diff --git a/geometry.cpp b/geometry.cpp
index 23eae267..5db944f0 100644
--- a/geometry.cpp
+++ b/geometry.cpp
@@ -206,6 +206,12 @@ struct geometry_information {
   /* convert the tangent space in actual coordinates to logical coordinates */
   transmatrix actual_to_logical;
 
+  /* convert the tangent space in logical coordinates to actual coordinates */
+  transmatrix logical_to_actual_units;
+
+  /* convert the tangent space in actual coordinates to logical coordinates */
+  transmatrix actual_to_logical_units;
+
   /** size of the Sword (from Orb of the Sword), used in the shmup mode */
   ld sword_size;
   /** scale factor for the graphics of most things*/
@@ -591,15 +597,18 @@ void geometry_information::prepare_lta() {
   if(b) geom3::light_flip(false);
   lta = Id;
   if(embedded_plane) {
-    if(geom3::euc_in_noniso()) {
-      lta[0][0] *= geom3::euclid_embed_scale;
-      lta[1][1] *= geom3::euclid_embed_scale * geom3::euclid_embed_scale_y;
-      lta = cspin(0, 1, geom3::euclid_embed_rotate * degree) * lta;
-      }
     if(geom3::euc_vertical()) lta = cspin90(2, 1) * lta;
     if(geom3::hyp_in_solnih()) lta = cspin90(0, 1) * cspin90(1, 2) * cspin90(0, 1) * lta;
     }
+  logical_to_actual_units = lta;
+  if(geom3::euc_in_noniso()) {
+    lta = Id;
+    lta[0][0] *= geom3::euclid_embed_scale;
+    lta[1][1] *= geom3::euclid_embed_scale * geom3::euclid_embed_scale_y;
+    lta = cspin(0, 1, geom3::euclid_embed_rotate * degree) * lta * logical_to_actual_units;
+    }
   actual_to_logical = inverse(lta);
+  actual_to_logical_units = inverse(logical_to_actual_units);
   if(b) geom3::light_flip(true);
   }
 
@@ -1142,8 +1151,8 @@ EX namespace geom3 {
   EX ld euclid_embed_scale_mean() { return euclid_embed_scale * sqrt(euclid_embed_scale_y); }
   EX void set_euclid_embed_scale(ld x) { euclid_embed_scale = x; euclid_embed_scale_y = 1; euclid_embed_rotate = 0; }
 
-  EX bool supports_flat() { return spatial_embedding == seDefault; }
-  EX bool supports_invert() { return among(spatial_embedding, seDefault, seLowerCurvature, seMuchLowerCurvature, seNil, seSol, seNIH, seSolN); }
+  EX bool supports_flat() { return among(spatial_embedding, seDefault, seProductH, seProductS); }
+  EX bool supports_invert() { return among(spatial_embedding, seDefault, seLowerCurvature, seMuchLowerCurvature, seNil, seSol, seNIH, seSolN, seProductH, seProductS); }
 
   EX vector<geometryinfo> ginf_backup;
 
@@ -1382,7 +1391,7 @@ EX void switch_always3() {
           vid.depth = 0.5;
           }
         }
-      if(spatial_embedding == seDefault && flat_embedding) {
+      if(supports_flat() && flat_embedding) {
         vid.eye += vid.depth / 2;
         vid.depth = 0;
         }
diff --git a/graph.cpp b/graph.cpp
index ddd39579..52e6c3f5 100644
--- a/graph.cpp
+++ b/graph.cpp
@@ -5101,13 +5101,13 @@ EX void make_actual_view() {
       transmatrix T = actual_view_transform * View;
       ld z = -tC0(view_inverse(T)) [2];
       transmatrix R = actual_view_transform;
-      R = cgi.logical_to_actual * R;
+      R = cgi.logical_to_actual_units * R;
       if(R[1][2] || R[2][2])
         R = cspin(1, 2, -atan2(R[1][2], R[2][2])) * R;
       if(R[0][2] || R[2][2])
         R = cspin(0, 2, -atan2(R[0][2], R[2][2])) * R;
       if(geom3::hyp_in_solnih()) R = Id;
-      R = cgi.actual_to_logical * R;
+      R = cgi.actual_to_logical_units * R;
       current_display->radar_transform = inverse(R) * zpush(-z);
       }
     else if(gproduct) {
diff --git a/hyperpoint.cpp b/hyperpoint.cpp
index 248fceac..796b5262 100644
--- a/hyperpoint.cpp
+++ b/hyperpoint.cpp
@@ -675,7 +675,7 @@ EX transmatrix spin(ld alpha) {
   }
 
 EX transmatrix unswap_spin(transmatrix T) {
-  return cgi.actual_to_logical * T * cgi.logical_to_actual;
+  return cgi.actual_to_logical_units * T * cgi.logical_to_actual_units;
   }
 
 /** rotate by 90 degrees in the XY plane */
diff --git a/hypgraph.cpp b/hypgraph.cpp
index b2421276..b13a365f 100644
--- a/hypgraph.cpp
+++ b/hypgraph.cpp
@@ -2036,11 +2036,11 @@ EX void adjust_eye(transmatrix& T, cell *c, ld sign) {
 
 /** achieve top-down perspective */
 EX transmatrix default_spin() {
-  return cspin90(0, 1) * cgi.actual_to_logical;
+  return cspin90(0, 1) * cgi.actual_to_logical_units;
   }
 
 EX void centerpc(ld aspd) {
-  
+
   if(subscreens::split([=] () {centerpc(aspd);})) return;
   if(dual::split([=] () { centerpc(aspd); })) return;
 
@@ -2207,12 +2207,6 @@ void ballgeometry() {
   queuereset(pmodel, PPR::CIRCLE);
   }
 
-EX transmatrix logical_to_actual_units() {
-  transmatrix T = cgi.logical_to_actual;
-  for(int i=0; i<3; i++) set_column(T, i, get_column(T, i) / hypot_d(3, get_column(T, i)));
-  return T;
-  }
-
 EX void resetview() {
   DEBBI(DF_GRAPH, ("reset view"));
   // EUCLIDEAN
@@ -2245,7 +2239,7 @@ EX void resetview() {
 
   if(WDIM == 2) vo = spin(M_PI + vid.fixed_facing_dir * degree) * vo;
   if(WDIM == 3) vo = cspin90(0, 2) * vo;
-  vo = inverse(logical_to_actual_units()) * vo;
+  vo = cgi.actual_to_logical_units * vo;
   if(embedded_plane) vo = cspin90(1, 2) * vo;
   if(embedded_plane && vid.wall_height < 0) vo = cspin180(0, 1) * vo;