diff --git a/lib/mpl_toolkits/axisartist/grid_finder.py b/lib/mpl_toolkits/axisartist/grid_finder.py
index 2276d28bae90..eb988b14b289 100644
--- a/lib/mpl_toolkits/axisartist/grid_finder.py
+++ b/lib/mpl_toolkits/axisartist/grid_finder.py
@@ -176,24 +176,26 @@ def _clip_grid_lines_and_find_ticks(self, lines, values, levs, bb):
         return gi
 
     def update_transform(self, aux_trans):
-        if isinstance(aux_trans, Transform):
-            def transform_xy(x, y):
-                ll1 = np.column_stack([x, y])
-                ll2 = aux_trans.transform(ll1)
-                lon, lat = ll2[:, 0], ll2[:, 1]
-                return lon, lat
-
-            def inv_transform_xy(x, y):
-                ll1 = np.column_stack([x, y])
-                ll2 = aux_trans.inverted().transform(ll1)
-                lon, lat = ll2[:, 0], ll2[:, 1]
-                return lon, lat
-
+        if not isinstance(aux_trans, Transform) and len(aux_trans) != 2:
+            raise TypeError("'aux_trans' must be either a Transform instance "
+                            "or a pair of callables")
+        self._aux_transform = aux_trans
+
+    def transform_xy(self, x, y):
+        aux_trf = self._aux_transform
+        if isinstance(aux_trf, Transform):
+            return aux_trf.transform(np.column_stack([x, y])).T
         else:
-            transform_xy, inv_transform_xy = aux_trans
+            transform_xy, inv_transform_xy = aux_trf
+            return transform_xy(x, y)
 
-        self.transform_xy = transform_xy
-        self.inv_transform_xy = inv_transform_xy
+    def inv_transform_xy(self, x, y):
+        aux_trf = self._aux_transform
+        if isinstance(aux_trf, Transform):
+            return aux_trf.inverted().transform(np.column_stack([x, y])).T
+        else:
+            transform_xy, inv_transform_xy = aux_trf
+            return inv_transform_xy(x, y)
 
     def update(self, **kw):
         for k in kw: