 10.0, Vec3::Z * 10.0];
+    let m = crate::mesh_body(&p, &[[0, 2, 1], [0, 1, 3], [0, 3, 2], [1, 2, 3]]).unwrap();
+    assert_eq!(m.classify_point(Vec3::new(1.0, 1.0, 1.0)).unwrap(), Inside);
+    assert_eq!(m.classify_point(Vec3::new(5.0, 5.0, 5.0)).unwrap(), Outside);
+    assert_eq!(m.classify_point(Vec3::new(1.0, 1.0, 0.0)).unwrap(), OnBoundary);
+    assert_eq!(m.classify_point(Vec3::X * 10.0).unwrap(), OnBoundary);
+}
diff --git a/crates/kernel/src/delete_face.rs b/crates/kernel/src/delete_face.rs
index 5a8461f4b3..d4655d0475 100644
--- a/crates/kernel/src/delete_face.rs
+++ b/crates/kernel/src/delete_face.rs
@@ -358,7 +358,7 @@ mod tests {
         assert_eq!(restored.face_count(), 6);
         assert_eq!(restored.unmeshed_faces(), 0);
         assert!((volume(&restored) - 6250.0).abs() < 1e-6);
-        assert!(restored.contains(Vec3::new(0.1, 0.1, 0.1)));
+        assert_eq!(restored.classify_point(Vec3::new(0.1, 0.1, 0.1)).unwrap(), crate::PointClass::Inside);
     }
 
     #[test]
diff --git a/crates/kernel/src/lib.rs b/crates/kernel/src/lib.rs
index 7821808a1e..6c457444ce 100644
--- a/crates/kernel/src/lib.rs
+++ b/crates/kernel/src/lib.rs
@@ -162,7 +162,8 @@ pub use blend::{
 pub use blend_corner::fillet_radii;
 pub use blend_variable::{RadiusSpline, Taper, fillet_chord, fillet_tapered, fillet_variable, fillet_variable_points, fillet_variable_spline, taper};
 pub use body::{
-    Body, CapSignature, DISPLAY_ANGLE, DisplayShare, EdgeInfo, FaceInfo, FacePaint, FaceSource, MeshTolerance, Paint, Refusal, SurfaceKind,
+    Body, CapSignature, DISPLAY_ANGLE, DisplayShare, EdgeInfo, FaceInfo, FacePaint, FaceSource, MeshTolerance, Paint, PointClass, Refusal,
+    SurfaceKind,
 };
 pub use boundary::{Boundary, boundary_patch};
 pub use boundary_network::{BoundaryCurve, CurveSource, MAX_CHAIN_CURVES, MAX_EDGE_PICKS, boundary_join, boundary_surface};
diff --git a/crates/kernel/src/sphere_lens_tests.rs b/crates/kernel/src/sphere_lens_tests.rs
index b44a7f1aab..d12e7a3422 100644
--- a/crates/kernel/src/sphere_lens_tests.rs
+++ b/crates/kernel/src/sphere_lens_tests.rs
@@ -48,8 +48,8 @@ fn projected_spherical_outline_does_not_block_a_lens_cut() {
     let measured = crate::measure(&result).unwrap();
     assert!((measured.volume - 8810.08973246).abs() < 2., "{measured:?}");
     assert!(measured.bbox.max.z < 20.01, "no thin lid may remain above the cut: {measured:?}");
-    assert!(result.contains(Vec3::new(0., -10., 2.)));
-    assert!(!result.contains(Vec3::new(0., -10., 15.)));
+    assert_eq!(result.classify_point(Vec3::new(0., -10., 2.)).unwrap(), crate::PointClass::Inside);
+    assert_eq!(result.classify_point(Vec3::new(0., -10., 15.)).unwrap(), crate::PointClass::Outside);
     assert_eq!(measured.merged.face_types.get("cylinder"), Some(&1));
     let history = result.history().expect("the cut has a face history");
     history.check(&result, &[&target, &cutter]).unwrap();
diff --git a/crates/kernel/tests/chamfer_curved_holder.rs b/crates/kernel/tests/chamfer_curved_holder.rs
index 96196bb205..e4fb8e2274 100644
--- a/crates/kernel/tests/chamfer_curved_holder.rs
+++ b/crates/kernel/tests/chamfer_curved_holder.rs
@@ -73,9 +73,9 @@ fn chamfer_the_top_lip_at_a_tangent_wall_junction() {
     let expected = expected_removed(2.0);
     eprintln!("removed {removed:.8} mm³, expected {expected:.8} mm³");
     assert!((removed - expected).abs() < 0.03, "removed {removed}, expected {expected}");
-    assert!(!made.contains(Vec3::new(10.0, 9.5, 59.5)), "the old corner was removed");
-    assert!(made.contains(Vec3::new(10.0, 7.0, 59.5)), "material behind the bevel remains");
-    assert!(made.contains(Vec3::new(10.0, 10.0, 55.0)), "material below the bevel remains");
+    assert_eq!(made.classify_point(Vec3::new(10.0, 9.5, 59.5)).unwrap(), solidcraft_kernel::PointClass::Outside, "the old corner was removed");
+    assert_eq!(made.classify_point(Vec3::new(10.0, 7.0, 59.5)).unwrap(), solidcraft_kernel::PointClass::Inside, "material behind the bevel remains");
+    assert_eq!(made.classify_point(Vec3::new(10.0, 10.0, 55.0)).unwrap(), solidcraft_kernel::PointClass::Inside, "material below the bevel remains");
     // A closed mesh's signed volume is independent of the origin.
     let volume_about = |origin: Vec3| {
         after
diff --git a/crates/kernel/tests/shell_barrel.rs b/crates/kernel/tests/shell_barrel.rs
index fc7422ae94..d424c71c8a 100644
--- a/crates/kernel/tests/shell_barrel.rs
+++ b/crates/kernel/tests/shell_barrel.rs
@@ -54,7 +54,10 @@ fn shell_of_a_revolved_barrel() {
     assert!((volume - full).abs() < full * 1e-3, "{volume} vs {full}");
     for turn in [0.0, 1.0, 2.5, 4.0] {
         let (c, s) = (f64::cos(turn), f64::sin(turn));
-        assert!(barrel.contains(Vec3::new(13.8 * c, 13.8 * s, HALF_HEIGHT)) && !barrel.contains(Vec3::new(14.2 * c, 14.2 * s, HALF_HEIGHT)));
+        assert!(
+            barrel.classify_point(Vec3::new(13.8 * c, 13.8 * s, HALF_HEIGHT)).unwrap() == solidcraft_kernel::PointClass::Inside
+                && barrel.classify_point(Vec3::new(14.2 * c, 14.2 * s, HALF_HEIGHT)).unwrap() == solidcraft_kernel::PointClass::Outside
+        );
     }
     let result = shell(&barrel, &[Vec3::new(0.0, 0.0, 2.0 * HALF_HEIGHT)], 2.0).unwrap();
     assert!(result.validity().is_empty(), "{:?}", result.validity());