diff --git a/pyproject.toml b/pyproject.toml
index 3dfefff..35e0b79 100644
--- a/pyproject.toml
+++ b/pyproject.toml
@@ -4,7 +4,7 @@ build-backend = "setuptools.build_meta"
 
 [project]
 name = "qten"
-version = "0.4.1"
+version = "0.4.2"
 description = "Torch-based tensors, lattices, symmetries, and band-structure tools for quantum lattice models."
 readme = "README.md"
 requires-python = ">=3.11"
diff --git a/src/qten/geometries/ops.py b/src/qten/geometries/ops.py
index 2b38068..7fefcb9 100644
--- a/src/qten/geometries/ops.py
+++ b/src/qten/geometries/ops.py
@@ -247,15 +247,14 @@ def get_strip_region_2d(
     origin: Offset[AffineSpace] | Offset[Lattice] | None = None,
 ) -> tuple[Offset[Lattice], ...]:
     r"""
-    Return a 2D rectangular strip region in primitive-strip lattice coordinates.
+    Return a 2D rectangular strip region in strip coordinates induced by `direction`.
 
     This helper is defined only for 2D lattices.
 
     Let `r0` be the supplied [`origin`][qten.geometries.spatials.AffineSpace.origin] (or the lattice origin when omitted).
-    Let \((d_x, d_y)\) be the supplied direction coordinates. Let
-    \(p = (p_x, p_y)\) be the associated primitive integer direction, and let
-    \(n = (-p_y, p_x)\) be the primitive integer normal. `side="lhs"` grows
-    toward positive \(n\) and `side="rhs"` grows toward negative \(n\).
+    Let \((d_x, d_y)\) be the supplied direction coordinates and let
+    \(n = (-d_y, d_x)\) be the rotated transverse direction. `side="lhs"`
+    grows toward positive \(n\) and `side="rhs"` grows toward negative \(n\).
 
     A lattice site belongs to the strip when some periodic image of that site
     satisfies both of the following:
@@ -265,25 +264,21 @@ def get_strip_region_2d(
       \le d_x(r_x-r_{0x}) + d_y(r_y-r_{0y})
       \le (\mathrm{length\_step}-1)(d_x^2+d_y^2)\).
     - Transverse bound:
-      \(0 \le s[-p_y(r_x-r_{0x}) + p_x(r_y-r_{0y})]
-      \le \mathrm{width\_step}-1\).
+      \(0 \le s[-d_y(r_x-r_{0x}) + d_x(r_y-r_{0y})]
+      \le (\mathrm{width\_step}-1)(d_x^2+d_y^2)\).
 
     where \(s = 1\) for `"lhs"` and \(s = -1\) for `"rhs"`.
 
-    For integer directions, \((d_x, d_y) = (p_x, p_y)\). For rational directions,
-    longitudinal shell spacing is computed from the supplied direction
-    `(dx, dy)`, while transverse shelling is computed from the primitive
-    integer direction `p`.
-
     `width_step` counts the transverse shell thickness including the main axis
-    row. `trim_step` is a tail trimmer only: it advances the strip start along
-    the longitudinal axis without affecting the transverse width.
+    row in the rotated, scaled direction induced by `direction`. `trim_step`
+    is a tail trimmer only: it advances the strip start along the
+    longitudinal axis without affecting the transverse width.
 
     Parameters
     ----------
     direction : Offset[Lattice]
-        Non-zero lattice translation on a 2D lattice whose primitive direction
-        defines the strip axis.
+        Non-zero lattice translation on a 2D lattice whose direction defines
+        both the strip axis and the scaled transverse direction.
     length_step : int
         Number of strip shells from the origin along the primitive direction.
     width_step : int
@@ -292,8 +287,8 @@ def get_strip_region_2d(
         Number of longitudinal shells trimmed from the tail near the origin.
     side : Literal["lhs", "rhs"]
         Side on which transverse width shells are accumulated relative to the
-        strip direction. `"lhs"` uses the positive lattice normal and `"rhs"`
-        uses the negative lattice normal.
+        strip direction. `"lhs"` uses the positive rotated direction and
+        `"rhs"` uses the negative one.
     origin : Offset[AffineSpace] | Offset[Lattice] | None
         Anchor point for the strip coordinates. If omitted, the zero offset in
         the lattice space is used. When provided, it is rebased into the
@@ -326,7 +321,7 @@ def get_strip_region_2d(
     if length_step == 0 or width_step == 0:
         return ()
 
-    lattice, dx, dy, px, py = _strip_direction_data(direction)
+    lattice, dx, dy, _px, _py = _strip_direction_data(direction)
     if origin is None:
         origin = lattice.origin()
     if origin.dim != lattice.dim:
@@ -338,13 +333,14 @@ def get_strip_region_2d(
     if len(all_sites) == 0:
         return ()
 
-    normal_x = -py
-    normal_y = px
+    normal_x = -dy
+    normal_y = dx
     normal_sign = 1 if side == "lhs" else -1
-    longitudinal_min = trim_step * (dx * dx + dy * dy)
-    longitudinal_max = (length_step - 1) * (dx * dx + dy * dy)
+    direction_norm_sq = dx * dx + dy * dy
+    longitudinal_min = trim_step * direction_norm_sq
+    longitudinal_max = (length_step - 1) * direction_norm_sq
     transverse_min = 0
-    transverse_max = width_step - 1
+    transverse_max = (width_step - 1) * direction_norm_sq
 
     boundary_basis = np.array(lattice.boundaries.basis.tolist(), dtype=int)
     image_shifts = [
diff --git a/tests/test_geometry_ops.py b/tests/test_geometry_ops.py
index 9845183..1645179 100644
--- a/tests/test_geometry_ops.py
+++ b/tests/test_geometry_ops.py
@@ -239,7 +239,7 @@ def test_get_strip_region_2d_supports_affine_origin():
     )
 
 
-def test_get_strip_region_2d_uses_primitive_direction():
+def test_get_strip_region_2d_inherits_direction_scale_transversely():
     lattice = Lattice(
         basis=ImmutableDenseMatrix.eye(2),
         boundaries=PeriodicBoundary(ImmutableDenseMatrix.diag(8, 8)),
@@ -256,9 +256,12 @@ def test_get_strip_region_2d_uses_primitive_direction():
     assert tuple(tuple(site.rep) for site in region) == (
         (0, 0),
         (0, 1),
+        (0, 2),
         (1, 1),
         (1, 2),
+        (1, 3),
         (2, 2),
+        (7, 1),
     )
 
 
@@ -277,10 +280,13 @@ def test_get_strip_region_2d_matches_diagonal_examples():
     )
     assert set(tuple(site.rep) for site in region) == {
         (0, 0),
-        (1, 1),
-        (2, 2),
         (0, 1),
+        (0, 2),
+        (1, 1),
         (1, 2),
+        (1, 3),
+        (2, 2),
+        (15, 1),
     }
 
     region = get_strip_region_2d(
@@ -291,13 +297,18 @@ def test_get_strip_region_2d_matches_diagonal_examples():
     )
     assert set(tuple(site.rep) for site in region) == {
         (0, 0),
-        (1, 1),
-        (2, 2),
         (0, 1),
-        (1, 2),
-        (15, 1),
         (0, 2),
+        (0, 3),
+        (0, 4),
+        (1, 1),
+        (1, 2),
         (1, 3),
+        (2, 2),
+        (14, 2),
+        (15, 1),
+        (15, 2),
+        (15, 3),
     }
 
     region = get_strip_region_2d(
@@ -309,10 +320,13 @@ def test_get_strip_region_2d_matches_diagonal_examples():
     )
     assert set(tuple(site.rep) for site in region) == {
         (0, 2),
+        (0, 3),
+        (0, 4),
         (1, 1),
         (2, 2),
         (1, 2),
         (1, 3),
+        (15, 3),
     }
 
     region = get_strip_region_2d(
@@ -323,10 +337,13 @@ def test_get_strip_region_2d_matches_diagonal_examples():
     )
     assert set(tuple(site.rep) for site in region) == {
         (0, 0),
-        (1, 1),
-        (2, 2),
         (1, 0),
+        (1, 1),
+        (1, 15),
+        (2, 0),
         (2, 1),
+        (2, 2),
+        (3, 1),
     }