diff --git a/graph.cpp b/graph.cpp
index 6ff2927f..77e76211 100644
--- a/graph.cpp
+++ b/graph.cpp
@@ -5042,11 +5042,11 @@ EX ld wall_radar(cell *c, transmatrix T, transmatrix LPe, ld max) {
   ld step = max / 20;
   ld fixed_yshift = 0;
   for(int i=0; i<20; i++) {
-    T = parallel_transport(T, ori, ztangent(-step));
+    T = shift_object(T, ori, ztangent(-step), shift_method(false));
     virtualRebase(c, T);
     color_t col;
     if(isWall3(c, col) || (WDIM == 2 && GDIM == 3 && tC0(T)[2] > cgi.FLOOR)) { 
-      T = parallel_transport(T, ori, ztangent(step));
+      T = shift_object(T, ori, ztangent(step), shift_method(false));
       step /= 2; i = 17; 
       if(step < 1e-3) break; 
       }
@@ -5126,7 +5126,7 @@ EX void precise_mouseover() {
     ld step = 0.2;
     cell *c = centerover;
     for(int i=0; i<100; i++) {
-      apply_parallel_transport(T, ori, ztangent(step));
+      apply_shift_object(T, ori, ztangent(step));
       int pd = through_wall(c, T * C0);
       if(pd != -1) {
         color_t col;
diff --git a/hyperpoint.cpp b/hyperpoint.cpp
index 88efd92d..89b8f787 100644
--- a/hyperpoint.cpp
+++ b/hyperpoint.cpp
@@ -1646,17 +1646,27 @@ EX eShiftMethod shift_method(bool automatic IS(false)) {
   return smGeodesic;
   }
 
-EX transmatrix parallel_transport(const transmatrix Position, const transmatrix& ori, const hyperpoint direction) {
-  if(nonisotropic) return nisot::parallel_transport(Position, direction);
-  else if(gproduct) {
-    hyperpoint h = product::direct_exp(ori * direction);
-    return Position * rgpushxto0(h);
+EX transmatrix shift_object(const transmatrix Position, const transmatrix& ori, const hyperpoint direction, eShiftMethod sm IS(shift_method(true))) {
+  switch(sm) {
+    case smGeodesic:
+      return nisot::parallel_transport(Position, direction);
+    case smLie:
+      return nisot::lie_transport(Position, direction);
+    case smProduct: {
+      hyperpoint h = product::direct_exp(ori * direction);
+      return Position * rgpushxto0(h);
+      }
+    case smIsometric: {
+      return Position * rgpushxto0(direct_exp(direction));
+      }
+    case smEmbedded: {
+      throw hr_exception("not implemented");
+      }
     }
-  else return Position * rgpushxto0(direct_exp(direction));
   }
 
-EX void apply_parallel_transport(transmatrix& Position, const transmatrix orientation, const hyperpoint direction) {
-  Position = parallel_transport(Position, orientation, direction);
+EX void apply_shift_object(transmatrix& Position, const transmatrix orientation, const hyperpoint direction, eShiftMethod sm IS(shift_method(true))) {
+  Position = shift_object(Position, orientation, direction, sm);
   }
 
 EX void rotate_object(transmatrix& Position, transmatrix& orientation, transmatrix R) {
diff --git a/nonisotropic.cpp b/nonisotropic.cpp
index d99b7cee..6eaf57a2 100644
--- a/nonisotropic.cpp
+++ b/nonisotropic.cpp
@@ -2846,10 +2846,16 @@ EX namespace nisot {
   EX transmatrix parallel_transport(const transmatrix Position, const hyperpoint direction) {
     auto P = Position;
     nisot::fixmatrix(P);  
-    if(!geodesic_movement) return eupush(Position * translate(-direction) * inverse(Position) * C0, -1) * Position;
     return parallel_transport_bare(P, direction);
     }
-  
+
+  EX transmatrix lie_transport(const transmatrix Position, const hyperpoint direction) {
+    transmatrix pshift = eupush( tC0(Position) );
+    transmatrix irot = iso_inverse(pshift) * Position;
+    hyperpoint tH = lie_exp(irot * direction);
+    return pshift * eupush(tH) * irot;
+    }
+
   EX transmatrix spin_towards(const transmatrix Position, const hyperpoint goal, flagtype prec IS(pNORMAL)) {
 
     hyperpoint at = tC0(Position);
diff --git a/shmup.cpp b/shmup.cpp
index f3eda845..b8279947 100644
--- a/shmup.cpp
+++ b/shmup.cpp
@@ -1059,7 +1059,7 @@ void movePlayer(monster *m, int delta) {
       playergoturn[cpid] = 0;
       if(igo) { go = false; break; }
       ld r = hypot_d(WDIM, avg_inertia);
-      apply_parallel_transport(nat.T, m->ori, lrspintox(avg_inertia) * xtangent(r * delta));
+      apply_shift_object(nat.T, m->ori, lrspintox(avg_inertia) * xtangent(r * delta));
       if(WDIM == 3) rotate_object(nat.T, m->ori, cspin(0, 2, playerturn[cpid]) * cspin(1, 2, playerturny[cpid]));
       m->vel = r * (600/SCALE);
       }
@@ -1070,7 +1070,7 @@ void movePlayer(monster *m, int delta) {
         playersmallspin[cpid] = cspin(0, 1, fspin) * cspin(2, 0, igospan[igo]);
         if(fspin < 360) igo--; else fspin = 0;
         }
-      nat.T = parallel_transport(nat1.T, m->ori, playersmallspin[cpid] * point3(playerstrafe[cpid], 0, playergo[cpid]));
+      nat.T = shift_object(nat1.T, m->ori, playersmallspin[cpid] * point3(playerstrafe[cpid], 0, playergo[cpid]));
       rotate_object(nat.T, m->ori, cspin(0, 2, playerturn[cpid]) * cspin(1, 2, playerturny[cpid]));
       m->inertia[0] = playerstrafe[cpid] / delta;
       m->inertia[1] = 0;
@@ -1078,7 +1078,7 @@ void movePlayer(monster *m, int delta) {
       }
     else if(playergo[cpid]) {
       playergoturn[cpid] = igospan[igo]+godir[cpid];    
-      nat.T = parallel_transport(nat1.T, m->ori, spin(playergoturn[cpid]) * xtangent(playergo[cpid]));
+      nat.T = shift_object(nat1.T, m->ori, spin(playergoturn[cpid]) * xtangent(playergo[cpid]));
       m->inertia = spin(playergoturn[cpid]) * xtangent(playergo[cpid] / delta);
       }
     
@@ -1168,7 +1168,7 @@ void movePlayer(monster *m, int delta) {
         int i0 = i;
         for(int a=0; a<3; a++) v[a] = (i0 % 3) - 1, i0 /= 3;
         v = v * .1 / hypot_d(3, v);
-        shiftmatrix T1 = (i == 13) ? nat : shiftless(parallel_transport(nat.T, m->ori, v), nat.shift);
+        shiftmatrix T1 = (i == 13) ? nat : shiftless(shift_object(nat.T, m->ori, v), nat.shift);
         cell *c3 = c2;
         while(true) {
           cell *c4 = findbaseAround(tC0(T1), c3, 1);
@@ -1478,7 +1478,7 @@ void moveMimic(monster *m) {
   // no need to care about Mirror images, as they already have their 'at' matrix reversed :|
 
   if(WDIM == 3) {
-    nat.T = parallel_transport(nat.T, m->ori, playersmallspin[cpid] * point3(playerstrafe[cpid], 0, playergo[cpid]));
+    nat.T = shift_object(nat.T, m->ori, playersmallspin[cpid] * point3(playerstrafe[cpid], 0, playergo[cpid]));
     rotate_object(nat.T, m->ori, cspin(0, 2, playerturn[cpid]) * cspin(1, 2, playerturny[cpid]));
     }
   else
@@ -1711,10 +1711,10 @@ void moveBullet(monster *m, int delta) {
     m->dead = true;
 
   if(inertia_based) {
-    nat.T = parallel_transport(nat.T, m->ori, m->inertia * delta);
+    nat.T = shift_object(nat.T, m->ori, m->inertia * delta);
     }
   else 
-    nat.T = parallel_transport(nat.T, m->ori, fronttangent(delta * SCALE * m->vel / speedfactor()));
+    nat.T = shift_object(nat.T, m->ori, fronttangent(delta * SCALE * m->vel / speedfactor()));
   cell *c2 = m->findbase(nat, fake::split() ? 10 : 1);
 
   if(m->parent && isPlayer(m->parent) && markOrb(itOrbLava) && c2 != m->base && !isPlayerOn(m->base)) 
@@ -2217,14 +2217,14 @@ void moveMonster(monster *m, int delta) {
   
   if(inertia_based) {
     if(igo) return;
-    nat.T = parallel_transport(nat.T, m->ori, m->inertia * delta);
+    nat.T = shift_object(nat.T, m->ori, m->inertia * delta);
     }
   else if(WDIM == 3 && igo) {
     ld fspin = rand() % 1000;  
-    nat.T = parallel_transport(nat0.T, m->ori, cspin(1,2,fspin) * spin(igospan[igo]) * xtangent(step));
+    nat.T = shift_object(nat0.T, m->ori, cspin(1,2,fspin) * spin(igospan[igo]) * xtangent(step));
     }
   else {
-    nat.T = parallel_transport(nat0.T, m->ori, spin(igospan[igo]) * xtangent(step));
+    nat.T = shift_object(nat0.T, m->ori, spin(igospan[igo]) * xtangent(step));
     }
 
   if(m->type != moRagingBull && !peace::on)