diff --git a/firedrake/mesh.py b/firedrake/mesh.py
index f0ec0debdf..268b38b348 100644
--- a/firedrake/mesh.py
+++ b/firedrake/mesh.py
@@ -3017,10 +3017,13 @@ def curve_field(self, order, permutation_tol=1e-8, cg_field=None):
         fiat_element = new_coordinates.function_space().finat_element.fiat_equivalent
         nodes = fiat_element.dual_basis()
         ref_pts = []
-        for node in nodes:
-            # Assert singleton point for each node.
-            pt, = node.get_point_dict().keys()
-            ref_pts.append(pt)
+        entity_ids = fiat_element.entity_dofs()
+        for dim in sorted(entity_ids):
+            for entity in sorted(entity_ids[dim]):
+                for i in entity_ids[dim][entity]:
+                    # Assert singleton point for each node.
+                    pt, = nodes[i].get_point_dict().keys()
+                    ref_pts.append(pt)
         reference_points = np.array(ref_pts)
 
         # Construct numpy arrays for physical domain data
@@ -3030,8 +3033,8 @@ def curve_field(self, order, permutation_tol=1e-8, cg_field=None):
         curved_points = np.zeros(
             (ng_dimension, reference_points.shape[0], self.geometric_dimension)
         )
+        self.netgen_mesh.Curve(1)
         self.netgen_mesh.CalcElementMapping(reference_points, physical_points)
-        # NOTE: This will segfault for MeshHierarchy on a netgen CSG geometry
         self.netgen_mesh.Curve(order)
         self.netgen_mesh.CalcElementMapping(reference_points, curved_points)
         curved = ng_element.NumPy()["curved"]
@@ -3056,7 +3059,6 @@ def curve_field(self, order, permutation_tol=1e-8, cg_field=None):
         permutation = find_permutation(
             own_physical_points,
             new_coordinates.dat.data_ro_with_halos[broken_indices].real,
-            tol=permutation_tol,
         )
         self.comm.Barrier()
         # Apply the permutation to each cell in turn
diff --git a/firedrake/netgen.py b/firedrake/netgen.py
index 57b442b83a..178dbb6a6d 100644
--- a/firedrake/netgen.py
+++ b/firedrake/netgen.py
@@ -80,8 +80,7 @@ def netgen_distribute(V: firedrake.functionspaceimpl.WithGeometryBase,
 
 
 @PETSc.Log.EventDecorator()
-def find_permutation(points_a: np.ndarray, points_b: np.ndarray,
-                     tol: float = 1e-5):
+def find_permutation(points_a: np.ndarray, points_b: np.ndarray):
     """ Find all permutations between a list of two sets of points.
 
     Given two numpy arrays of shape (ncells, npoints, dim) containing
@@ -95,7 +94,18 @@ def find_permutation(points_a: np.ndarray, points_b: np.ndarray,
     if points_a.shape != points_b.shape:
         raise ValueError("`points_a` and `points_b` must have the same shape.")
 
-    p = [np.where(cdist(a, b).T < tol)[1] for a, b in zip(points_a, points_b)]
+    dim = points_a.shape[-1]
+    vids = list(range(dim+1))
+
+    bs = points_a[:, vids[0], :]
+    As = points_a[:, vids[1:], :]
+    As -= bs[:, None, :]
+    Ainvs = np.linalg.inv(As)
+
+    ref_points_a = np.matmul(points_a - bs[:, None, :], Ainvs)
+    ref_points_b = np.matmul(points_b - bs[:, None, :], Ainvs)
+
+    p = [np.argmin(cdist(a, b), axis=0) for a, b in zip(ref_points_a, ref_points_b)]
 
     if len(p) == 0:
         return p