diff --git a/graph.cpp b/graph.cpp
index c05d2046..9d7d6d09 100644
--- a/graph.cpp
+++ b/graph.cpp
@@ -810,7 +810,8 @@ EX transmatrix face_the_player(const transmatrix V) {
   if(GDIM == 2) return V;
   if(sl2) return V * zpush(cos(ptick(750)) * cgi.plevel / 16);
   if(prod) return mscale(V, cos(ptick(750)) * cgi.plevel / 16);
-  if(nonisotropic) return spin_towards(V, C0, 2, 0);
+  transmatrix dummy; /* used only in prod anyways */
+  if(nonisotropic) return spin_towards(V, dummy, C0, 2, 0);
   return rgpushxto0(tC0(V));
   }
 
@@ -7182,14 +7183,16 @@ EX transmatrix actual_view_transform;
 
 EX ld wall_radar(cell *c, transmatrix T, transmatrix LPe, ld max) {
   if(!in_perspective() || !vid.use_wall_radar) return max;
+  transmatrix ori;
+  if(prod) ori = inverse(LPe);
   ld step = max / 20;
   ld fixed_yshift = 0;
   for(int i=0; i<20; i++) {
-    T = solmul_pt(T, LPe, ztangent(-step));
+    T = parallel_transport(T, ori, ztangent(-step));
     virtualRebase(c, T, true);
     color_t col;
     if(isWall3(c, col) || (WDIM == 2 && GDIM == 3 && tC0(T)[2] > cgi.FLOOR)) { 
-      T = solmul_pt(T, LPe, ztangent(step));
+      T = parallel_transport(T, ori, ztangent(step));
       step /= 2; i = 17; 
       if(step < 1e-3) break; 
       }
diff --git a/hyperpoint.cpp b/hyperpoint.cpp
index 3aea3cd1..6dac36e2 100644
--- a/hyperpoint.cpp
+++ b/hyperpoint.cpp
@@ -239,11 +239,13 @@ EX ld volume_auto(ld r) {
     }
   }
 
+EX ld asin_clamp(ld x) { return x>1 ? M_PI/2 : x<-1 ? -M_PI/2 : std::isnan(x) ? 0 : asin(x); }
+
 EX ld asin_auto_clamp(ld x) {
   switch(cgclass) {
     case gcEuclid: return x;
     case gcHyperbolic: return asinh(x);
-    case gcSphere: return x>1 ? M_PI/2 : x<-1 ? -M_PI/2 : std::isnan(x) ? 0 : asin(x);
+    case gcSphere: return asin_clamp(x);
     default: return x;
     }
   }
@@ -980,21 +982,43 @@ bool asign(ld y1, ld y2) { return signum(y1) != signum(y2); }
 
 ld xcross(ld x1, ld y1, ld x2, ld y2) { return x1 + (x2 - x1) * y1 / (y1 - y2); }
 
-EX transmatrix solmul_pt(const transmatrix Position, const transmatrix T) {
-  if(nonisotropic) return nisot::parallel_transport(Position, Id, tC0(T));
-  else return Position * T;
+EX transmatrix parallel_transport(const transmatrix Position, const transmatrix& ori, const hyperpoint direction) {
+  if(nonisotropic) return nisot::parallel_transport(Position, direction);
+  else if(prod) {
+    hyperpoint h = product::direct_exp(ori * direction);
+    return Position * rgpushxto0(h);
+    }
+  else return Position * rgpushxto0(direct_exp(direction, 100));
   }
 
-EX transmatrix solmul_pt(const transmatrix Position, const transmatrix LPe, const hyperpoint h) {
-  if(nonisotropic || prod) return nisot::parallel_transport(Position, LPe, h);
-  else return Position * rgpushxto0(direct_exp(h, 100));
+EX void apply_parallel_transport(transmatrix& Position, const transmatrix orientation, const hyperpoint direction) {
+  Position = parallel_transport(Position, orientation, direction);
   }
 
-EX transmatrix spin_towards(const transmatrix Position, const hyperpoint goal, int dir, int back) {
-  transmatrix T = 
-    nonisotropic ? nisot::spin_towards(Position, goal) : 
-    rspintox(inverse(Position) * goal);
-  if(back < 0) T = T * spin(M_PI);
+EX void rotate_object(transmatrix& Position, transmatrix& orientation, transmatrix R) {
+  if(prod) orientation = orientation * R;
+  else Position = Position * R;
+  }
+
+EX transmatrix spin_towards(const transmatrix Position, transmatrix& ori, const hyperpoint goal, int dir, int back) {
+  transmatrix T;
+  ld alpha = 0;
+  if(nonisotropic)
+    T = nisot::spin_towards(Position, goal);
+  else { 
+    hyperpoint U = inverse(Position) * goal;
+    if(prod) {
+      hyperpoint h = product::inverse_exp(U);
+      alpha = asin_clamp(h[2] / hypot_d(3, h));
+      U = product_decompose(U).second;
+      }
+    T = rspintox(U);
+    }
+  if(back < 0) T = T * spin(M_PI), alpha = -alpha;
+  if(prod) {
+    if(dir == 0) ori = cspin(2, 0, alpha);
+    if(dir == 2) ori = cspin(2, 0, alpha - M_PI/2), dir = 0;
+    }
   if(dir) T = T * cspin(dir, 0, -M_PI/2);
   T = Position * T;
   return T;
diff --git a/hypgraph.cpp b/hypgraph.cpp
index 8833081d..3275dfdf 100644
--- a/hypgraph.cpp
+++ b/hypgraph.cpp
@@ -1315,6 +1315,7 @@ EX void centerpc(ld aspd) {
     int sl = snakelevel(cwt.at);
     if(sl && WDIM == 2) T = T * zpush(cgi.SLEV[sl] - cgi.FLOOR);
     View = inverse(T);
+    if(prod) nisot::local_perspective = inverse(pc->ori);
     if(WDIM == 2) rotate_view( cspin(0, 1, M_PI) * cspin(2, 1, M_PI/2 + shmup::playerturny[id]) * spin(-M_PI/2) );
     return;
     }
@@ -2086,7 +2087,7 @@ EX transmatrix get_shift_view_of(const hyperpoint H, const transmatrix V) {
     return V1 * eupush(IV * eupush(H) * V1 * C0);
     }
   else {
-    return inverse(nisot::parallel_transport(inverse(V), Id, -H));
+    return inverse(nisot::parallel_transport(inverse(V), -H));
     }
   }
 
diff --git a/nonisotropic.cpp b/nonisotropic.cpp
index df3dfe00..cd05b1ca 100644
--- a/nonisotropic.cpp
+++ b/nonisotropic.cpp
@@ -1193,15 +1193,11 @@ EX namespace nisot {
     T = push * gtl;
     }
 
-  EX transmatrix parallel_transport(const transmatrix Position, const transmatrix LPe, const hyperpoint h) {
-    if(prod) {
-      hyperpoint h = product::direct_exp(inverse(LPe) * product::inverse_exp(h));
-      return Position * rgpushxto0(h);
-      }
+  EX transmatrix parallel_transport(const transmatrix Position, const hyperpoint direction) {
     auto P = Position;
     nisot::fixmatrix(P);  
-    if(!geodesic_movement) return inverse(eupush(Position * translate(-h) * inverse(Position) * C0)) * Position;
-    return parallel_transport_bare(P, h);
+    if(!geodesic_movement) return inverse(eupush(Position * translate(-direction) * inverse(Position) * C0)) * Position;
+    return parallel_transport_bare(P, direction);
     }
   
   EX transmatrix spin_towards(const transmatrix Position, const hyperpoint goal) {
diff --git a/shmup.cpp b/shmup.cpp
index 02baa40f..10bb8e8f 100644
--- a/shmup.cpp
+++ b/shmup.cpp
@@ -46,6 +46,8 @@ struct monster {
     // butterflies: last position
   transmatrix at;
   transmatrix pat;
+  /** orientation for the product geometry */
+  transmatrix ori;
   eMonster stk;
   bool dead;
   bool notpushed;
@@ -67,7 +69,7 @@ struct monster {
   monster() { 
     dead = false; inBoat = false; parent = NULL; nextshot = 0; 
     stunoff = 0; blowoff = 0; footphase = 0; no_targetting = false;
-    swordangle = 0; inertia = Hypc;
+    swordangle = 0; inertia = Hypc; if(prod) ori = Id;
     }
   
   void store();
@@ -363,6 +365,7 @@ void shootBullet(monster *m) {
   bullet->base = m->base;
   bullet->at = m->at;
   if(WDIM == 3) bullet->at = bullet->at * cpush(2, 0.15 * SCALE);
+  if(prod) bullet->ori = m->ori;
   bullet->type = moBullet;
   bullet->parent = m;
   bullet->pid = m->pid;
@@ -381,6 +384,7 @@ void shootBullet(monster *m) {
     monster* bullet = new monster;
     bullet->base = m->base;
     bullet->at = m->at * cspin(0, WDIM-1, M_PI/4*i);
+    if(prod) bullet->ori = m->ori;
     if(WDIM == 3) bullet->at = bullet->at * cpush(2, 0.15 * SCALE);
     bullet->type = moBullet;
     bullet->parent = m;
@@ -831,7 +835,7 @@ void movePlayer(monster *m, int delta) {
     
   if(playerturn[cpid] && canmove && !blown && WDIM == 2) {
     m->swordangle -= playerturn[cpid];
-    nat = nat * spin(playerturn[cpid]);
+    rotate_object(nat, m->ori, spin(playerturn[cpid]));
     if(inertia_based) m->inertia = spin(-playerturn[cpid]) * m->inertia;
     }
   transmatrix nat0 = nat;
@@ -976,19 +980,20 @@ void movePlayer(monster *m, int delta) {
     if(inertia_based) {
       if(igo) { go = false; break; }
       ld r = hypot_d(WDIM, avg_inertia);
-      nat = solmul_pt(nat, rspintox(avg_inertia) * xpush(r * delta)) * spintox(avg_inertia);
-      if(WDIM == 3) nat = nat * cspin(0, 2, playerturn[cpid]) * cspin(1, 2, playerturny[cpid]);
+      apply_parallel_transport(nat, m->ori, rspintox(avg_inertia) * xtangent(r * delta));
+      if(WDIM == 3) rotate_object(nat, m->ori, cspin(0, 2, playerturn[cpid]) * cspin(1, 2, playerturny[cpid]));
       m->vel = r * (600/SCALE);
       }
     else if(WDIM == 3) {
-      nat = solmul_pt(nat1, cpush(0, playerstrafe[cpid]) * cpush(2, playergo[cpid])) * cspin(0, 2, playerturn[cpid]) * cspin(1, 2, playerturny[cpid]);
+      nat = parallel_transport(nat1, m->ori, point3(playerstrafe[cpid], 0, playergo[cpid]));
+      rotate_object(nat, m->ori, cspin(0, 2, playerturn[cpid]) * cspin(1, 2, playerturny[cpid]));
       m->inertia[0] = playerstrafe[cpid] / delta;
       m->inertia[1] = 0;
       m->inertia[2] = playergo[cpid] / delta;
       }
     else if(playergo[cpid]) {
-      nat = solmul_pt(nat1, spin(playergoturn[cpid]) * xpush(playergo[cpid])) * spin(-playergoturn[cpid]);
-      m->inertia = spin(playergoturn[cpid]) * xpush0(playergo[cpid] / delta);
+      nat = parallel_transport(nat1, m->ori, spin(playergoturn[cpid]) * xtangent(playergo[cpid]));
+      m->inertia = spin(playergoturn[cpid]) * xtangent(playergo[cpid] / delta);
       }
     
     // spin(span[igo]) * xpush(playergo[cpid]) * spin(-span[igo]);
@@ -1417,6 +1422,7 @@ void shoot(eItem it, monster *m) {
   monster* bullet = new monster;
   bullet->base = m->base;
   bullet->at = m->at * rspintox(inverse(m->pat) * mouseh);
+  /* ori */
   if(WDIM == 3) bullet->at = bullet->at * cpush(2, 0.15 * SCALE);
   bullet->type = it == itOrbDragon ? moFireball : it == itOrbAir ? moAirball : moBullet;
   bullet->parent = m;
@@ -1509,9 +1515,9 @@ int bulletdir() {
   return (WDIM == 2) ? 0 : 2;
   }
   
-transmatrix frontpush(ld x) {
-  if(WDIM == 2) return xpush(x);
-  else return cpush(2, x);
+hyperpoint fronttangent(ld x) {
+  if(WDIM == 2) return xtangent(x);
+  else return ztangent(x);
   }
 
 ld collision_distance(monster *bullet, monster *target) {
@@ -1530,6 +1536,7 @@ void spawn_asteroids(monster *bullet, monster *target) {
     monster* child = new monster;
     child->base = target->base;
     child->at = target->at;
+    child->ori = target->ori;
     child->type = target->type;
     child->parent = NULL;
     child->pid = target->pid;
@@ -1564,7 +1571,7 @@ void moveBullet(monster *m, int delta) {
     m->vel -= delta  / speedfactor() / 600000.0;
     if(m->vel < 0 && m->parent) {
       // return to the flailer!
-      nat = spin_towards(m->pat, m->parent->pat * C0, bulletdir(), -1);
+      nat = spin_towards(m->pat, m->ori, m->parent->pat * C0, bulletdir(), -1);
       }
     }
   else m->vel = bullet_velocity(m->type);
@@ -1573,11 +1580,10 @@ void moveBullet(monster *m, int delta) {
     m->dead = true;
 
   if(inertia_based) {
-    ld r = hypot_d(WDIM, m->inertia);
-    nat = solmul_pt(nat, rspintox(m->inertia) * xpush(r * delta) * spintox(m->inertia));
+    nat = parallel_transport(nat, m->ori, m->inertia * delta);
     }
   else 
-    nat = solmul_pt(nat, frontpush(delta * SCALE * m->vel / speedfactor()));
+    nat = parallel_transport(nat, m->ori, fronttangent(delta * SCALE * m->vel / speedfactor()));
   cell *c2 = m->findbase(nat);
 
   if(m->parent && isPlayer(m->parent) && markOrb(itOrbLava) && c2 != m->base && !isPlayerOn(m->base)) 
@@ -1682,7 +1688,7 @@ void moveBullet(monster *m, int delta) {
       if((m2->type == moPalace || m2->type == moFatGuard || m2->type == moSkeleton ||
         m2->type == moVizier || isMetalBeast(m2->type) || m2->type == moTortoise || m2->type == moBrownBug || 
         m2->type == moReptile || m2->type == moSalamander || m2->type == moTerraWarrior) && m2->hitpoints > 1 && !slayer) {
-        m2->rebasePat(spin_towards(m2->pat, nat0 * C0, 0, 1));
+        m2->rebasePat(spin_towards(m2->pat, m->ori, nat0 * C0, 0, 1));
         if(m2->type != moSkeleton && !isMetalBeast(m2->type) && m2->type != moReptile && m2->type != moSalamander && m2->type != moBrownBug) 
           m2->hitpoints--;
         m->dead = true;
@@ -1733,7 +1739,7 @@ void moveBullet(monster *m, int delta) {
       // Knights reflect bullets
       if(m2->type == moKnight) {
         if(m->parent && m->parent != &arrowtrap_fakeparent) {
-          nat = spin_towards(nat, tC0(m->parent->pat), bulletdir(), 1);
+          nat = spin_towards(nat, m->ori, tC0(m->parent->pat), bulletdir(), 1);
           m->rebasePat(nat);
           }
         m->parent = m2;
@@ -1769,7 +1775,7 @@ EX bool dragonbreath(cell *dragon) {
   int randplayer = hrand(numplayers());
   monster* bullet = new monster;
   bullet->base = dragon;
-  bullet->at = spin_towards(Id, inverse(gmatrix[dragon]) * tC0(pc[randplayer]->pat), bulletdir(), 1);
+  bullet->at = spin_towards(Id, bullet->ori, inverse(gmatrix[dragon]) * tC0(pc[randplayer]->pat), bulletdir(), 1);
   bullet->type = moFireball;
   bullet->parent = bullet;
   bullet->pid = randplayer;
@@ -2003,7 +2009,7 @@ void moveMonster(monster *m, int delta) {
       if(h[0] < fabsl(h[1])) return;
       }
     else if(!peace::on) {
-      nat = spin_towards(m->pat, tC0(goal), 0, 1);
+      nat = spin_towards(m->pat, m->ori, tC0(goal), 0, 1);
       }
     }
   
@@ -2051,15 +2057,14 @@ void moveMonster(monster *m, int delta) {
   
   if(inertia_based) {
     if(igo) return;
-    ld r = hypot_d(WDIM, m->inertia);
-    nat = solmul_pt(nat, rspintox(m->inertia) * xpush(r * delta) * spintox(m->inertia));
+    nat = parallel_transport(nat, m->ori, m->inertia * delta);
     }
   else if(WDIM == 3 && igo) {
     ld fspin = rand() % 1000;  
-    nat = solmul_pt(nat0, cspin(1,2,fspin) * spin(igospan[igo]) * xpush(step) * spin(-igospan[igo]) * cspin(2,1,fspin));
+    nat = parallel_transport(nat0, m->ori, cspin(1,2,fspin) * spin(igospan[igo]) * xtangent(step));
     }
   else {
-    nat = solmul_pt(nat0, spin(igospan[igo]) * xpush(step) * spin(-igospan[igo])); // * spintox(wherePC);
+    nat = parallel_transport(nat0, m->ori, spin(igospan[igo]) * xtangent(step));
     }
 
   if(m->type != moRagingBull && !peace::on)
@@ -2134,6 +2139,7 @@ void moveMonster(monster *m, int delta) {
       monster* bullet = new monster;
       bullet->base = m->base;
       bullet->at = m->at;
+      bullet->ori = m->ori;
       bullet->type = moTongue;
       bullet->parent = m;
       bullet->parenttype = m->type;
@@ -2287,6 +2293,7 @@ void moveMonster(monster *m, int delta) {
       monster* bullet = new monster;
       bullet->base = m->base;
       bullet->at = m->at;
+      bullet->ori = m->ori;
       bullet->type = moFireball;
       bullet->parent = m;
       additional.push_back(bullet);
@@ -2300,6 +2307,7 @@ void moveMonster(monster *m, int delta) {
       monster* bullet = new monster;
       bullet->base = m->base;
       bullet->at = m->at;
+      bullet->ori = m->ori;
       bullet->type = moBullet;
       bullet->parent = m;
       bullet->parenttype = moOutlaw;
@@ -2312,6 +2320,7 @@ void moveMonster(monster *m, int delta) {
       monster* bullet = new monster;
       bullet->base = m->base;
       bullet->at = m->at;
+      bullet->ori = m->ori;
       bullet->type = moAirball;
       bullet->parent = m;
       bullet->pid = i;
@@ -2331,6 +2340,7 @@ void moveMonster(monster *m, int delta) {
       monster* bullet = new monster;
       bullet->base = m->base;
       bullet->at = m->at;
+      bullet->ori = m->ori;
       bullet->type = moFlailBullet;
       bullet->parent = m;
       bullet->vel = 1/400.0;
@@ -2344,6 +2354,7 @@ void moveMonster(monster *m, int delta) {
       monster* bullet = new monster;
       bullet->base = m->base;
       bullet->at = m->at;
+      bullet->ori = m->ori;
       bullet->type = moCrushball;
       bullet->parent = m;
       bullet->pid = i;
diff --git a/system.cpp b/system.cpp
index 05e305aa..6217ecc9 100644
--- a/system.cpp
+++ b/system.cpp
@@ -1189,7 +1189,6 @@ EX void set_geometry(eGeometry target) {
     if(GDIM == 2 && among(pmodel, mdPerspective, mdGeodesic)) pmodel = mdDisk;
     if(nonisotropic && old_DIM == 2 && vid.texture_step < 4) vid.texture_step = 4;
     if(prod) { pmodel = mdPerspective; if(vid.texture_step < 4) vid.texture_step = 4; }
-    if(prod && shmup::on) shmup::on = false;
     }
   }
 
@@ -1288,7 +1287,6 @@ EX void switch_game_mode(char switchWhat) {
       chaosmode = false;
       princess::challenge = false;
       if(sol || bounded) set_geometry(gNormal);
-      if(prod) set_geometry(hybrid::underlying);
       dual::disable();
       break;
 #endif
@@ -1310,7 +1308,6 @@ EX void switch_game_mode(char switchWhat) {
       shmup::on = !shmup::on;
       princess::challenge = false;
       if(!shmup::on) racing::on = false;
-      if(prod) set_geometry(hybrid::underlying);
       break;
     
     case rg::randpattern: