diff --git a/crates/geom/src/brep.rs b/crates/geom/src/brep.rs
index d81dd31ea2..ea2738e707 100644
--- a/crates/geom/src/brep.rs
+++ b/crates/geom/src/brep.rs
@@ -15,6 +15,7 @@ pub mod eval;
 pub mod geometry;
 pub mod history;
 pub mod pcurve;
+pub mod placement;
 mod topology;
 pub use curve2::Curve2;
 
diff --git a/crates/geom/src/brep/placement.rs b/crates/geom/src/brep/placement.rs
new file mode 100644
index 0000000000..97d5e816da
--- /dev/null
+++ b/crates/geom/src/brep/placement.rs
@@ -0,0 +1,151 @@
+//! A record moved by a rigid motion: a rotation and a translation. Every point, axis and control
+//! point moves; every parameter, pcurve, tolerance, orientation and the canonical order stay as
+//! they are, because a rigid motion keeps distances, angles and handedness. The moved record is
+//! the same B-rep in other coordinates, exactly. Reflections and scalings change parameters or
+//! geometry kinds; they are a kernel operation, not this.
+use super::{Brep, Curve, Frame, Surface};
+
+/// A rotation (rows of an orthonormal matrix with determinant +1) and a translation:
+/// `p ↦ rotation · p + translation`.
+#[derive(Clone, Copy, Debug, PartialEq)]
+pub struct Rigid {
+    pub rotation: [[f64; 3]; 3],
+    pub translation: [f64; 3],
+}
+
+/// How far the rotation may be from orthonormal (per entry of RᵀR − I) and its determinant from 1.
+const ORTHONORMAL: f64 = 1e-9;
+
+impl Rigid {
+    /// The rigid motion of a column-major affine matrix (columns 0–2 the images of x, y and z,
+    /// column 3 the translation), or `None` when its linear part scales, shears or reflects.
+    pub fn of_matrix(m: &[[f64; 4]; 4]) -> Option<Rigid> {
+        if m.iter().flatten().any(|x| !x.is_finite()) || m[0][3] != 0.0 || m[1][3] != 0.0 || m[2][3] != 0.0 || m[3][3] != 1.0 {
+            return None;
+        }
+        let rotation = [[m[0][0], m[1][0], m[2][0]], [m[0][1], m[1][1], m[2][1]], [m[0][2], m[1][2], m[2][2]]];
+        let column = |j: usize| [rotation[0][j], rotation[1][j], rotation[2][j]];
+        let dot = |a: [f64; 3], b: [f64; 3]| a[0] * b[0] + a[1] * b[1] + a[2] * b[2];
+        for i in 0..3 {
+            for j in 0..3 {
+                let want = if i == j { 1.0 } else { 0.0 };
+                if (dot(column(i), column(j)) - want).abs() > ORTHONORMAL {
+                    return None;
+                }
+            }
+        }
+        let [a, b, c] = rotation;
+        let det = a[0] * (b[1] * c[2] - b[2] * c[1]) - a[1] * (b[0] * c[2] - b[2] * c[0]) + a[2] * (b[0] * c[1] - b[1] * c[0]);
+        ((det - 1.0).abs() <= ORTHONORMAL).then_some(Rigid { rotation, translation: [m[3][0], m[3][1], m[3][2]] })
+    }
+
+    fn direction(&self, d: [f64; 3]) -> [f64; 3] {
+        self.rotation.map(|row| row[0] * d[0] + row[1] * d[1] + row[2] * d[2])
+    }
+
+    fn point(&self, p: [f64; 3]) -> [f64; 3] {
+        let r = self.direction(p);
+        [r[0] + self.translation[0], r[1] + self.translation[1], r[2] + self.translation[2]]
+    }
+
+    /// A homogeneous control point `[x w, y w, z w, w]`: the point moves, its weight stays.
+    fn homogeneous(&self, h: [f64; 4]) -> [f64; 4] {
+        let r = self.direction([h[0], h[1], h[2]]);
+        let w = h[3];
+        [r[0] + w * self.translation[0], r[1] + w * self.translation[1], r[2] + w * self.translation[2], w]
+    }
+
+    /// The column-major affine matrix of this motion.
+    fn matrix(&self) -> [[f64; 4]; 4] {
+        let (r, t) = (self.rotation, self.translation);
+        [[r[0][0], r[1][0], r[2][0], 0.0], [r[0][1], r[1][1], r[2][1], 0.0], [r[0][2], r[1][2], r[2][2], 0.0], [t[0], t[1], t[2], 1.0]]
+    }
+
+    fn frame(&self, f: &Frame) -> Frame {
+        Frame { origin: self.point(f.origin), x_axis: self.direction(f.x_axis), y_axis: self.direction(f.y_axis) }
+    }
+
+    fn curve(&self, c: &Curve) -> Curve {
+        match c {
+            Curve::Line { start, end } => Curve::Line { start: self.point(*start), end: self.point(*end) },
+            Curve::Circle { frame, radius } => Curve::Circle { frame: self.frame(frame), radius: *radius },
+            Curve::Ellipse { frame, radii } => Curve::Ellipse { frame: self.frame(frame), radii: *radii },
+            Curve::Polyline { points } => Curve::Polyline { points: points.iter().map(|p| self.point(*p)).collect() },
+            Curve::Spline { knots, points } => Curve::Spline { knots: knots.clone(), points: points.iter().map(|p| self.point(*p)).collect() },
+            Curve::RationalSpline { knots, homogeneous } => {
+                Curve::RationalSpline { knots: knots.clone(), homogeneous: homogeneous.iter().map(|h| self.homogeneous(*h)).collect() }
+            }
+            Curve::SurfaceIntersection { surfaces, leader, tolerance } => Curve::SurfaceIntersection {
+                surfaces: [Box::new(self.surface(&surfaces[0])), Box::new(self.surface(&surfaces[1]))],
+                leader: Box::new(self.curve(leader)),
+                tolerance: *tolerance,
+            },
+        }
+    }
+
+    fn surface(&self, s: &Surface) -> Surface {
+        match s {
+            Surface::Plane { origin, u_axis, v_axis } => {
+                Surface::Plane { origin: self.point(*origin), u_axis: self.direction(*u_axis), v_axis: self.direction(*v_axis) }
+            }
+            Surface::PlaneThrough { origin, u_point, v_point } => {
+                Surface::PlaneThrough { origin: self.point(*origin), u_point: self.point(*u_point), v_point: self.point(*v_point) }
+            }
+            Surface::Cylinder { frame, radius } => Surface::Cylinder { frame: self.frame(frame), radius: *radius },
+            Surface::Cone { frame, radius_at_origin, half_angle } => {
+                Surface::Cone { frame: self.frame(frame), radius_at_origin: *radius_at_origin, half_angle: *half_angle }
+            }
+            Surface::Sphere { frame, radius } => Surface::Sphere { frame: self.frame(frame), radius: *radius },
+            Surface::Torus { frame, major_radius, minor_radius } => {
+                Surface::Torus { frame: self.frame(frame), major_radius: *major_radius, minor_radius: *minor_radius }
+            }
+            Surface::Spline { u_knots, v_knots, points } => Surface::Spline {
+                u_knots: u_knots.clone(),
+                v_knots: v_knots.clone(),
+                points: points.iter().map(|row| row.iter().map(|p| self.point(*p)).collect()).collect(),
+            },
+            Surface::RationalSpline { u_knots, v_knots, homogeneous } => Surface::RationalSpline {
+                u_knots: u_knots.clone(),
+                v_knots: v_knots.clone(),
+                homogeneous: homogeneous.iter().map(|row| row.iter().map(|h| self.homogeneous(*h)).collect()).collect(),
+            },
+            // The revolution's own placement is composed with the motion; profile and axis stay
+            // where the producer put them.
+            Surface::PlacedRevolution { profile, origin, axis, matrix, forward } => Surface::PlacedRevolution {
+                profile: profile.clone(),
+                origin: *origin,
+                axis: *axis,
+                matrix: compose(&self.matrix(), matrix),
+                forward: *forward,
+            },
+        }
+    }
+}
+
+/// `a · b` of column-major 4×4 matrices.
+fn compose(a: &[[f64; 4]; 4], b: &[[f64; 4]; 4]) -> [[f64; 4]; 4] {
+    let mut out = [[0.0; 4]; 4];
+    for (col, out_col) in out.iter_mut().enumerate() {
+        for (row, cell) in out_col.iter_mut().enumerate() {
+            *cell = (0..4).map(|k| a[k][row] * b[col][k]).sum();
+        }
+    }
+    out
+}
+
+impl Brep {
+    /// This record moved by `motion` (module docs): topology, parameters and pcurves unchanged.
+    pub fn moved(&self, motion: &Rigid) -> Brep {
+        let mut out = self.clone();
+        for f in &mut out.faces {
+            f.surface = motion.surface(&f.surface);
+        }
+        for e in &mut out.edges {
+            e.curve = motion.curve(&e.curve);
+        }
+        for v in &mut out.vertices {
+            v.point = motion.point(v.point);
+        }
+        out
+    }
+}