hyperrogue/nonisotropic.cpp

// Hyperbolic Rogue -- nonisotropic spaces (Solv and Nil)
// Copyright (C) 2011-2019 Zeno Rogue, see 'hyper.cpp' for details

/** \file nonisotropic.cpp
 *  \brief nonisotropic spaces (Solv and Nil)
 */

#include "hyper.h"
namespace hr {

EX namespace nisot {

  EX bool local_perspective_used;
  
  EX bool geodesic_movement = true;

  EX transmatrix translate(hyperpoint h, ld co IS(1)) {
    if(sl2 || sphere)
      return co > 0 ? stretch::translate(h) : stretch::itranslate(h);
    transmatrix T = Id;
    for(int i=0; i<GDIM; i++) T[i][LDIM] = h[i];
    if(sol && nih) {
      T[0][0] = pow(2, -h[2]);
      T[1][1] = pow(3, h[2]);
      }
    else if(sol) {
      T[0][0] = exp(-h[2]);
      T[1][1] = exp(+h[2]);
      }
    else if(nih) {
      T[0][0] = pow(2, h[2]);
      T[1][1] = pow(3, h[2]);
      }
    if(nil) {
      T[2][1] = h[0] * (nilv::model_used + 1) / 2;
      T[2][0] = h[1] * (nilv::model_used - 1) / 2;
      }
    if(co < 0) return iso_inverse(T);
    return T;
    }

  EX }

#if !CAP_SOLV
EX namespace sn {
  EX always_false in() { return always_false(); }
EX }
#endif

#if CAP_SOLV
EX namespace sn {

  EX bool in() { return among(cgclass, gcSol, gcNIH, gcSolN); }

  EX eGeometryClass geom() {
    return cgclass;
    }

  #if HDR
  typedef array<float, 3> compressed_point;
  
  inline hyperpoint decompress(compressed_point p) { return point3(p[0], p[1], p[2]); }
  inline compressed_point compress(hyperpoint h) { return make_array<float>(h[0], h[1], h[2]); }

  struct tabled_inverses {
    int PRECX, PRECY, PRECZ;
    vector<compressed_point> tab;
    string fname;
    bool loaded;
    
    void load();
    hyperpoint get(ld ix, ld iy, ld iz, bool lazy);
    
    compressed_point& get_int(int ix, int iy, int iz) { return tab[(iz*PRECY+iy)*PRECX+ix]; }
  
    GLuint texture_id;
    bool toload;
    
    GLuint get_texture_id();
  
    tabled_inverses(string s) : fname(s), texture_id(0), toload(true) {}  
    };
  #endif
  
  void tabled_inverses::load() {
    if(loaded) return;
    FILE *f = fopen(fname.c_str(), "rb");
    if(!f) f = fopen((rsrcdir + fname).c_str(), "rb");
    if(!f) { addMessage(XLAT("geodesic table missing")); pmodel = mdPerspective; return; }
    hr::ignore(fread(&PRECX, 4, 1, f));
    hr::ignore(fread(&PRECY, 4, 1, f));
    hr::ignore(fread(&PRECZ, 4, 1, f));
    tab.resize(PRECX * PRECY * PRECZ);
    hr::ignore(fread(&tab[0], sizeof(compressed_point) * PRECX * PRECY * PRECZ, 1, f));
    fclose(f);
    loaded = true;    
    }
  
  hyperpoint tabled_inverses::get(ld ix, ld iy, ld iz, bool lazy) {
    ix *= PRECX-1;
    iy *= PRECY-1;
    iz *= PRECZ-1;
    
    hyperpoint res;
    
    if(lazy) {
      if(isnan(ix) || isnan(iy) || isnan(iz)) return Hypc;
      return decompress(get_int(int(ix+.5), int(iy+.5), int(iz+.5)));
      }
    
    else {
  
      if(ix >= PRECX-1 || isnan(ix)) ix = PRECX-2;
      if(iy >= PRECX-1 || isnan(iy)) iy = PRECX-2;
      if(iz >= PRECZ-1 || isnan(iz)) iz = PRECZ-2;
      
      int ax = ix, bx = ax+1;
      int ay = iy, by = ay+1;
      int az = iz, bz = az+1;
      
      #define S0(x,y,z) get_int(x, y, z)[t]
      #define S1(x,y) (S0(x,y,az) * (bz-iz) + S0(x,y,bz) * (iz-az))
      #define S2(x) (S1(x,ay) * (by-iy) + S1(x,by) * (iy-ay))
  
      for(int t=0; t<3; t++)
        res[t] = S2(ax) * (bx-ix) + S2(bx) * (ix-ax);
      
      res[3] = 0;
      }
    
    return res;
    }
  
  GLuint tabled_inverses::get_texture_id() {
    #if CAP_GL
    if(!toload) return texture_id;
  
    load();
    if(!loaded) return 0;
  
    println(hlog, "installing table");
    toload = false;
    
    if(texture_id == 0) glGenTextures(1, &texture_id);
  
    glBindTexture( GL_TEXTURE_3D, texture_id);
  
    glTexParameteri(GL_TEXTURE_3D, GL_TEXTURE_MIN_FILTER, GL_LINEAR);
    glTexParameteri(GL_TEXTURE_3D, GL_TEXTURE_MAG_FILTER, GL_LINEAR);
  
    glTexParameteri(GL_TEXTURE_3D, GL_TEXTURE_WRAP_S, GL_CLAMP_TO_EDGE);
    glTexParameteri(GL_TEXTURE_3D, GL_TEXTURE_WRAP_T, GL_CLAMP_TO_EDGE);
    glTexParameteri(GL_TEXTURE_3D, GL_TEXTURE_WRAP_R, GL_CLAMP_TO_EDGE);
    
    auto xbuffer = new glvertex[PRECZ*PRECY*PRECX];
    
    for(int z=0; z<PRECZ*PRECY*PRECX; z++) {
      auto& t = tab[z];
      xbuffer[z] = glhr::makevertex(t[0], t[1], t[2]);
      }
    
    #if !ISWEB
    glTexImage3D(GL_TEXTURE_3D, 0, 34836 /*GL_RGBA32F*/, PRECX, PRECX, PRECZ, 0, GL_RGBA, GL_FLOAT, xbuffer);
    #else
    // glTexStorage3D(GL_TEXTURE_3D, 1, 34836 /*GL_RGBA32F*/, PRECX, PRECX, PRECZ);
    // glTexSubImage3D(GL_TEXTURE_3D, 0, 0, 0, 0, PRECX, PRECY, PRECZ, GL_RGBA, GL_FLOAT, xbuffer);
    #endif
    delete[] xbuffer;
    #endif
    return texture_id;
    }
  
  EX ld x_to_ix(ld u) {
    if(u == 0.) return 0.;
    ld diag = u*u/2.;
 
    ld x = diag;
    ld y = u;
    ld z = diag+1.;
  
    x /= (1.+z);
    y /= (1.+z);
  
    return 0.5 - atan((0.5-x) / y) / M_PI;
    }

  EX ld ix_to_x(ld ix) {
    ld minx = 0;
    while(x_to_ix(minx) <= ix) minx++;
    ld maxx = minx; minx--;
    for(int it=0; it<20; it++) {
      ld x = (minx + maxx) / 2;
      if(x_to_ix(x) < ix) minx = x;
      else maxx = x;
      }
    return minx;
    }

  EX ld z_to_iz(ld z) { 
    z = sinh(z) / (1 + cosh(z));
    if(nih) z = (z+1) / 2;
    return z;
    }

  EX ld iz_to_z(ld iz) {
    ld minz = 0;
    while(z_to_iz(minz) <= iz) minz++;
    while(z_to_iz(minz) > iz) minz--;
    ld maxz = minz + 1;
    for(int it=0; it<20; it++) {
      ld z = (minz + maxz) / 2;
      if(z_to_iz(z) < iz) minz = z;
      else maxz = z;
      }
    return (minz+maxz) / 2;
    }

  EX hyperpoint azeq_to_table(hyperpoint x) {
    // azimuthal equidistant to Poincare
    ld r = hypot_d(3, x);
    if(r == 0) return point3(0,0,0);
    ld make_r = sinh(r) / (1 + cosh(r));
    ld d = make_r / r;
    return x * d;
    }

  EX hyperpoint table_to_azeq(hyperpoint x) {
    // Poincare to azimuthal equidistant  
    ld hr = sqhypot_d(3, x);
    if(hr < 1e-5) return x * 2;
    if(hr >= 1) return x * 60;
    ld hz = (1 + hr) / (1 - hr);
    ld d = (hz+1) * acosh(hz) / sinh(acosh(hz));    
    return x * d;
    }

  EX void queueline_lie(shiftpoint h1, shiftpoint h2, color_t col, int prec, PPR prio) {
    hyperpoint h = gpushxto0(h1.h) * h2.h;
    hyperpoint z = lie_log(shiftless(h));
    int steps = ceil(hypot_d(3, z) * pow(2, prec));
    for(int i=0; i<=steps; i++) curvepoint(lie_exp(z * i / steps).h);
    queuecurve(rgpushxto0(h1), col, 0, prio);
    }

  struct hrmap_solnih : hrmap {
    hrmap *binary_map;
    hrmap *ternary_map; /* nih only */
    map<pair<heptagon*, heptagon*>, heptagon*> at;
    map<heptagon*, pair<heptagon*, heptagon*>> coords;
    
    heptagon *origin;
    
    heptagon *getOrigin() override { return origin; }
    
    heptagon *get_at(heptagon *x, heptagon *y) {
      auto& h = at[make_pair(x, y)];
      if(h) return h;
      h = init_heptagon(S7);
      h->c7 = newCell(S7, h);
      coords[h] = make_pair(x, y);
      h->distance = x->distance;
      h->zebraval = x->emeraldval;
      h->emeraldval = y->emeraldval;
      return h;      
      }

    hrmap_solnih() {
    
      heptagon *alt;
      heptagon *alt3;
    
      if(true) {
        dynamicval<eGeometry> g(geometry, gBinary4); 
        alt = init_heptagon(S7);
        alt->s = hsOrigin;
        alt->alt = alt;
        binary_map = bt::new_alt_map(alt);
        }
      
      if(nih) {
        dynamicval<eGeometry> g(geometry, gTernary); 
        alt3 = init_heptagon(S7);
        alt3->s = hsOrigin;
        alt3->alt = alt3;
        ternary_map = bt::new_alt_map(alt3);
        }
      else {
        alt3 = alt;
        ternary_map = nullptr;
        }
      
      origin = get_at(alt, alt3);
      }
    
    heptagon *altstep(heptagon *h, int d) {
      dynamicval<eGeometry> g(geometry, gBinary4); 
      dynamicval<hrmap*> cm(currentmap, binary_map);
      return h->cmove(d);
      }
    
    heptagon *altstep3(heptagon *h, int d) {
      dynamicval<eGeometry> g(geometry, gTernary); 
      dynamicval<hrmap*> cm(currentmap, ternary_map);
      return h->cmove(d);
      }
    
    heptagon *create_step(heptagon *parent, int d) override {
      auto p = coords[parent];
      auto pf = p.first, ps = p.second;
      auto rule = [&] (heptagon *c1, heptagon *c2, int d1) {
        auto g = get_at(c1, c2);
        parent->c.connect(d, g, d1, false);
        return g;
        };
      
      switch(geometry){

      case gSol: switch(d) {
        case 0: // right
          return rule(altstep(pf, 2), ps, 4);
        case 1: // up
          return rule(pf, altstep(ps, 2), 5);
        case 2: // front left
          return rule(altstep(pf, 0), altstep(ps, 3), ps->zebraval ? 7 : 6);
        case 3: // front right
          return rule(altstep(pf, 1), altstep(ps, 3), ps->zebraval ? 7 : 6);
        case 4: // left
          return rule(altstep(pf, 4), ps, 0);
        case 5: // down
          return rule(pf, altstep(ps, 4), 1);
        case 6: // back down
          return rule(altstep(pf, 3), altstep(ps, 0), pf->zebraval ? 3 : 2);
        case 7: // back up
          return rule(altstep(pf, 3), altstep(ps, 1), pf->zebraval ? 3 : 2);
        default:
          return NULL;
        }

      case gNIH: switch(d) {
        case 0: // right
          return rule(altstep(pf, 2), ps, 2);
        case 1: // up
          return rule(pf, altstep3(ps, 3), 3);
        case 2: // left
          return rule(altstep(pf, 4), ps, 0);
        case 3: // down
          return rule(pf, altstep3(ps, 5), 1);
        case 4: // back
          return rule(altstep(pf, 3), altstep3(ps, 4), 5 + pf->zebraval + 2 * ps->zebraval);
        default:
          return rule(altstep(pf, (d-5) % 2), altstep3(ps, (d-5)/2), 4);
        }

      case gSolN: switch(d) {
        case 0: // right
          return rule(altstep(pf, 2), ps, 2);
        case 1: // up
          return rule(pf, altstep3(ps, 3), 3);
        case 2: // left
          return rule(altstep(pf, 4), ps, 0);
        case 3: // down
          return rule(pf, altstep3(ps, 5), 1);
        case 4: case 5: 
          return rule(altstep(pf, d-4), altstep3(ps, 4), ps->zebraval + 6);
        case 6: case 7: case 8: 
          return rule(altstep(pf, 3), altstep3(ps, d-6), pf->zebraval + 4);
        default:
          return NULL;
        }

        default: throw hr_exception("not solnihv");
        }
      }

    ~hrmap_solnih() {
      delete binary_map;
      if(ternary_map) delete ternary_map;
      for(auto& p: at) clear_heptagon(p.second);
      }

    transmatrix adjmatrix(int i, int j) {
      switch(geometry) {
      case gSol: {
        ld z = log(2);
        ld bw = vid.binary_width * z;
        switch(i) {
          case 0: return xpush(+bw);
          case 1: return ypush(+bw);
          case 2: case 3:
            return ypush(bw*(6.5-j)) * zpush(+z) * xpush(bw*(i-2.5));
          case 4: return xpush(-bw);
          case 5: return ypush(-bw);
          case 6: case 7:
            return xpush(bw*(2.5-j)) * zpush(-z) * ypush(bw*(i-6.5));
          default:return Id;
          }
        }
      case gNIH: {
        ld bw = vid.binary_width;
        switch(i) {          
          case 0: return xpush(+bw);
          case 1: return ypush(+bw);
          case 2: return xpush(-bw);
          case 3: return ypush(-bw);
          case 4: return xpush(-((j-5)%2-.5)*bw) * ypush(-((j-5)/2-1)*bw) * zpush(1);
          default:
            return zpush(-1) * xpush(((i-5)%2-.5)*bw) * ypush(((i-5)/2-1)*bw);
          }
        }
      case gSolN: {
        ld bw = vid.binary_width;
        switch(i) {          
          case 0: return xpush(+bw);
          case 1: return ypush(+bw);
          case 2: return xpush(-bw);
          case 3: return ypush(-bw);
          case 4: 
          case 5:
            return ypush(bw*(7-j)) * zpush(+1) * xpush(bw*(i-4.5));
          case 6:
          case 7:
          case 8:
            return xpush(bw*(4.5-j)) * zpush(-1) * ypush(bw*(i-7));
          default:
            throw hr_exception("wrong dir");
          }
        }

      default: throw hr_exception("wrong geometry");
      }
      }

    transmatrix adj(heptagon *h, int d) override {
      h->cmove(d); return adjmatrix(d, h->c.spin(d));
      }
    
    transmatrix relative_matrixh(heptagon *h2, heptagon *h1, const hyperpoint& hint) override { 
      for(int i=0; i<h1->type; i++) if(h1->move(i) == h2) return adjmatrix(i, h1->c.spin(i));
      if(gmatrix0.count(h2->c7) && gmatrix0.count(h1->c7))
        return inverse_shift(gmatrix0[h1->c7], gmatrix0[h2->c7]);
      
      transmatrix front = Id, back = Id;
      
      int up, down;

      switch(geometry) {
        case gSol: up = 2; down = 6; break;
        case gSolN: up = 4; down = 7; break;
        case gNIH: up = 4; down = 4; break;
        default: throw hr_exception("not nihsolv");
        }
      
      while(h1->distance > h2->distance) front = front * adj(h1, down), h1 = h1->cmove(down);
      while(h1->distance < h2->distance) back = iadj(h2, down) * back, h2 = h2->cmove(down);
      while(coords[h1].first != coords[h2].first) front = front * adj(h1, down), back = iadj(h2, down) * back, h1 = h1->cmove(down), h2 = h2->cmove(down);
      while(coords[h1].second != coords[h2].second) front = front * adj(h1, up), back = iadj(h2, up) * back, h1 = h1->cmove(up), h2 = h2->cmove(up);
      return front * back;
      }
    };

  EX pair<heptagon*,heptagon*> getcoord(heptagon *h) {
    return ((hrmap_solnih*)currentmap)->coords[h];
    }

  EX heptagon *get_at(heptagon *h1, heptagon *h2, bool gen) {
    auto m = ((hrmap_solnih*)currentmap);
    if(!gen && !m->at.count(make_pair(h1, h2))) return nullptr;
    return m->get_at(h1, h2);
    }
  
  EX string common = 
    "uniform mediump sampler3D tInvExpTable;"    
    "uniform mediump float PRECX, PRECY, PRECZ;"

    "float x_to_ix(float u) {"
    "  if(u < 1e-6) return 0.;"
    "  float diag = u*u/2.;"
   
    "  float x = diag;"
    "  float y = u;"
    "  float z = diag+1.;"
    
    "  x /= (1.+z);"
    "  y /= (1.+z);"
    
    "  return 0.5 - atan((0.5-x) / y) / 3.1415926535897932384626433832795;"
    "  }"
    
    "float z_to_iz_s(float z) {"
      "return sinh(z) / (1. + cosh(z));"
      "}"
    
    "float z_to_iz_ns(float z) {"
      "z = sinh(z) / (1. + cosh(z));"
      "return (z+1.)/2.;"
      "}";    

  hyperpoint christoffel(const hyperpoint at, const hyperpoint velocity, const hyperpoint transported) {
    const ld l2 = log(2);
    const ld l3 = log(3);
    switch(geom()) {
      case gcSolN:
        return hpxyz3(
          -(velocity[2] * transported[0] + velocity[0] * transported[2]) * l2,
           (velocity[2] * transported[1] + velocity[1] * transported[2]) * l3,
           velocity[0] * transported[0] * exp(2*l2*at[2]) * l2 - velocity[1] * transported[1] * exp(-2*l3*at[2]) * l3,
           0
           );
      case gcSol:
        return hpxyz3(
          -velocity[2] * transported[0] - velocity[0] * transported[2],
           velocity[2] * transported[1] + velocity[1] * transported[2],
           velocity[0] * transported[0] * exp(2*at[2]) - velocity[1] * transported[1] * exp(-2*at[2]),
           0
           );
      case gcNIH:
        return hpxyz3(
           (velocity[2] * transported[0] + velocity[0] * transported[2]) * l2,
           (velocity[2] * transported[1] + velocity[1] * transported[2]) * l3,
           -(velocity[0] * transported[0] * exp(-2*l2*at[2]) * l2 + velocity[1] * transported[1] * exp(-2*l3*at[2]) * l3),
           0
           );
      default:
        throw hr_exception("christoffel not in solnihv");
      }
    }
  
  EX hyperpoint get_inverse_exp_symsol(hyperpoint h, flagtype flags) {
    auto& s = get_tabled();
    s.load();
    
    ld ix = h[0] >= 0. ? sn::x_to_ix(h[0]) : sn::x_to_ix(-h[0]);
    ld iy = h[1] >= 0. ? sn::x_to_ix(h[1]) : sn::x_to_ix(-h[1]);
    ld iz = sn::z_to_iz(h[2]);

    if(h[2] < 0.) { iz = -iz; swap(ix, iy); }
    
    hyperpoint res = s.get(ix, iy, iz, flags & pfNO_INTERPOLATION);
  
    if(h[2] < 0.) { swap(res[0], res[1]); res[2] = -res[2]; }
    if(h[0] < 0.) res[0] = -res[0];
    if(h[1] < 0.) res[1] = -res[1];
    
    if(flags & pfNO_DISTANCE) return res;
    return table_to_azeq(res);
    }

  EX hyperpoint get_inverse_exp_nsym(hyperpoint h, flagtype flags) {
    auto& s = get_tabled();
    s.load();
    
    ld ix = h[0] >= 0. ? sn::x_to_ix(h[0]) : sn::x_to_ix(-h[0]);
    ld iy = h[1] >= 0. ? sn::x_to_ix(h[1]) : sn::x_to_ix(-h[1]);
    ld iz = sn::z_to_iz(h[2]);
    
    hyperpoint res = s.get(ix, iy, iz, flags & pfNO_INTERPOLATION);
  
    if(h[0] < 0.) res[0] = -res[0];
    if(h[1] < 0.) res[1] = -res[1];
    
    if(flags & pfNO_DISTANCE) return res;
    return table_to_azeq(res);
    }

  EX string shader_symsol = sn::common +

    "vec4 inverse_exp(vec4 h) {"
    
    "float ix = h[0] >= 0. ? x_to_ix(h[0]) : x_to_ix(-h[0]);"
    "float iy = h[1] >= 0. ? x_to_ix(h[1]) : x_to_ix(-h[1]);"
    "float iz = z_to_iz_s(h[2]);"
    
    "if(h[2] < 1e-6) { iz = -iz; float s = ix; ix = iy; iy = s; }"
    "if(iz < 0.) iz = 0.;"
    
    "vec4 res;"

    "float cx = ix*(1.-1./PRECX) + .5/PRECX;"
    "float cy = iy*(1.-1./PRECY) + .5/PRECY;"
    "float cz = iz*(1.-1./PRECZ) + .5/PRECZ;"

    // "if(ix > .5 && iy > .6 && ix < iy + .05 && iz < .2 && iz < (iy - 0.5) * 0.6)"
    "\n#ifndef SOLV_ALL\n"

    "bool ok = true;"

    // hard to tell which triangles fall on the other sides
    "if(iz < .03 && ix > .65 && iy > .65) ok = false;"
    "if(iz < .013 && ix > .55 && iy > .55) ok = false;"
    "if(iz < .0075 && ix > .45 && iy > .45) ok = false;"  
    "if(iz > 0.004 && ix > 0.4 && iy > 0.4 && ix < .6 && iy < .6) ok = true;"
    "if(iz > 0.000004 && ix > 0.4 && ix < 0.7 && iy > 0.4 && iy < 0.7) ok = true;"
    "if(iz < 0.04 && ix > 0.70 && ix < 0.8 && iy > 0.5 && iy < 0.7) ok = false;"
    "if(iz < 0.05 && ix > .45 && iy > .75 && ix < .55 && iy < .95) ok = false;"
    "if(iz < 0.05 && ix > .85 && iy > .45 && iy < .75) ok = false;"
    "if(iz < 0.025 && ix > .65 && iy > .65 && ix < .8 && iy < .8) ok = false;"

    "if(!ok) res = vec4(0./0.,0./0.,0./0.,1);"
    "else "

    "\n#endif\n"
  
      "res = texture3D(tInvExpTable, vec3(cx, cy, cz));"

    "if(h[2] < 1e-6) { res.xy = res.yx; res[2] = -res[2]; }"
    "if(h[0] < 0.) res[0] = -res[0];"
    "if(h[1] < 0.) res[1] = -res[1];"
    
    "return res;"
    "}";

  EX string shader_nsymsol = sn::common + R"*(

    vec4 inverse_exp(vec4 h) {
      
    float ix = h[0] >= 0. ? x_to_ix(h[0]) : x_to_ix(-h[0]);
    float iy = h[1] >= 0. ? x_to_ix(h[1]) : x_to_ix(-h[1]);
    float iz = z_to_iz_ns(h[2]);
      
    vec4 res;
  
    float cx = ix*(1.-1./PRECX) + .5/PRECX;
    float cy = iy*(1.-1./PRECY) + .5/PRECY;
    float cz = iz*(1.-1./PRECZ) + .5/PRECZ;
  
    if(ix > .65 && iy > .5 && iz > .45 && iz < .55)
      res = vec4(0.,0.,0.,1.);
    else if(ix > .55 && iy > .75 && ix < .7 && iz > .45 && iz < .55)
      res = vec4(0.,0.,0.,1.);
    else if(ix > .45 && iy > .75 && ix < .7 && iz > .4 && iz < .5)
      res = vec4(0.,0.,0.,1.);
    else if(ix > .85 && iy > .5 && iz > .55 && iz < .75)
      res = vec4(0.,0.,0.,1.);
    else if(ix > .7 && iy > .55 && iz > .42 && iz < .58)
      res = vec4(0.,0.,0.,1.);
    else if(iz > 0.45 && ix > 0.8 && iy > 0.3 && iy < 0.6)
      res = vec4(0.,0.,0.,1.);
    else if(iz > 0.45 && ix > 0.8 && iy > 0.3 && iy < 0.6)
      res = vec4(0.,0.,0.,1.);
    else if(iz > .4 && iz < .55 && ix > .7 && iy > .36 && iy < .5 && ix < .8 && ix+iy > 1.2)
      res = vec4(0.,0.,0.,1.);
    else res = texture3D(tInvExpTable, vec3(cx, cy, cz));
  
    if(h[0] < 0.) res[0] = -res[0];
    if(h[1] < 0.) res[1] = -res[1];
      
    return res;
    })*";

  EX string shader_nsym = sn::common +

    "vec4 inverse_exp(vec4 h) {"
    
    "float ix = h[0] >= 0. ? x_to_ix(h[0]) : x_to_ix(-h[0]);"
    "float iy = h[1] >= 0. ? x_to_ix(h[1]) : x_to_ix(-h[1]);"
    "float iz = z_to_iz_ns(h[2]);"
    
    "vec4 res;"

    "float cx = ix*(1.-1./PRECX) + .5/PRECX;"
    "float cy = iy*(1.-1./PRECY) + .5/PRECY;"
    "float cz = iz*(1.-1./PRECZ) + .5/PRECZ;"

    "res = texture3D(tInvExpTable, vec3(cx, cy, cz));"

    "if(h[0] < 0.) res[0] = -res[0];"
    "if(h[1] < 0.) res[1] = -res[1];"
    
    "return res;"
    "}";

  EX ld solrange_xy = 15;
  EX ld solrange_z = 4;
  
  EX bool in_table_range(hyperpoint h) {
    return abs(h[0]) < solrange_xy && abs(h[1]) < solrange_xy && abs(h[2]) < solrange_z;
    }

  EX tabled_inverses solt = sn::tabled_inverses("solv-geodesics.dat");
  EX tabled_inverses niht = sn::tabled_inverses("shyp-geodesics.dat");
  EX tabled_inverses sont = sn::tabled_inverses("ssol-geodesics.dat");
  
  EX tabled_inverses& get_tabled() {
    switch(geom()) {
      case gcSol: return solt;
      case gcNIH: return niht;
      case gcSolN: return sont;
      default: throw hr_exception("not solnih");
      }
    }

  EX int approx_distance(heptagon *h1, heptagon *h2) {
    auto m = (sn::hrmap_solnih*) currentmap;
    dynamicval<eGeometry> g(geometry, gBinary4); 
    dynamicval<hrmap*> cm(currentmap, m->binary_map);
    int d1 = bt::celldistance3_approx(m->coords[h1].first, m->coords[h2].first);
    int d2 = bt::celldistance3_approx(m->coords[h1].second, m->coords[h2].second);
    return d1 + d2 - abs(h1->distance - h2->distance);
    }
  
  EX void create_faces() {
    if(geometry == gSol) {
      ld zstep = -log(2) / 2;
      ld bwh = vid.binary_width * zstep;
      auto pt = [&] (int x, int y, int z) { return xpush(bwh*x) * ypush(bwh*y) * zpush(zstep*z) * C0; };
      add_wall(0, {pt(-1,-1,-1), pt(-1,-1,+1), pt(-1,00,+1), pt(-1,+1,+1), pt(-1,+1,-1)});
      add_wall(1, {pt(-1,-1,-1), pt(00,-1,-1), pt(+1,-1,-1), pt(+1,-1,+1), pt(-1,-1,+1)});
      add_wall(2, {pt(+1,+1,-1), pt(+1,-1,-1), pt(00,-1,-1), pt(00,+1,-1)});
      add_wall(3, {pt(00,+1,-1), pt(00,-1,-1), pt(-1,-1,-1), pt(-1,+1,-1)});
      add_wall(4, {pt(+1,-1,-1), pt(+1,-1,+1), pt(+1,00,+1), pt(+1,+1,+1), pt(+1,+1,-1)});
      add_wall(5, {pt(-1,+1,-1), pt(00,+1,-1), pt(+1,+1,-1), pt(+1,+1,+1), pt(-1,+1,+1)});
      add_wall(6, {pt(-1,+1,+1), pt(+1,+1,+1), pt(+1,00,+1), pt(-1,00,+1)});
      add_wall(7, {pt(-1,00,+1), pt(+1,00,+1), pt(+1,-1,+1), pt(-1,-1,+1)});
      }
  
    if(geometry == gNIH) {
      ld zstep = .5;
      ld bwh = vid.binary_width / 6;
      auto pt = [&] (int x, int y, int z) { return xpush(bwh*x) * ypush(bwh*y) * zpush(zstep*z) * C0; };
      add_wall(0, {pt(+3,-3,-1), pt(+3,-3,+1), pt(+3,+3,+1), pt(+3,+3,-1), pt(+3,+1,-1), pt(+3,-1,-1) });
      add_wall(1, {pt(-3,+3,-1), pt(-3,+3,+1), pt(+3,+3,+1), pt(+3,+3,-1), pt(+0,+3,-1) });
      add_wall(2, {pt(-3,-3,-1), pt(-3,-3,+1), pt(-3,+3,+1), pt(-3,+3,-1), pt(-3,+1,-1), pt(-3,-1,-1) });
      add_wall(3, {pt(-3,-3,-1), pt(-3,-3,+1), pt(+3,-3,+1), pt(+3,-3,-1), pt(+0,-3,-1)});
      
      add_wall(4, {pt(-3,-3,+1), pt(-3,+3,+1), pt(+3,+3,+1), pt(+3,-3,+1)});
      
      for(int i=0; i<6; i++) {
        int x = -3 + (i%2) * 3;
        int y = -3 + (i/2) * 2;
        add_wall(5+i, {pt(x,y,-1), pt(x+3,y,-1), pt(x+3,y+2,-1), pt(x,y+2,-1)});
        }
      }
    
    if(geometry == gSolN) {
      ld zstep = -.5;
      ld bwh = vid.binary_width / 6;
      auto pt = [&] (int x, int y, int z) { return xpush(bwh*x) * ypush(bwh*y) * zpush(zstep*z) * C0; };
      add_wall(0, {pt(+3,-3,-1), pt(+3,-3,+1), pt(+3,-1,+1), pt(+3,+1,+1), pt(+3,+3,+1), pt(+3,+3,-1)});
      add_wall(1, {pt(-3,+3,-1), pt(00,+3,-1), pt(+3,+3,-1), pt(+3,+3,+1), pt(-3,+3,+1)});
      add_wall(2, {pt(-3,-3,-1), pt(-3,-3,+1), pt(-3,-1,+1), pt(-3,+1,+1), pt(-3,+3,+1), pt(-3,+3,-1)});
      add_wall(3, {pt(-3,-3,-1), pt(00,-3,-1), pt(+3,-3,-1), pt(+3,-3,+1), pt(-3,-3,+1)});
      add_wall(4, {pt(-3,+3,-1), pt(-3,-3,-1), pt(00,-3,-1), pt(00,+3,-1)});
      add_wall(5, {pt(00,+3,-1), pt(00,-3,-1), pt(+3,-3,-1), pt(+3,+3,-1)});
      add_wall(6, {pt(-3,-3,+1), pt(+3,-3,+1), pt(+3,-1,+1), pt(-3,-1,+1)});
      add_wall(7, {pt(-3,-1,+1), pt(+3,-1,+1), pt(+3,+1,+1), pt(-3,+1,+1)});
      add_wall(8, {pt(-3,+1,+1), pt(+3,+1,+1), pt(+3,+3,+1), pt(-3,+3,+1)});
      }
    
    get_hsh().compute_hept();
    }
EX }
#endif

EX namespace nilv {

  #if HDR
  /** nmSym is the rotationally symmetric model of Nil, while nmHeis is the Heisenberg model. */
  constexpr ld nmSym = 0, nmHeis = 1;
  #endif

  /** HyperRogue currently uses nmSym by default, but some parts are still written in nmHeis */
  EX ld model_used = nmSym;

  /** a helper function for model conversions */
  EX ld sym_to_heis_bonus(const hyperpoint& H) {
    return H[0] * H[1] / 2;
    }

  EX hyperpoint convert(hyperpoint H, ld from, ld to) {
    H[2] += sym_to_heis_bonus(H) * (to - from);
    return H;
    }

  EX void convert_ref(hyperpoint& H, ld from, ld to) {
    H[2] += sym_to_heis_bonus(H) * (to - from);
    }

  EX void convert_tangent_ref(hyperpoint at, hyperpoint& v, ld from, ld to) {
    v[2] += (at[0] * v[1] + at[1] * v[0]) * (to - from) / 2;
    }

  EX void convert_ref(transmatrix& T, ld from, ld to) {
    auto T1 = transpose(T);
    convert_ref(T1[3], from, to);
    for(int i: {0, 1, 2})
      convert_tangent_ref(T1[3], T1[i], from, to);
    T = transpose(T1);
    }

  EX hyperpoint checked_convert(hyperpoint H, ld from, ld to) {
    if(nil) return convert(H, from, to);
    return H;
    }

  hyperpoint christoffel(const hyperpoint Position, const hyperpoint Velocity, const hyperpoint Transported, ld model = model_used) {
    hyperpoint c; c[3] = 0;
    ld x = Position[0]; ld y = Position[1];
    auto mu = model;
    c[ 0 ] = 0
     + Velocity[ 0 ] * Transported[ 1 ] * ( y*(mu - 1)/4 )
     + Velocity[ 1 ] * Transported[ 0 ] * ( y*(mu - 1)/4 )
     + Velocity[ 1 ] * Transported[ 1 ] * ( x*(mu + 1)/2 )
     + Velocity[ 1 ] * Transported[ 2 ] * ( -1/2. )
     + Velocity[ 2 ] * Transported[ 1 ] * ( -1/2. );
    c[ 1 ] = 0
     + Velocity[ 0 ] * Transported[ 0 ] * ( y*(1 - mu)/2 )
     + Velocity[ 0 ] * Transported[ 1 ] * ( -x*(mu + 1)/4 )
     + Velocity[ 0 ] * Transported[ 2 ] * ( 1/2. )
     + Velocity[ 1 ] * Transported[ 0 ] * ( -x*(mu + 1)/4 )
     + Velocity[ 2 ] * Transported[ 0 ] * ( 1/2. );
    c[ 2 ] = 0
     + Velocity[ 0 ] * Transported[ 0 ] * ( x*y*(1 - mu*mu)/4 )
     + Velocity[ 0 ] * Transported[ 1 ] * ( -mu*mu*x*x/8 + mu*mu*y*y/8 - mu*x*x/4 - mu*y*y/4 + mu/2 - x*x/8 + y*y/8 )
     + Velocity[ 0 ] * Transported[ 2 ] * ( x*(mu + 1)/4 )
     + Velocity[ 1 ] * Transported[ 0 ] * ( -mu*mu*x*x/8 + mu*mu*y*y/8 - mu*x*x/4 - mu*y*y/4 + mu/2 - x*x/8 + y*y/8 )
     + Velocity[ 1 ] * Transported[ 1 ] * ( x*y*(mu*mu - 1)/4 )
     + Velocity[ 1 ] * Transported[ 2 ] * ( y*(1 - mu)/4 )
     + Velocity[ 2 ] * Transported[ 0 ] * ( x*(mu + 1)/4 )
     + Velocity[ 2 ] * Transported[ 1 ] * ( y*(1 - mu)/4 );
    return c;
    }

  EX hyperpoint formula_exp(hyperpoint v) {
    // copying Modelling Nil-geometry in Euclidean Space with Software Presentation
    // v[0] = c cos alpha
    // v[1] = c sin alpha
    // v[2] = w
    
    if(v[0] == 0 && v[1] == 0) return point31(v[0], v[1], v[2]);
    
    if(v[2] == 0) return convert(point31(v[0], v[1], 0), nmSym, model_used);
    
    ld alpha = atan2(v[1], v[0]);
    ld w = v[2];
    ld c = hypot(v[0], v[1]) / v[2];
    
    return convert(point31(
      2 * c * sin(w/2) * cos(w/2 + alpha), 
      2 * c * sin(w/2) * sin(w/2 + alpha), 
      w * (1 + (c*c/2) * ((1 - sin(w)/w) + (1-cos(w))/w * sin(w + 2 * alpha)))
      ), nmHeis, model_used);
    }
  
  EX hyperpoint get_inverse_exp(hyperpoint h, flagtype prec IS(pNORMAL)) {
    ld wmin, wmax;
    
    ld side = convert(h, model_used, nmSym)[2];

    convert_ref(h, model_used, nmHeis);
    
    if(hypot_d(2, h) < 1e-6) return point3(h[0], h[1], h[2]);
    else if(side > 1e-6) {
      wmin = 0, wmax = TAU;
      }
    else if(side < -1e-6) {
      wmin = - TAU, wmax = 0;
      }
    else return point3(h[0], h[1], 0);
    
    ld alpha_total = h[0] ? atan(h[1] / h[0]) : 90._deg;
  
    ld b;
    if(abs(h[0]) > abs(h[1]))
      b = h[0] / 2 / cos(alpha_total);
    else
      b = h[1] / 2 / sin(alpha_total);  
    
    ld s = sin(2 * alpha_total);
    
    int max_iter = (prec & pfLOW_BS_ITER) ? 5 : 20;
    
    for(int it=0;; it++) {
      ld w = (wmin + wmax) / 2;
      ld z = b * b * (s + (sin(w) - w)/(cos(w) - 1)) + w;

      if(it == max_iter) {
        ld alpha = alpha_total - w/2;
        ld c = b / sin(w/2);
        return point3(c * w * cos(alpha), c * w * sin(alpha), w);
        }
      if(h[2] > z) wmin = w;
      else wmax = w;    
      }
    }
  
  EX string nilshader() {
    return "vec4 inverse_exp(vec4 h) {"
      "float wmin, wmax;"
      "h[2] += h[0] * h[1] / 2. * " + glhr::to_glsl(1-model_used) + ";"
      "float side = h[2] - h[0] * h[1] / 2.;"
      "if(h[0]*h[0] + h[1]*h[1] < 1e-12) return vec4(h[0], h[1], h[2], 1);"
      "if(side > 1e-6) { wmin = 0.; wmax = 2.*PI; }"
      "else if(side < -1e-6) { wmin = -2.*PI; wmax = 0.; }"
      "else return vec4(h[0], h[1], 0., 1.);"
      "float at = h[0] != 0. ? atan(h[1] / h[0]) : PI/2.;"
      "float b = abs(h[0]) > abs(h[1]) ? h[0] / 2. / cos(at) : h[1] / 2. / sin(at);"
      "float s = sin(2. * at);"

      "for(int it=0; it<50; it++) {"
        "float w = (wmin + wmax) / 2.;"
        // the formula after ':' produces visible numerical artifacts for w~0
        "float z = b * b * (s + (abs(w) < .1 ? w/3. + w*w*w/90. + w*w*w*w*w/2520.: (sin(w) - w)/(cos(w) - 1.))) + w;"
        "if(h[2] > z) wmin = w;"
        "else wmax = w;"
        "}"

      "float w = (wmin + wmax) / 2.;"
      "float alpha = at - w/2.;"
      "float c = b / sin(w/2.);"
      "return vec4(c*w*cos(alpha), c*w*sin(alpha), w, 1.);"
      "}";
    }
  
  #if HDR
  bool mvec_uses_hex();

  /** This type is used for indexed Nil cells. See mvec_to_point to convert from this to hyperpoint the tile is centered at.
      In non-hex Nils, these correspond exactly to coordinates in nmHeis model (with nilwidth equal 1).
      In hex Nils, these would be valid in nmSym models if the 'z' coordinates were halved. (But, they are sheared, as for usual hex coordinates in HyperRogue.)
   **/
  struct mvec : array<int, 3> {
    explicit mvec() = default;
    constexpr explicit mvec(int x, int y, int z) : array<int, 3>{{x, y, z}} {}
    mvec inverse() {  
      auto& a = *this;
      if(mvec_uses_hex()) return mvec(-a[0], -a[1], -a[2]);
      return mvec(-a[0], -a[1], -a[2]+a[1] * a[0]); 
      }
    mvec operator * (const mvec b) {
      auto& a = *this;
      if(mvec_uses_hex())
        return mvec(a[0]+b[0], a[1]+b[1], a[2]+b[2]+a[0]*b[1]-a[1]*b[0]);
      return mvec(a[0] + b[0], a[1] + b[1], a[2] + b[2] + a[0] * b[1]);
      }
    };
  #endif

  static constexpr mvec mvec_zero = mvec(0, 0, 0);

  EX ld nilwidth = 1;
      
  EX bool mvec_uses_hex() { return current_ns().mvec_hex; }

  EX hyperpoint mvec_to_point(mvec m) {
    if(mvec_uses_hex())
      return convert(hpxy3((m[0] + m[1] / 2.) * nilwidth, (m[1] * sqrt(3)/2) * nilwidth, m[2] * sqrt(3) / 4 * nilwidth * nilwidth), nmSym, model_used);
    return convert(hpxy3(m[0] * nilwidth, m[1] * nilwidth, m[2] * nilwidth * nilwidth), nmHeis, model_used);
    }
  
  #if HDR
  struct nilstructure {
    vector<mvec> movevectors;
    vector<vector<hyperpoint>> facevertices;
    bool mvec_hex;
    vector<int> other_side;
    string name;
    };
  #endif
  
  EX hyperpoint heis(ld x, ld y, ld z) { return convert(point31(x, y, z), nmHeis, model_used); }

  EX hyperpoint hexrota(ld x, ld y, ld z) { return convert(point31(x/2, y * sqrt(3)/6, z / 24 * sqrt(3)), nmSym, model_used); }

  EX int nil_structure_index;

  /** this returns nil_structure_index if it is used, otherwise -1 (if not nil) or -2 (if nil but not using nil_structure_index */
  EX int get_nsi() {
    if(!nil) return -1;
    if(mtwisted) return -2;
    return nil_structure_index;
    }

  nilstructure ns6 = {
    {{ mvec(-1,0,0), mvec(0,-1,0), mvec(0,0,-1), mvec(1,0,0), mvec(0,1,0), mvec(0,0,1) }},
  
    {{
    { heis(-0.5,-0.5,-0.25), heis(-0.5,-0.5,0.75), heis(-0.5,0.5,0.25), heis(-0.5,0.5,-0.75), },
    { heis(0.5,-0.5,-0.5), heis(0.5,-0.5,0.5), heis(-0.5,-0.5,0.5), heis(-0.5,-0.5,-0.5), },
    { heis(0,0,-0.5), heis(-0.5,0.5,-0.75), heis(-0.5,-0.5,-0.25), heis(0,0,-0.5), heis(-0.5,-0.5,-0.25), heis(-0.5,-0.5,-0.5), heis(0,0,-0.5), heis(-0.5,-0.5,-0.5), heis(0.5,-0.5,-0.5), heis(0,0,-0.5), heis(0.5,-0.5,-0.5), heis(0.5,-0.5,-0.75), heis(0,0,-0.5), heis(0.5,-0.5,-0.75), heis(0.5,0.5,-0.25), heis(0,0,-0.5), heis(0.5,0.5,-0.25), heis(0.5,0.5,-0.5), heis(0,0,-0.5), heis(0.5,0.5,-0.5), heis(-0.5,0.5,-0.5), heis(0,0,-0.5), heis(-0.5,0.5,-0.5), heis(-0.5,0.5,-0.75), },
    { heis(0.5,0.5,-0.25), heis(0.5,0.5,0.75), heis(0.5,-0.5,0.25), heis(0.5,-0.5,-0.75), },
    { heis(-0.5,0.5,-0.5), heis(-0.5,0.5,0.5), heis(0.5,0.5,0.5), heis(0.5,0.5,-0.5), },
    { heis(0,0,0.5), heis(-0.5,0.5,0.25), heis(-0.5,-0.5,0.75), heis(0,0,0.5), heis(-0.5,-0.5,0.75), heis(-0.5,-0.5,0.5), heis(0,0,0.5), heis(-0.5,-0.5,0.5), heis(0.5,-0.5,0.5), heis(0,0,0.5), heis(0.5,-0.5,0.5), heis(0.5,-0.5,0.25), heis(0,0,0.5), heis(0.5,-0.5,0.25), heis(0.5,0.5,0.75), heis(0,0,0.5), heis(0.5,0.5,0.75), heis(0.5,0.5,0.5), heis(0,0,0.5), heis(0.5,0.5,0.5), heis(-0.5,0.5,0.5), heis(0,0,0.5), heis(-0.5,0.5,0.5), heis(-0.5,0.5,0.25), },
    }},
    false,
    {3,4,5,0,1,2},
    "six sides"
    };
  
  nilstructure ns8 = {
    {{ mvec(-1,0,0), mvec(-1,0,1), mvec(0,-1,0), mvec(0,0,-1), mvec(1,0,0), mvec(1,0,-1), mvec(0,1,0), mvec(0,0,1) }},
  
    {{
      { heis(-0.5,-0.5,-0.25), heis(-0.5,-0.5,0.75), heis(-0.5,0.5,-0.25), },
      { heis(-0.5,-0.5,0.75), heis(-0.5,0.5,0.75), heis(-0.5,0.5,-0.25), },
      { heis(-0.5,-0.5,-0.25), heis(-0.5,-0.5,0.75), heis(0.5,-0.5,0.25), heis(0.5,-0.5,-0.75), },
      { heis(-0.5,-0.5,-0.25), heis(-0.5,0.5,-0.25), heis(0.5,0.5,-0.75), heis(0.5,-0.5,-0.75), },
      { heis(0.5,0.5,0.25), heis(0.5,-0.5,0.25), heis(0.5,-0.5,-0.75), },
      { heis(0.5,0.5,-0.75), heis(0.5,0.5,0.25), heis(0.5,-0.5,-0.75), },
      { heis(-0.5,0.5,0.75), heis(-0.5,0.5,-0.25), heis(0.5,0.5,-0.75), heis(0.5,0.5,0.25), },
      { heis(-0.5,-0.5,0.75), heis(-0.5,0.5,0.75), heis(0.5,0.5,0.25), heis(0.5,-0.5,0.25), },
      }},
    false,
    {4,5,6,7,0,1,2,3},
    "eight sides"
    };

  nilstructure nshex = {
    {{ mvec(1,0,0), mvec(1,-1,0), mvec(0,-1,0), mvec(-1,0,0), mvec(-1,1,0), mvec(0,1,0), mvec(0,0,-1), mvec(0,0,1) }}, 

    {{
{hexrota(1,-1,2),hexrota(1,-1,-4),hexrota(1,1,-2),hexrota(1,1,4)},
{hexrota(0,-2,2),hexrota(0,-2,-4),hexrota(1,-1,-2),hexrota(1,-1,4)},
{hexrota(-1,-1,2),hexrota(-1,-1,-4),hexrota(0,-2,-2),hexrota(0,-2,4)},
{hexrota(-1,1,2),hexrota(-1,1,-4),hexrota(-1,-1,-2),hexrota(-1,-1,4)},
{hexrota(0,2,2),hexrota(0,2,-4),hexrota(-1,1,-2),hexrota(-1,1,4)},
{hexrota(1,1,2),hexrota(1,1,-4),hexrota(0,2,-2),hexrota(0,2,4)},
{hexrota(0,0,-3),hexrota(1,1,-2),hexrota(1,-1,-4),hexrota(0,0,-3),hexrota(1,-1,-4),hexrota(1,-1,-2),hexrota(0,0,-3),hexrota(1,-1,-2),hexrota(0,-2,-4),hexrota(0,0,-3),hexrota(0,-2,-4),hexrota(0,-2,-2),hexrota(0,0,-3),hexrota(0,-2,-2),hexrota(-1,-1,-4),hexrota(0,0,-3),hexrota(-1,-1,-4),hexrota(-1,-1,-2),hexrota(0,0,-3),hexrota(-1,-1,-2),hexrota(-1,1,-4),hexrota(0,0,-3),hexrota(-1,1,-4),hexrota(-1,1,-2),hexrota(0,0,-3),hexrota(-1,1,-2),hexrota(0,2,-4),hexrota(0,0,-3),hexrota(0,2,-4),hexrota(0,2,-2),hexrota(0,0,-3),hexrota(0,2,-2),hexrota(1,1,-4),hexrota(0,0,-3),hexrota(1,1,-4),hexrota(1,1,-2)},
{hexrota(0,0,3),hexrota(1,1,4),hexrota(1,-1,2),hexrota(0,0,3),hexrota(1,-1,2),hexrota(1,-1,4),hexrota(0,0,3),hexrota(1,-1,4),hexrota(0,-2,2),hexrota(0,0,3),hexrota(0,-2,2),hexrota(0,-2,4),hexrota(0,0,3),hexrota(0,-2,4),hexrota(-1,-1,2),hexrota(0,0,3),hexrota(-1,-1,2),hexrota(-1,-1,4),hexrota(0,0,3),hexrota(-1,-1,4),hexrota(-1,1,2),hexrota(0,0,3),hexrota(-1,1,2),hexrota(-1,1,4),hexrota(0,0,3),hexrota(-1,1,4),hexrota(0,2,2),hexrota(0,0,3),hexrota(0,2,2),hexrota(0,2,4),hexrota(0,0,3),hexrota(0,2,4),hexrota(1,1,2),hexrota(0,0,3),hexrota(1,1,2),hexrota(1,1,4)}
       }},

    true,
    {3,4,5,0,1,2,7,6},
    "hex"
    };

  EX vector<nilstructure*> nil_structures = { &ns6, &ns8, &nshex };
  
  EX nilstructure& current_ns() { return *nil_structures[nil_structure_index]; }

  EX array<int,3> nilperiod, nilperiod_edit;
  int S7_edit;
  
  EX transmatrix adjmatrix(int i) {
     return nisot::translate(mvec_to_point(current_ns().movevectors[i]));
     }
    
  struct hrmap_nil : hrmap {
    map<mvec, heptagon*> at;
    map<heptagon*, mvec> coords;
    
    heptagon *getOrigin() override { return get_at(mvec_zero); }
    
    ~hrmap_nil() {
      for(auto& p: at) clear_heptagon(p.second);
      }

    heptagon *get_at(mvec c) {
      auto& h = at[c];
      if(h) return h;
      h = init_heptagon(S7);
      h->c7 = newCell(S7, h);
      coords[h] = c;
      h->zebraval = c[0];
      h->emeraldval = c[1];
      h->fieldval = c[2];
      return h;      
      }

    heptagon *create_step(heptagon *parent, int d) override {
      auto p = coords[parent];
      auto q = p * current_ns().movevectors[d];
      for(int a=0; a<3; a++) {
        auto oq = q[a];
        q[a] = zgmod(q[a], nilperiod[a]);
        if(mvec_uses_hex()) {
          if(a == 0) q[2] -= (oq - q[a]) * q[1];
          if(a == 1) q[2] += (oq - q[a]) * q[0];
          }
        else {
          if(a == 0) q[2] -= (oq - q[a]) * q[1];
          }
        }
      auto child = get_at(q);
      parent->c.connect(d, child, current_ns().other_side[d], false);
      return child;
      }

    transmatrix adj(heptagon *h, int i) override { return adjmatrix(i); }
  
    transmatrix relative_matrixh(heptagon *h2, heptagon *h1, const hyperpoint& hint) override { 
      for(int a=0; a<S7; a++) if(h2 == h1->move(a)) return adjmatrix(a);
      auto p = coords[h1].inverse() * coords[h2];
      for(int a=0; a<3; a++) p[a] = szgmod(p[a], nilperiod[a]);     
      return nisot::translate(mvec_to_point(p));
      }
    };

  EX mvec get_coord(heptagon *h) { return ((hrmap_nil*)currentmap)->coords[h]; }

  EX heptagon *get_heptagon_at(mvec m) { return ((hrmap_nil*)currentmap)->get_at(m); }

  EX void set_flags() {
    ginf[gNil].sides = isize(current_ns().movevectors);
    int coords = 0;
    for(int a=0; a<3; a++) if(nilperiod[a]) coords++;
    set_flag(ginf[gNil].flags, qANYQ, coords);
    set_flag(ginf[gNil].flags, qCLOSED, coords == 3);
    set_flag(ginf[gNil].flags, qSMALL, coords == 3 && nilperiod[0] * nilperiod[1] * nilperiod[2] <= 4096);
    }

  EX hyperpoint on_geodesic(hyperpoint s0, hyperpoint s1, ld x) {
    hyperpoint local = nisot::translate(s0, -1) * s1;
    hyperpoint h = get_inverse_exp(local);
    return nisot::translate(s0) * formula_exp(h * x);
    }

EX color_t colorize(cell *c, char whichCanvas) {
  mvec at = ((hrmap_nil*)currentmap)->coords[c->master];
  color_t res = 0;
  
  auto setres = [&] (int z, color_t which) {
    if(zgmod(at[2] - z, nilperiod[2]) == 0) res = which;
    if(zgmod(at[2] - z-1, nilperiod[2]) == 0) res = which;
    };

  if(at[1] == 0 && at[0] >=0 && at[0] < 4)
    setres(-at[0], gradient(0x1FF0000, 0x10000FF, 0, at[0], 4));
  else if(at[0] == 4 && at[1] >= 0 && at[1] < 4)
    setres(at[1]*3-4, gradient(0x10000FF, 0x100FF00, 0, at[1], 4));
  else if(at[1] == 4 && at[0] >= 0 && at[0] <= 4)
    setres(4+at[0], gradient(0x100FF00, 0x1FFFF00, 4, at[0], 0));
  else if(at[0] == 0 && at[1] >= 0 && at[1] <= 4)
    setres(at[1], gradient(0x1FFFF00, 0x1FF0000, 4, at[1], 0));
  
  return res;
  }

EX void prepare_niltorus3() {
  nilperiod_edit = nilperiod;
  S7_edit = nil_structure_index;
  }

EX void show_niltorus3() {
  cmode = sm::SIDE | sm::MAYDARK;
  gamescreen();
  dialog::init(XLAT("Nil quotient spaces"));
  for(int a=0; a<3; a++) {
    string title = XLAT("%1 period", s0+char('X'+a));
    dialog::addSelItem(title, its(nilperiod_edit[a]), 'x');
    dialog::add_action([=] { 
      dialog::editNumber(nilperiod_edit[a], 0, 60, 1, 0, title, 
        XLAT("Set to 0 to make it non-periodic.")
        ); 
      dialog::bound_low(0);
      });      
    }
  dialog::addSelItem(XLAT("honeycomb"), XLAT(nil_structures[S7_edit]->name), 'h');
  dialog::add_action([] { S7_edit = (S7_edit+1)%isize(nil_structures);  });

  int hx = nil_structures[S7_edit]->mvec_hex ? 2 : 1;
  bool ok = !zgmod(nilperiod_edit[0]*nilperiod_edit[1]*hx, nilperiod_edit[2]);

  dialog::addBreak(50);

  if(ok) {
    dialog::addItem(XLAT("activate"), 'a');
    dialog::add_action([] {
      stop_game();
      nilperiod = nilperiod_edit;
      nil_structure_index = S7_edit;
      set_flags();
      geometry = gNil;
      start_game();
      });
    }
  else dialog::addInfo(XLAT("Y period * X period (*2 for hex) must be divisible by Z period"));
    
  dialog::addBreak(50);
  dialog::addBack();
  dialog::display();
  }

EX void create_faces() {
  for(int i=0; i<S7; i++) {
    vector<hyperpoint> fvs = nilv::current_ns().facevertices[i];
    using nilv::nilwidth;
    for(auto& h: fvs) h[0] *= nilwidth, h[1] *= nilwidth, h[2] *= nilwidth * nilwidth;
    add_wall(i, fvs);
    }
  get_hsh().compute_hept();
  }
  
EX }

EX bool in_s2xe() { return gproduct && hybrid::under_class() == gcSphere; }
EX bool in_h2xe() { return gproduct && hybrid::under_class() == gcHyperbolic; }
EX bool in_e2xe() { return gproduct && hybrid::under_class() == gcEuclid; }

EX namespace hybrid {

  EX eGeometry underlying;
  EX geometry_information *underlying_cgip;

  EX eGeometryClass under_class() {
    if(embedded_plane) {
      auto c = geom3::ginf_backup[geometry].cclass;
      if(c == gcEuclid) c = cginf.g.sig[2] > 0 ? gcSphere : gcHyperbolic;
      return c;
      }
    return ginf[hybrid::underlying].cclass;
    }

  EX int csteps;
  
  EX int disc_quotient = 0;
  
  EX map<heptagon*, short> altmap_heights;

  EX void configure(eGeometry g) {
    if(WDIM == 3) return;
    ray::reset_raycaster();
    check_cgi();
    cgi.require_basics();
    underlying = geometry;
    underlying_cgip = cgip;
    bool sph = sphere;
    bool euc = euclid;
    auto keep = ginf[g].menu_displayed_name;
    ginf[g] = ginf[underlying];
    ginf[g].menu_displayed_name = keep;
    if(g == gTwistedProduct) {
      ginf[g].g = euc ? giNil : sph ? giSphere3 : giSL2;
      ginf[g].tiling_name = "twisted " + ginf[g].tiling_name + "xZ";
      string& qn = ginf[g].quotient_name;
      if(csteps && csteps != (sph ? cgi.psl_steps*2 : 0)) {
        string qplus;
        if(csteps == cgi.psl_steps)
          qplus = sph ? "elliptic" : "PSL";
        else if(csteps == 2 * cgi.psl_steps && !sph)
          qplus = "SL";
        else qplus = its(csteps);
        if(qn == "none") qn = qplus;
        else qn = qn + "/" + qplus;
        }
      if(sph) ginf[g].flags |= qELLIPTIC;
      if(csteps && csteps != cgi.psl_steps && csteps != 2*cgi.psl_steps) 
        ginf[g].flags |= qANYQ;
      }
    else {
      ginf[g].cclass = gcProduct;
      ginf[g].g.gameplay_dimension++;
      ginf[g].g.graphical_dimension++;
      ginf[g].tiling_name += "xZ";
      if(csteps) ginf[g].flags |= qANYQ, ginf[g].tiling_name += its(csteps);
      }
    ginf[g].flags |= qHYBRID;
    }
  
  EX void reconfigure() {
    if(!mhybrid) return;
    stop_game();
    auto g = geometry;
    geometry = underlying;
    configure(g);
    geometry = g;
    }

  EX void enable_rotspace() {
    if(euclid) {
      start_game();
      int q = cwt.at->type;
      stop_game();
      hybrid::csteps = q;
      set_plevel(TAU / q);
      set_geometry(gProduct);
      hybrid::reconfigure();
      }
    else {
      stop_game();
      set_geometry(gTwistedProduct);
      check_cgi(); cgi.require_basics();
      hybrid::csteps = cgi.psl_steps;
      hybrid::reconfigure();
      }
    }

  EX void fixup_csteps() {
    check_cgi(); cgi.require_basics();
    if(!hybrid::csteps || gmod(cgi.psl_steps, hybrid::csteps)) {
      hybrid::csteps = cgi.psl_steps;
      if(PIU(fake::in() ? FPIU(euclid) : euclid)) {
        auto& T = euc::eu_input.user_axes;
        hybrid::csteps = abs(T[0][0] * T[1][1] - T[0][1] * T[1][0]);
        if(S3 == 3) hybrid::csteps *= 2;
        if(BITRUNCATED) hybrid::csteps *= S3;
        if(INVERSE) hybrid::csteps *= 2;
        }
      hybrid::reconfigure();
      }
    }

  EX hrmap *pmap;
  EX geometry_information *pcgip;
  EX eGeometry actual_geometry;
  
  #if HDR
  template<class T> auto in_actual(const T& t) -> decltype(t()) {
    if(pmap == nullptr) return t();
    dynamicval<eGeometry> g(geometry, actual_geometry);
    dynamicval<geometry_information*> gc(cgip, pcgip);
    dynamicval<hrmap*> gu(currentmap, pmap);
    dynamicval<hrmap*> gup(pmap, NULL);
    return t();  
    }

  #define PIA(x) hr::hybrid::in_actual([&] { return (x); })
  #endif
  
  EX void set_plevel(ld lev) {
    stop_game();
    vid.plevel_factor = 1;
    check_cgi(); cgi.prepare_basics();
    vid.plevel_factor = lev / cgi.plevel;
    check_cgi();
    }

  struct hrmap_hybrid : hrmap {
    
    hrmap *underlying_map;
    
    bool twisted;
    map<cell*, pair<cellwalker, cellwalker>> spins;
    
    map<pair<cell*, int>, cell*> at;
    map<cell*, pair<cell*, int>> where;
    
    heptagon *getOrigin() override { return underlying_map->getOrigin(); }

    const int SHIFT_UNKNOWN = 30000;
    map<cell*, vector<int>> shifts;
    map<cell*, ld> orig_height;
  
    EX vector<int>& make_shift(cell *c) {
      auto& res = shifts[c];
      if(res.empty()) res = vector<int> (c->type+1, SHIFT_UNKNOWN);
      return res;
      }
    
    EX int& get_shift_current(cellwalker cw) {
      return make_shift(cw.at)[cw.spin];
      }
    
    EX bool have_shift(cellwalker cw) {
      return shifts.count(cw.at) && get_shift_current(cw) != SHIFT_UNKNOWN;
      }
    
    EX int get_shift(cellwalker cw0) {
      if(S3 >= OINF) return 0;
      auto& v = get_shift_current(cw0);
      if(v != SHIFT_UNKNOWN) return v;

      /** note: will not be done in fix_bounded_cycles because then it is already called in underlying map, but that is fine */
      if(nil) {
        /** we prevent possible 'two candidate' errors by computing the correct value, we know all the positions in the underlying map, so we can do that */
        transmatrix uT, uU, uT1;
        in_underlying([&] {
          uT = currentmap->relative_matrix(cw0.at, currentmap->gamestart(), C0);
          uU = currentmap->adj(cw0.at, cw0.spin);
          uT1 = currentmap->relative_matrix(cw0.cpeek(), currentmap->gamestart(), C0);
          });
        transmatrix lT = twist::lift_matrix(uT);
        transmatrix lU = twist::lift_matrix(uU);
        transmatrix lT1 = twist::lift_matrix(uT1);
        if(!orig_height.count(cw0.at)) orig_height[cw0.at] = (lT*C0) [2] / nilv::nilwidth / nilv::nilwidth;
        ld diff = (lT * lU * iso_inverse(lT1) * C0)[2] / nilv::nilwidth / nilv::nilwidth - orig_height[cw0.at];
        if(!orig_height.count(cw0.peek())) orig_height[cw0.peek()] = -diff;
        diff += orig_height[cw0.peek()];
        if(abs(frac(diff / cgi.plevel + 0.5) - 0.5) > 1e-6) throw hr_exception("not an integer in get_shift");
        v = floor(diff / cgi.plevel + 0.5);
        return v;
        }

      vector<int> candidates;
      
      for(int a: {1, -1}) {
        cellwalker cw = cw0;
        cw += wstep; cw += a;
        int s = 0;
        while(cw != cw0) {
          if(!have_shift(cw)) goto next;
          s += shifts[cw.at][cw.spin];
          cw += wstep;
          cw += a;
          }
        candidates.push_back(-a * cgi.single_step * (sphere ? -1 : 1) - s);
        next: ;
        }
      
      if(candidates.size() == 2 && candidates[0] != candidates[1]) {
        bool over_closed = in_underlying([] { return quotient || sphere || closed_manifold; });
        if(!over_closed) println(hlog, "two candidates in shift : ", candidates);
        int val = candidates[0] - candidates[1];
        int old_disc_quotient = disc_quotient;
        if(disc_quotient == 0) disc_quotient = val;
        disc_quotient = gcd(val, disc_quotient);
        if(disc_quotient < 0) disc_quotient = -disc_quotient;
        if(old_disc_quotient != disc_quotient && !over_closed)
          addMessage(XLAT("ERROR: failed to solve the twist values, the map will be incorrect", its(disc_quotient)));
        }
  
      int val = 0;
      
      auto cw1 = cw0+wstep;
      if(1) {
        /* the value from PSL, helps to draw the underlying space correctly */
        auto ps = cgi.psl_steps;
        val = cw0.spin*ps / cw0.at->type - cw1.spin*ps / cw1.at->type + ps/2;
        }
      if(!candidates.empty()) val = candidates[0];
      
      v = val;
      get_shift_current(cw1) = -val;
            
      return val;
      }  
    
    EX void ensure_shifts(cell *c) {
      if(S3 >= OINF) return;
      if(!make_shift(c)[c->type]) return;

      if(geometry == gNormal) {
        forCellEx(c1, c) if(celldist(c1) < celldist(c)) { ensure_shifts(c1); }
        }

      for(int a=0; a<c->type; a++) {
        cellwalker cw0(c, a);
        cellwalker cw = cw0;
        do {
          get_shift(cw);
          cw += wstep;
          cw ++;
          }
        while(cw != cw0);
        }

      make_shift(c)[c->type] = 0;
      }
    
    EX int cycle_discrepancy(cellwalker cw0) {
      int total = 0;
      auto cw = cw0;
      do {
        total += get_shift(cw);
        cw += wstep;
        cw++;
        }
      while(cw != cw0);
      return total + cgi.single_step * ((sphere || euclid) ? -1 : 1);
      }
    
    EX void fix_bounded_cycles() {
      if(!mtwisted) return;
      if(!closed_manifold) return;
      in_underlying([&] {
        cellwalker final(currentmap->gamestart(), 0);
        auto& ac = currentmap->allcells();
        for(cell *c: ac) for(int i=0; i<c->type; i++) {
          cellwalker cw(c, i);
          int cd = cycle_discrepancy(cw);
          if(!cd) continue;
          while(cw != final) {
            if(celldist(cw.peek()) < celldist(cw.at)) {
              cw += wstep;
              cw++;
              }              
            else {
              get_shift(cw);
              get_shift_current(cw) -= cd;
              get_shift_current(cw+wstep) += cd;
              cw++;
              }
            }
          }

        disc_quotient = abs(cycle_discrepancy(final));

        if(debugflags & DF_GEOM) for(cell *c: ac) for(int i=0; i<c->type; i++) {
          cellwalker cw(c, i);
          if(cycle_discrepancy(cw)) println(hlog, cw, " ", cycle_discrepancy(cw));
          }
        if(debugflags & DF_GEOM) for(cell *c: ac) for(int i=0; i<c->type; i++) {
          auto err = get_shift(cellwalker(c, i)) + get_shift(cellwalker(c, i)+wstep);
          if(err)
            println(hlog, "two-side error: ", err, " on ", cellwalker(c, i));
          }
        });
      }
    
    template<class T> auto in_underlying(const T& t) -> decltype(t()) {
      pcgip = cgip; 
      dynamicval<hrmap*> gpm(pmap, this);
      dynamicval<eGeometry> gag(actual_geometry, geometry);
      dynamicval<eGeometry> g(geometry, underlying);
      dynamicval<int> gss(underlying_cgip->single_step, cgi.single_step);
      dynamicval<int> gsp(underlying_cgip->psl_steps, cgi.psl_steps);
      dynamicval<geometry_information*> gc(cgip, underlying_cgip);
      dynamicval<hrmap*> gu(currentmap, underlying_map);
      return t();
      }
    
    cell *getCell(cell *u, int h) {
      if(twisted) {
        if(!spins.count(u))
          println(hlog, "link missing: ", u);
        else {
          while(h >= csteps) h -= csteps, u = spins[u].first.at;
          while(h < 0) h += csteps, u = spins[u].second.at;
          }
        }
      h = zgmod(h, csteps);
      cell*& c = at[make_pair(u, h)];
      if(!c) { c = newCell(u->type+2, u->master); where[c] = {u, h}; }
      return c;
      }
  
    cell* gamestart() override { return getCell(underlying_map->gamestart(), 0); }
  
    hrmap_hybrid() {
      underlying_map = nullptr;
      twisted = false;
      disc_quotient = 0;
      in_underlying([this] { initcells(); underlying_map = currentmap; });
      for(hrmap*& m: allmaps) if(m == underlying_map) m = NULL;
      fix_bounded_cycles();
      }
    
    ~hrmap_hybrid() {
      in_underlying([] { delete currentmap; });
      for(auto& p: at) destroy_cell(p.second);
      }
    
    void find_cell_connection(cell *c, int d) override { 
      hybrid::find_cell_connection(c, d); 
      }

    int shvid(cell *c) override {
      cell *c1 = hybrid::get_where(c).first; 
      return PIU( hr::shvid(c1) );      
      }

    int full_shvid(cell *c) override {
      cell *c1 = hybrid::get_where(c).first;
      return PIU( currentmap->full_shvid(c1) );
      }

    transmatrix spin_to(cell *c, int d, ld bonus) override { if(d >= c->type-2) return Id; c = get_where(c).first; return fix4_f( in_underlying([&] { return currentmap->spin_to(c, d, bonus); }) ); }
    transmatrix spin_from(cell *c, int d, ld bonus) override { if(d >= c->type-2) return Id; c = get_where(c).first; return fix4_f( in_underlying([&] { return currentmap->spin_from(c, d, bonus); }) ); }

    subcellshape& get_cellshape(cell *c) override {      
      int id = full_shvid(c);
      return generate_subcellshape_if_needed(c, id);      
      }
    };
  
  hrmap_hybrid* hmap() { return (hrmap_hybrid*) currentmap; }
  
  EX cell *get_at(cell *base, int level) {
    return hmap()->getCell(base, level);
    }
  
  EX pair<cell*, int> get_where(cell *c) { return hmap()->where[c]; }
  
  EX void find_cell_connection(cell *c, int d) {
    auto m = hmap();
    if(d >= c->type - 2) {
      int s = cgi.single_step;
      int lev = m->where[c].second + (d == c->type-1 ? s : -s);
      cell *c1 = get_at(m->where[c].first, lev);
      c->c.connect(d, c1, c1->type - 3 + c->type - d, false);
      }
    else {
      auto cu = m->where[c].first;
      auto cu1 = m->in_underlying([&] { return cu->cmove(d); });
      int d1 = cu->c.spin(d);
      int s = 0;
      if(geometry == gTwistedProduct) {
        auto cm = (hrmap_hybrid*)currentmap;
        m->in_underlying([&] { cm->ensure_shifts(cu); });
        s = ((hrmap_hybrid*)currentmap)->get_shift(cellwalker(cu, d));
        }
      cell *c1 = get_at(cu1, m->where[c].second + s);
      c->c.connect(d, c1, d1, cu->c.mirror(d));
      }
    }  

  EX hrmap* get_umap() { if(!dynamic_cast<hrmap_hybrid*>(currentmap)) return nullptr; else return ((hrmap_hybrid*)currentmap)->underlying_map; }
  
  #if HDR
  template<class T> auto in_underlying_geometry(const T& f) -> decltype(f()) {
    if(!mhybrid && !gproduct) return f();
    if(embedded_plane) {
      if(cgi.emb->is_euc_in_product()) {
        dynamicval<eGeometryClass> dgc(cginf.g.kind, cginf.g.sig[2] < 0 ? gcHyperbolic : gcSphere);
        return f();
        }
      if(cgi.emb->is_cylinder()) {
        dynamicval<eGeometryClass> dgc(cginf.g.kind, cginf.g.sig[2] < 0 ? gcHyperbolic : gcSphere);
        return f();
        }
      geom3::light_flip(true);
      finalizer ff([] { geom3::light_flip(false); });
      return f();
      }
    if(geom3::flipped) throw hr_exception("called in_underlying_geometry in flipped");
    pcgip = cgip;
    dynamicval<eGeometry> gag(actual_geometry, geometry);
    dynamicval<eGeometry> g(geometry, underlying);
    dynamicval<int> gss(underlying_cgip->single_step, cgi.single_step);
    dynamicval<int> gsp(underlying_cgip->psl_steps, cgi.psl_steps);
    dynamicval<geometry_information*> gc(cgip, underlying_cgip);
    dynamicval<hrmap*> gpm(pmap, currentmap);
    dynamicval<hrmap*> gm(currentmap, get_umap());
    return f();
    }
  
  #define PIU(x) hr::hybrid::in_underlying_geometry([&] { return (x); })
  #endif

  /** like in_underlying_geometry but does not return */
  EX void switch_to_underlying() {
    if(!mhybrid && !gproduct) return;
    if(embedded_plane) throw hr_exception("switch_to_underlying in embedded_plane");
    auto m = hmap();
    pmap = m;
    actual_geometry = geometry;
    geometry = underlying;
    underlying_cgip->single_step = cgi.single_step;
    underlying_cgip->psl_steps = cgi.psl_steps;
    pcgip = cgip;
    cgip = underlying_cgip;
    currentmap = m->underlying_map;
    }

  /** like in_actual but does not return */
  EX void switch_to_actual() {
    if(!pmap) return;
    geometry = actual_geometry;
    cgip = pcgip;
    currentmap = pmap;
    pmap = nullptr;
    }

  // next: 0 = i-th corner, 1 = next corner, 2 = center of the wall
  EX hyperpoint get_corner(cell *c, int i, int next, ld z) {
    ld lev = cgi.plevel * z / 2;
    if(WDIM == 2) {
      ld zz = lerp(cgi.FLOOR, cgi.WALL, (1+z) / 2);
      hyperpoint h = orthogonal_move(get_corner_position(c, i+next), zz);
      return h;
      }
    if(gproduct) {
      dynamicval<eGeometry> g(geometry, hybrid::underlying);
      dynamicval<geometry_information*> gc(cgip, hybrid::underlying_cgip);
      dynamicval<hrmap*> gm(currentmap, ((hrmap_hybrid*)currentmap)->underlying_map);
      return scale_point(get_corner_position(c, i+next), exp(lev));
      }
    else {
      #if MAXMDIM >= 4
      ld tf, he, alpha;
      in_underlying_geometry([&] {
        hyperpoint h1 = get_corner_position(c, i);
        hyperpoint h2 = get_corner_position(c, i+1);
        hyperpoint hm;
        if(next == 2) {
          hm = h1;
          he = 0;
          }
        else {
          hyperpoint hm = mid(h1, h2);
          he = hdist(hm, h2)/2;
          if(next) he = -he;
          }
        tf = hdist0(hm)/2;
        alpha = atan2(hm[1], hm[0]);
        });
      return spin(alpha) * twist::uxpush(tf) * twist::uypush(he) * twist::uzpush(lev) * C0;
      #else
      throw hr_exception();
      #endif
      }
    }
  
  auto clear_samples = addHook(hooks_clearmemory, 40, [] () {
    for(auto& c: cgis) for(auto& v: c.second.walloffsets)
      v.second = nullptr;
    altmap_heights.clear();
    });
  
  EX vector<pair<int, cell*>> gen_sample_list() {
    if(!mhybrid && WDIM != 2 && PURE)
      return {make_pair(0, centerover), make_pair(centerover->type, nullptr)};
    vector<pair<int, cell*>> result;
    for(auto& v: cgi.walloffsets) if(v.first >= 0) result.push_back(v);
    sort(result.begin(), result.end());
    int last = 0;
    for(auto& r: result) if(r.second) last = r.first + r.second->type  + (WDIM == 2 ? 2 : 0);
    result.emplace_back(last, nullptr);
    return result;
    }

  vector<cell*> to_link;
  
  EX void will_link(cell *c) { if(pmap && ((hrmap_hybrid*) pmap)->twisted) to_link.push_back(c); }
  
  EX bool in_link = false;
  
  EX void link() {
    if(in_link) return;
    dynamicval<bool> b(in_link, true);
    auto pm = (hrmap_hybrid*) pmap;
    if(!pm) return;
    auto& ss = pm->spins;
    int success = -1;
    while(success) {
      vector<cell*> xlink = std::move(to_link);
      success = 0;
      for(cell *c: xlink) {
        bool success_here = ss.count(c);
        if(!success_here) forCellIdEx(c2, i, c) if(ss.count(c2)) {
          ss[c].first = ss[c2].first + c->c.spin(i) + wstep - i;
          ss[c].second = ss[c2].second + c->c.spin(i) + wstep - i;
          success++;
          success_here = true;
          break;
          }
        if(!success_here) to_link.push_back(c);
        }
      }
    }

  EX int celldistance(cell *c1, cell *c2) {
    if(sl2) {
      auto w1 = hybrid::get_where(c1), w2 = hybrid::get_where(c2); 
      return PIU (hr::celldistance(w1.first, w2.first));
      }
    else if(csteps == 0) {
      auto w1 = hybrid::get_where(c1), w2 = hybrid::get_where(c2); 
      return PIU (hr::celldistance(w1.first, w2.first)) + abs(w1.second - w2.second);
      }
    else {
      int s = 0;
      int a = 999999, b = -999999;
      auto c = c1;
      do {
        auto w1 = hybrid::get_where(c), w2 = hybrid::get_where(c2); 
        if(w1.second == w2.second) {
          int d = PIU(hr::celldistance(w1.first, w2.first));
          a = min(s+d, a);
          b = max(s-d, b);
          }
        c = c->cmove(c1->type-1); s++;
        }
      while(c != c1);
      return min(a, s-b);
      }
    }

  EX void configure_period() {
    static int s;
    s = csteps / cgi.single_step;
    string str = "";
    if(mtwisted)
      str = XLAT(
        "If the 2D underlying manifold is bounded, the period should be a divisor of the 'rotation space' "
        "value (PSL(2,R)) times the Euler characteristics of the underlying manifold. "
        "For unbounded underlying manifold, any value should work (theoretically, "
        "the current implementation in HyperRogue is not perfect).\n\n"
        "We won't stop you from trying illegal numbers, but they won't work correctly.");
    dialog::editNumber(s, 0, 16, 1, 0, XLAT("%1 period", "Z"), str);
    dialog::bound_low(0);
    auto set_s = [] (int s1, bool ret) {
      return [s1,ret] {
        if(ret) popScreen();
        if(csteps == s1) return;
        stop_game();
        csteps = s1 * cgi.single_step;
        hybrid::reconfigure();
        start_game();
        };
      };
    dialog::get_di().extra_options = [=] () {
      if(mtwisted) {
        int e_steps = cgi.psl_steps / gcd(cgi.single_step, cgi.psl_steps); 
        bool ubounded = PIU(closed_manifold);
        dialog::addSelItem( sphere ? XLAT("elliptic") : XLAT("PSL(2,R)"), its(e_steps), 'P');
        dialog::add_action(set_s(e_steps, true));
        dialog::addSelItem( sphere ? XLAT("sphere") : XLAT("SL(2,R)"), its(2*e_steps), 'P');
        dialog::add_action(set_s(2*e_steps, true));
        if(sl2 && !ubounded) {
          dialog::addSelItem( XLAT("universal cover"), its(0), 'P');
          dialog::add_action(set_s(0, true));
          }
        dialog::addSelItem(ubounded ? XLAT("maximum") : XLAT("works correctly so far"), its(disc_quotient), 'Q');
        dialog::add_action(set_s(disc_quotient, true));
        }
      else {
        dialog::addSelItem( XLAT("non-periodic"), its(0), 'N');
        dialog::add_action(set_s(0, true));
        }
      dialog::get_di().reaction_final = set_s(s, false);
      };
    }

  EX ld underlying_scale = 0;
  EX bool drawing_underlying = false;
  EX bool underlying_as_pc = true;

  EX void draw_underlying(bool cornermode) {
    if(underlying_scale <= 0) return;
    ld d = hybrid::get_where(centerover).second;
    d *= cgi.plevel;
    transmatrix T = twist::uzpush(-d) * spin(-2*d);

    if(det(T) < 0) T = centralsym * T;

    ld orig_d = d;
    if(mproduct) d = 0;

    hyperpoint h = inverse(View * spin(master_to_c7_angle()) * T) * C0;

    auto g = std::move(gmatrix);
    auto g0 = std::move(gmatrix0);

    ld alpha = atan2(ortho_inverse(NLP) * point3(1, 0, 0));

    dynamicval<transmatrix> dn(NLP);
    dynamicval<transmatrix> dv(View);

    bool inprod = mproduct;
    transmatrix pView = View;

    if(inprod) {
      ld z = zlevel(tC0(View));

      /* special case when we are actually in E2xE as a rotation space */
      if(in_e2xe() && abs(cgi.plevel * hybrid::csteps - TAU) < 1e-6) alpha = orig_d - z;

      println(hlog, "depth = ", cgi.plevel * hybrid::csteps, " orig_d = ", orig_d, " z = ", z);

      pView = spin(alpha) * View;

      for(int a=0; a<3; a++) pView[a] *= exp(-z);
      }

    cell *co = hybrid::get_where(centerover).first;

    hybrid::in_underlying_geometry([&] {
      cgi.require_shapes();
      dynamicval<int> pcc(corner_centering, cornermode ? 1 : 2);
      dynamicval<bool> pf(playerfound, true);
      dynamicval<cell*> m5(centerover, co);
      dynamicval<transmatrix> m2(View, inprod ? pView : ypush(0) * twist::qtm(h));
      if(PURE && !inprod) View = View * pispin;
      View = inverse(stretch::mstretch_matrix) * spin(2*d) * View;
      dynamicval<shiftmatrix> m3(playerV, shiftless(Id));
      dynamicval<transmatrix> m4(actual_view_transform, Id);
      dynamicval<shiftmatrix> m6(cwtV, shiftless(Id));
      dynamicval<eModel> pm(pmodel, mdDisk);
      dynamicval<ld> pss(pconf.scale, (sphere ? 10 : euclid ? .4 : 1) * underlying_scale);
      dynamicval<ld> psa(pconf.alpha, sphere ? 10 : 1);
      dynamicval<hrmap*> p(hybrid::pmap, NULL);
      dynamicval<int> psr(sightrange_bonus, 0);

      dynamicval<int> psx(vid.use_smart_range, 2);
      dynamicval<ld> psy(vid.smart_range_detail, 1);
      dynamicval<bool> pdu(drawing_underlying, true);

      calcparam();
      reset_projection(); current_display->set_all(0, 0);
      ptds.clear();
      drawthemap();
      if(underlying_as_pc) drawPlayer(moPlayer, centerover, sphere ? shiftless(xpush(M_PI) * spin90()) : shiftless(spin90()), 0xFFFFFFFF, 0);
      drawqueue();
      if(!underlying_as_pc) displaychr(current_display->xcenter, current_display->ycenter, 0, 24 * mapfontscale / 100, '+', 0xFFFFFFFF);
      glflush();
      });
    gmatrix = std::move(g);
    gmatrix0 = std::move(g0);
    calcparam();
    reset_projection(); current_display->set_all(0, 0); make_actual_view();
    }

EX }
  
EX namespace product {

  int z0;
  
  struct hrmap_product : hybrid::hrmap_hybrid {
    transmatrix relative_matrixc(cell *c2, cell *c1, const hyperpoint& hint) override {
      return in_underlying([&] { return calc_relative_matrix(where[c2].first, where[c1].first, hint); }) * cpush(2, cgi.plevel * szgmod(where[c2].second - where[c1].second, hybrid::csteps));
      }

    transmatrix adj(cell *c, int i) override {
      if(twisted && i == c->type-1 && where[c].second == hybrid::csteps-1) {
        auto b = spins[where[c].first].first;
        transmatrix T = cpush(2, cgi.plevel);
        T = T * spin(TAU * b.spin / b.at->type);
        if(b.mirrored) T = T * Mirror;
        return T;
        }
      if(twisted && i == c->type-2 && where[c].second == 0) {
        auto b = spins[where[c].first].second;
        transmatrix T = cpush(2, -cgi.plevel);
        T = T * spin(TAU * b.spin / b.at->type);
        if(b.mirrored) T = T * Mirror;
        return T;
        }
      if(i == c->type-1) return cpush(2, cgi.plevel);
      else if(i == c->type-2) return cpush(2, -cgi.plevel);
      c = where[c].first;
      return PIU(currentmap->adj(c, i));
      }
  
    hrmap_product() {
      current_spin_invalid = false;
      using hybrid::csteps;
      if((cspin || cmirror) && csteps) {
        in_underlying([&] {
          twisted = validate_spin();
          if(!twisted) { current_spin_invalid = true; return; }
          auto ugs = currentmap->gamestart();
          spins[ugs] = make_pair(
            cellwalker(ugs, gmod(+cspin, ugs->type), cmirror),
            cellwalker(ugs, gmod(-cspin, ugs->type), cmirror)
            );
          manual_celllister cl;
          cl.add(ugs);
          for(int i=0; i<isize(cl.lst); i++) {
            cell *c = cl.lst[i];
            hybrid::will_link(c);
            forCellEx(c2, c) cl.add(c2);
            }
          hybrid::link();
          });
        }
      }

    transmatrix ray_iadj(cell *c, int i) override {
      if(i == c->type-2) return (cpush(2, +cgi.plevel));
      if(i == c->type-1) return (cpush(2, -cgi.plevel));
      transmatrix T;
      cell *cw = hybrid::get_where(c).first;
      hybrid::in_underlying_geometry([&] {
        T = currentmap->ray_iadj(cw, i);
        });
      return T;
      }
    };

  EX bool current_spin_invalid, cmirror;
  EX int cspin;
    
  /* todo might need a shiftpoint version */
  EX hyperpoint inverse_exp(hyperpoint h) {
    hyperpoint res;
    res[2] = zlevel(h);
    h = h * exp(-res[2]);
    ld r = hypot_d(2, h);
    if(hybrid::under_class() == gcEuclid) {
      res[0] = h[0];
      res[1] = h[1];
      }
    else if(r < 1e-6) {
      res[0] = h[0];
      res[1] = h[1];
      }
    else {
      auto c = acos_auto_clamp(h[2]);
      r = c / r;
      res[0] = h[0] * r;
      res[1] = h[1] * r;
      }
    return res;
    }
  
  EX hyperpoint direct_exp(hyperpoint h) {
    hyperpoint res;
    ld d = hypot_d(2, h);
    ld cd = d == 0 ? 0 : sin_auto(d) / d;
    res[0] = h[0] * cd;
    res[1] = h[1] * cd;
    res[2] = cos_auto(d);
    return res * exp(h[2]);
    }

  EX bool validate_spin() {
    if(mproduct) return hybrid::in_underlying_geometry(validate_spin);
    if(aperiodic) return false;
    if(!quotient && !arcm::in()) return true;
    map<cell*, cellwalker> cws;
    manual_celllister cl;
    cell *start = currentmap->gamestart();
    cl.add(start);
    cws[start] = cellwalker(start, gmod(cspin, start->type), cmirror);
    for(int i=0; i<isize(cl.lst); i++) {
      cell *c = cl.lst[i];
      cellwalker cwc = cws.at(c);
      forCellIdEx(c2, j, c) {
        cellwalker cwc2 = cwc + j + wstep - c->c.spin(j);
        if(!cws.count(c2)) cws[c2] = cwc2;
        else if(cws[c2] != cwc2) return false;
        cl.add(c2);
        }
      }
    return true;
    }

  EX void show_config() {
    cmode = sm::SIDE | sm::MAYDARK;
    gamescreen();
    dialog::init(XLAT("quotient product spaces"));
    dialog::addSelItem(XLAT("%1 period", "Z"), its(hybrid::csteps), 'z');
    dialog::add_action(hybrid::configure_period);
    dialog::addSelItem(XLAT("rotation"), its(cspin), 'r');
    dialog::add_action([] {
      static int s;
      dialog::editNumber(s, 0, 16, 1, 0, XLAT("rotation", "Z"), 
        XLAT("Works if the underlying space is symmetric.")
        );
      dialog::get_di().reaction_final = [] {
        if(s == cspin) return;
        stop_game();
        cspin = s;
        start_game();
        };
      });
    dialog::addBoolItem(XLAT("reflect"), cmirror, 'f');
    dialog::add_action([]{
      stop_game();
      cmirror = !cmirror;
      start_game();
      });
    if(current_spin_invalid) 
      dialog::addInfo("the current rotation is invalid");
    else
      dialog::addBreak(100);

    dialog::addBreak(50);
    dialog::addBack();
    dialog::display();
    } 
  
EX }

EX namespace slr {

  /** in what range are we rendering SL(2,R) */
  EX ld range_xy = 2;

  /** in what Z range are we rendering SL(2,R) */
  EX ld range_z = 2;
  
  /** the number of steps for inverse_exp in the shader */
  EX int shader_iterations = 15;

  EX transmatrix translate(hyperpoint h) {
    return matrix4(
      h[3], -h[2],  h[1],  h[0],
      h[2],  h[3], -h[0],  h[1],
      h[1], -h[0],  h[3],  h[2],
      h[0],  h[1], -h[2],  h[3]
      );
    }
  
  EX hyperpoint polar(ld r, ld theta, ld phi) {
    return hyperpoint(sinh(r) * cos(theta-phi), sinh(r) * sin(theta-phi), cosh(r) * sin(phi), cosh(r) * cos(phi));
    }

  EX hyperpoint xyz_point(ld x, ld y, ld z) {
    ld r = hypot(x, y);
    ld f = r ? sinh(r) / r : 1;
    return hyperpoint(x * f * cos(z) + y * f * sin(z), y * f * cos(z) - x * f * sin(z), cosh(r) * sin(z), cosh(r) * cos(z));
    }
  
  EX hyperpoint get_inverse_exp(shiftpoint h) {
    ld xy = hypot_d(2, h.h);
    ld phi = atan2(h[2], h[3]) + h.shift;

    if(xy < 1e-6) return point31(0.,0.,phi);

    bool flipped = phi > 0;
    if(flipped) phi = -phi;
    
    ld SV = stretch::not_squared();
    ld K = -1;
    
    ld alpha = flipped ? atan2(h[1], h[0]) - h.shift : atan2(h[1], -h[0]) + h.shift;
    
    hyperpoint res;
    
    ld fiber_barrier = atan(1/SV);
    
    ld flip_barrier = atan( 1 / tanh(asinh(xy)) / SV);
    
    // test the side of the flip barrier
    
    int part = -1;
    
    if(1) {
      ld kk = flip_barrier;
      
      ld x_part = cos(kk);
      ld z_part = sin(kk);
  
      ld rparam = x_part / z_part / SV;
      
      ld r = atanh(rparam);
      
      ld cr = cosh(r);
      ld sr = sinh(r);
      
      // sinh(r) = xy
      // r = tanh(sinh(xy))
      
  
      ld z = cr * (K - 1/SV/SV);
  
      ld k = 90._deg;
      ld a = k / K;
      ld zw = xy * cr / sr;
      ld u = z * a;
  
      ld phi1 = atan2(zw, cos(k)) - u;
      
      if(phi < phi1) part = 2;
      }
    
    if(part == -1) {
      ld zw = xy;
  
      ld u = xy * (K - 1/SV/SV) / K;
      ld phi1 = atan2(zw, 1) - u;
      
      if(phi > phi1) part = 0; else part = 1;
      }
    
    if(part == 2) {
      ld min_k = fiber_barrier;
      ld max_k = flip_barrier;
  
      for(int it=0; it<30; it++) {
        ld kk = (min_k + max_k) / 2;
        
        ld x_part = cos(kk);
        ld z_part = sin(kk);
        
        ld rparam = x_part / z_part / SV;
        
        assert(rparam <= 1);
  
        ld r = atanh(rparam);      
        ld cr = cosh(r);
        ld sr = sinh(r);
  
        ld z = cr * (K - 1/SV/SV);
  
        ld k = M_PI - asin(xy / sr);
        ld a = k / K;
        ld len = a * hypot(sr, cr/SV);
        ld zw = xy * cr / sr;
        ld u = z * a;
  
        ld phi1 = atan2(zw, cos(k)) - u;
  
        if(phi < phi1) max_k = kk;
        else min_k = kk;
  
        ld r_angle = alpha + u;
        res = point3(cos(r_angle) * x_part * len, -sin(r_angle) * x_part * len, z_part * len);
        }
      }
    
    if(part == 0) {
      ld min_k = 0;
      ld max_k = fiber_barrier;
  
      for(int it=0; it<30; it++) {
        ld kk = (min_k + max_k) / 2;
        
        ld x_part = cos(kk);
        ld z_part = sin(kk);
    
        ld rparam = x_part / z_part / SV;
        
        ld cr = 1 / sqrt(rparam*rparam - 1);
        ld sr = rparam * cr;
  
        ld z = cr * (K - 1/SV/SV);
        
        ld k = asinh(xy / sr);
        ld a = k / K;
        ld len = a * hypot(sr, cr/SV);
  
        ld zw = xy * cr / sr;
  
        ld u = z * a;
        ld phi1 = atan2(zw, cosh(k)) - u;
        
        if(phi > phi1) max_k = kk; else min_k = kk;
        
        ld r_angle = alpha + u;
        res = point3(cos(r_angle) * x_part * len, -sin(r_angle) * x_part * len, z_part * len);
        }
      }
    
    if(part == 1) {
      ld min_k = fiber_barrier;
      ld max_k = flip_barrier;
  
      for(int it=0; it<30; it++) {
        ld kk = (min_k + max_k) / 2;
        
        ld x_part = cos(kk);
        ld z_part = sin(kk);
    
        ld rparam = x_part / z_part / SV;
        
        ld r = atanh(rparam);      
        ld cr = cosh(r);
        ld sr = sinh(r);
  
        ld z = cr * (K - 1/SV/SV);
  
        ld k = asin(xy / sr);
        ld a = k / K;
        ld len = a * hypot(sr, cr/SV);
        ld zw = xy * cr / sr;
        ld u = z * a;
  
        ld phi1 = atan2(zw, cos(k)) - u;
        
        if(isnan(phi1)) max_k = kk;
        else if(phi > phi1) max_k = kk;
        else min_k = kk;
  
        ld r_angle = alpha + u;
        res = point3(cos(r_angle) * x_part * len, -sin(r_angle) * x_part * len, z_part * len);
        }      
      }
    
    if(flipped) res[0] *= -1, res[2] *= -1;

    return res;
    }

#if ISWEB
#define ITERATE "  for(int it=0; it<50; it++) { if(it >= uIterations) break; "
#else
#define ITERATE "  for(int it=0; it<uIterations; it++) {"
#endif

  EX string slshader = 

    "uniform mediump float uIndexSL;"
    "uniform mediump int uIterations;"
    "uniform mediump float uSV;"
    
    "vec4 inverse_exp(vec4 h) {"

      "float xy = length(h.xy);"
      "float phi = atan2(h[2], h[3]) + uIndexSL;"

      "if(xy < 1e-6) return vec4(0.,0.,phi,1.);"

      "vec4 res = vec4(sqrt(-1.),sqrt(-1.),sqrt(-1.),sqrt(-1.));"
      
      "bool flipped = phi > 0.;"
      
      "if(flipped) phi = -phi;"
      
      "float alpha = flipped ? atan2(h[1], h[0]) - uIndexSL : atan2(h[1], -h[0]) + uIndexSL;"
      
      "float fiber_barrier = atan(1./uSV);"
    
      "float flip_barrier = atan(1. / tanh(asinh(xy)) / uSV);"      
      
      "int part = 0;"
      
      "if(true) {"
        "float x_part = cos(flip_barrier);"
        "float z_part = sin(flip_barrier);"
        "float rparam = x_part / z_part / uSV;"
        "float r = atanh(rparam);"
        "float cr = cosh(r);"
        "float sr = sinh(r);"
        "float z = cr * (-1.-1./uSV/uSV);"
        "float k = PI/2.;"
        "float a = -k;"
        "float zw = xy * cr / sr;"
        "float u = z * a;"
        "float phi1 = atan2(zw, cos(k)) - u;"
        "if(phi < phi1) part = 2;"
        "}\n"
      
      "if(part == 0) {"
        "float zw = xy;"
        "float u = xy * (1. + 1./uSV/uSV);"
        "float phi1 = atan2(zw, 1.) - u;"
        "if(phi > phi1) part = 0; else part = 1;"
        "}\n"
      
      "if(part == 2) {"
        "float min_k = fiber_barrier;"
        "float max_k = flip_barrier;"
  
        ITERATE
          "float kk = (min_k + max_k) / 2.;"
          "float x_part = cos(kk);"
          "float z_part = sin(kk);"        
          "float rparam = x_part / z_part / uSV;"
          "float r = atanh(rparam);"
          "float cr = cosh(r);"
          "float sr = sinh(r);"
          
          "float z = cr * (-1. - 1./uSV/uSV);"
          "float k = PI - asin(xy / sr);"
          "float a = -k;"
          "float len = a * length(vec2(sr, cr/uSV));"
          "float zw = xy * cr / sr;"
          "float u = z * a;"
          "float phi1 = atan2(zw, cos(k)) - u;"
          "if(phi < phi1) max_k = kk; else min_k = kk;"
          "float r_angle = alpha + u;"
          "res = vec4(cos(r_angle) * x_part * len, -sin(r_angle) * x_part * len, z_part * len, 1);"
          "}"
        "}\n"
      
      "if(part == 0) {"
        "float min_k = 0.;"
        "float max_k = fiber_barrier;"
  
        ITERATE
          "float kk = (min_k + max_k) / 2.;"
          "float x_part = cos(kk);"
          "float z_part = sin(kk);"    
          "float rparam = x_part / z_part / uSV;"        
          "float cr = 1. / sqrt(rparam*rparam - 1.);"
          "float sr = rparam * cr;"  
          "float z = cr * (-1. - 1./uSV/uSV);"
          "float k = asinh(xy / sr);"
          "float a = -k;"
          "float len = a * length(vec2(sr, cr/uSV));"  
          "float zw = xy * cr / sr;"  
          "float u = z * a;"
          "float phi1 = atan2(zw, cosh(k)) - u;"
        
          "if(phi > phi1) max_k = kk; else min_k = kk;"
        
          "float r_angle = alpha + u;"
          "res = vec4(cos(r_angle) * x_part * len, -sin(r_angle) * x_part * len, z_part * len, 1);"
          "}"
        "}\n"
    
      "if(part == 1) {"
        "float min_k = fiber_barrier;"
        "float max_k = flip_barrier;"
  
        ITERATE
          "float kk = (min_k + max_k) / 2.;"
        
          "float x_part = cos(kk);"
          "float z_part = sin(kk);"
    
          "float rparam = x_part / z_part / uSV;"
        
          "float r = atanh(rparam);"
          "float cr = cosh(r);"
          "float sr = sinh(r);"
  
          "float z = cr * (-1. - 1./uSV/uSV);"
  
          "float k = asin(xy / sr);"
          "float a = -k;"
          "float len = a * length(vec2(sr, cr/uSV));"
          "float zw = xy * cr / sr;"
          "float u = z * a;"
  
          "float phi1 = atan2(zw, cos(k)) - u;"
        
          "if(phi > phi1) max_k = kk;"
          "else min_k = kk;"
  
          "float r_angle = alpha + u;"
          "res = vec4(cos(r_angle) * x_part * len, -sin(r_angle) * x_part * len, z_part * len, 1);"
          "}"
        "}\n"
    
      "if(flipped) res[0] *= -1., res[2] *= -1.;"

      "return res;"
      "}";

EX }

EX namespace twist {
  
#if MAXMDIM >= 4
  EX transmatrix uxpush(ld x) { 
    if(sl2) return xpush(x);
    if(nil) return xpush(x*2*nilv::nilwidth);
    return cspin(1, 3, x) * cspin(0, 2, x);
    }

  EX transmatrix uypush(ld y) { 
    if(sl2) return ypush(y);
    if(nil) return ypush(y*2*nilv::nilwidth);
    return cspin(0, 3, -y) * cspin(1, 2, y);
    }
     
  EX transmatrix uzpush(ld z) {
    if(sl2) return zpush(z);
    if(nil) return zpush(z*nilv::nilwidth*nilv::nilwidth) * spin(-2*z);
    return cspin(3, 2, -z) * cspin(0, 1, -z);
    }
  
  EX transmatrix lift_matrix(const transmatrix& T) {
    hyperpoint d;
    ld alpha, beta, distance;
    transmatrix Spin;
    hybrid::in_underlying_geometry([&] {
      hyperpoint h = tC0(T);
      Spin = iso_inverse(gpushxto0(h) * T);
      d = hr::inverse_exp(shiftless(h));
      alpha = atan2(Spin[0][1], Spin[0][0]);
      distance = hdist0(h);
      beta = atan2(h[1], h[0]);
      });
    for(int k=0; k<3; k++) Spin[3][k] = Spin[k][3] = 0; Spin[3][3] = 1;
    return spin(beta) * uxpush(distance/2) * spin(-beta+alpha);
    }
  
  EX std::map<int, transmatrix> saved_matrices_ray;

  struct hrmap_twisted : hybrid::hrmap_hybrid {

    std::map<int, transmatrix> saved_matrices;

    /** let relative_matrixc work in short distances at least */
    std::map<cell*, shiftmatrix> recorded_matrices;

    void find_cell_connection(cell *c, int d) override {
      hybrid::find_cell_connection(c, d);
      cell *c1 = c->move(d);
      if(!recorded_matrices.count(c1)) {
        recorded_matrices[c1] = recorded_matrices[c] * adj(c, d);
        optimize_shift(recorded_matrices[c1]);
        }
      }

    hrmap_twisted() : hybrid::hrmap_hybrid() {
      recorded_matrices[gamestart()] = shiftless(Id);
      }

    transmatrix adj(cell *c1, int i) override {
      if(i == c1->type-2) return uzpush(-cgi.plevel) * spin(-2*cgi.plevel);
      if(i == c1->type-1) return uzpush(+cgi.plevel) * spin(+2*cgi.plevel);
      cell *c2 = c1->cmove(i);
      #if CAP_ARCM
      int id1 = hybrid::underlying == gArchimedean ? arcm::id_of(c1->master) + 20 * arcm::parent_index_of(c1->master) : shvid(c1);
      int id2 = hybrid::underlying == gArchimedean ? arcm::id_of(c2->master) + 20 * arcm::parent_index_of(c2->master) : shvid(c2);
      #else
      int id1 = shvid(c1);
      int id2 = shvid(c2);
      #endif
      int j = c1->c.spin(i);
      int id = id1 + (id2 << 10) + (i << 20) + (j << 26);
      auto &M = saved_matrices[id];
      if(M[3][3]) return M;
      
      cell *cw = where[c1].first;
      return M = lift_matrix(PIU(currentmap->adj(cw, i)));      
      }
    
    shiftmatrix relative_shiftmatrix(cell *c2, cell *c1) {
      return nmul(ninverse(recorded_matrices[c1]), recorded_matrices[c2]);
      }

    transmatrix relative_matrixc(cell *c2, cell *c1, const hyperpoint& hint) override { 
      if(c1 == c2) return Id;
      if(gmatrix0.count(c2) && gmatrix0.count(c1))
        return inverse_shift(gmatrix0[c1], gmatrix0[c2]);
      for(int i=0; i<c1->type; i++) if(c1->move(i) == c2) return adj(c1, i);
      return inverse_shift(recorded_matrices[c1], recorded_matrices[c2]);
      }

    transmatrix ray_iadj(cell *c1, int i) override {
      if(i == c1->type-1) return uzpush(-cgi.plevel) * spin(-2*cgi.plevel);
      if(i == c1->type-2) return uzpush(+cgi.plevel) * spin(+2*cgi.plevel);
      cell *c2 = c1->cmove(i);
      #if CAP_ARCM
      int id1 = hybrid::underlying == gArchimedean ? arcm::id_of(c1->master) + 20 * arcm::parent_index_of(c1->master) : shvid(c1);
      int id2 = hybrid::underlying == gArchimedean ? arcm::id_of(c2->master) + 20 * arcm::parent_index_of(c2->master) : shvid(c2);
      #else
      int id1 = shvid(c1);
      int id2 = shvid(c2);
      #endif
      int j = c1->c.spin(i);
      int id = id1 + (id2 << 10) + (i << 20) + (j << 26);
      auto &M = saved_matrices_ray[id];
      if(M[3][3]) return M;
      
      cell *cw = hybrid::get_where(c1).first;
      
      transmatrix T;
      hybrid::in_underlying_geometry([&] {
        hyperpoint h0 = get_corner_position(cw, i);
        hyperpoint h1 = get_corner_position(cw, (i+1));
        T = to_other_side(h0, h1);
        });
  
      return M = lift_matrix(T);
      }
    };

  EX void clear_twisted_matrices() {
    saved_matrices_ray.clear();
    for(auto& m: allmaps) {
      auto m1 = dynamic_cast<hrmap_twisted*> (m);
      if(m1) m1->saved_matrices.clear();
      }
    }

  EX shiftmatrix relative_shiftmatrix(cell *c2, cell *c1) {
    auto hmap = dynamic_cast<hrmap_twisted*> (currentmap);
    return hmap->relative_shiftmatrix(c2, c1);
    }

  /** reinterpret the given point of rotspace as a rotation matrix in the underlying geometry (note: this is the inverse)
   *  note: you should already be in underlying geometry */
  EX transmatrix qtm(hyperpoint h) {

    ld& x = h[0];
    ld& y = h[1];
    ld& z = h[2];
    ld& w = h[3];
    
    ld xx = x*x;
    ld yy = y*y;
    ld zz = z*z;
    ld ww = w*w;
    
    ld xy = x*y;
    ld xz = x*z;
    ld xw = x*w;
    ld yz = y*z;
    ld yw = y*w;
    ld zw = z*w;
    
    transmatrix M;  
  
    M[0][0] = +xx - yy - zz + ww;
    M[1][1] = -xx + yy - zz + ww;
    M[2][2] = -xx - yy + zz + ww;
    
    M[0][1] = -2 * (xy + zw);
    M[1][0] = -2 * (xy - zw);
    
    M[0][2] = 2 * (xz - yw);
    M[2][0] = 2 * (xz + yw);

    M[1][2] = -2 * (yz + xw);
    M[2][1] = -2 * (yz - xw);
  
    if(hyperbolic) {
      swap(M[0][2], M[1][2]);
      swap(M[2][0], M[2][1]);
      M[1][2] *= -1;
      M[2][0] *= -1;
      M[2][2] = xx + yy + zz + ww;
      return M;
      }


    return M;
    }

  /** return the nearest multiple of TAU */
  EX ld get_shift_cycles(ld shift) {
    return floor(shift / TAU + .5) * TAU;
    }

  EX ld get_half_shift_cycles(ld shift) {
    return floor(shift / M_PI + .5) * M_PI;
    }

  EX transmatrix chg_shift(ld x) {
    return cspin(2, 3, x) * cspin(0, 1, x);
    }

  /** multiply a SLR-shiftmatrix by a SLR-shiftpoint */
  EX shiftpoint nmul(const shiftmatrix& T, shiftpoint h) {
    optimize_shift(h);
    ld sh = get_half_shift_cycles(h.shift);
    h.shift -= sh;
    auto res0 = T;
    optimize_shift(res0);
    auto res1 = res0 * chg_shift(h.shift);
    optimize_shift(res1);
    res1.shift += get_shift_cycles(res0.shift - res1.shift - 1e-9);
    auto res2 = res1 * h.h;
    optimize_shift(res2);
    res2.shift += get_shift_cycles(res1.shift - res2.shift);
    res2.shift += sh;
    return res2;
    }

  /** multiply a SLR-shiftmatrix by a SLR-shiftmatrix */
  EX shiftmatrix nmul(const shiftmatrix& T, shiftmatrix h) {
    optimize_shift(h);
    ld sh = get_half_shift_cycles(h.shift);
    h.shift -= sh;

    auto res0 = T;
    optimize_shift(res0);
    auto res1 = res0 * chg_shift(h.shift);
    optimize_shift(res1);
    res1.shift += get_shift_cycles(res0.shift - res1.shift - 1e-9);
    auto res2 = res1 * h.T;
    optimize_shift(res2);
    res2.shift += sh;
    return res2;
    }

  /** invert a SLR-shiftmatrix */
  EX shiftmatrix ninverse(const shiftmatrix& T) {
    shiftmatrix res;
    res.T = inverse(unshift(T));
    res.shift = 0;
    shiftmatrix m = nmul(res, T);
    optimize_shift(m);
    res.shift -= m.shift;
    return res;
    }

  EX void queueline_correct(shiftpoint h1, shiftpoint h2, color_t col, int prec, PPR prio) {

    optimize_shift(h1);
    optimize_shift(h2);

    shiftmatrix T(rgpushxto0(h1.h), h1.shift);

    shiftpoint h = nmul(ninverse(T), h2);

    hyperpoint z = inverse_exp(h);

    ld bonus = 0;

    int steps = ceil(hypot_d(3, z) * pow(2, prec));

    for(int i=0; i<=steps; i++) {
      shiftpoint next = formula_exp(z * i / steps);
      curvepoint(unshift(next, bonus));
      if(abs(next.shift - bonus) > 1) {
        queuecurve(T, col, 0, prio);
        bonus = next.shift; T.shift = h1.shift + bonus;
        curvepoint(unshift(next, bonus));
        }
      }

    queuecurve(T, col, 0, prio);
    }

  /** @brief exponential function for both slr and Berger sphere */

  EX shiftpoint formula_exp(hyperpoint vel) {
    bool sp = sphere;
    ld K = sp ? 1 : -1;

    if(vel[0] == 0 && vel[1] == 0 && vel[2] == 0) return shiftpoint(C0,0);
  
    ld len = hypot_d(3, vel);
  
    if(vel[2] < 0) len = -len;
    
    ld z_part = vel[2]/len;
    ld x_part = sqrt(max<ld>(1 - z_part * z_part, 0));
  
    ld SV = stretch::not_squared();
    
    ld rparam = x_part / z_part / SV;
    
    ld beta = atan2(vel[1], vel[0]);
    if(len < 0) beta += M_PI;
    
    if(sl2 && rparam > 1) {  
      ld cr = 1 / sqrt(rparam*rparam - 1); // *i
      ld sr = rparam * cr;                 // *i
      
      if(z_part == 0) cr = 0, sr = 1;
  
      ld z = cr * (K - 1/SV/SV); // *i
  
      ld a = len / hypot(sr, cr/SV); // /i
  
      ld k = K*a; // /i
      ld u = z*a;
  
      ld xy = sr * sinh(k);
      ld zw = cr * sinh(k);

      return shiftpoint(hyperpoint(K*xy * cos(beta), K*xy * sin(beta), zw, cosh(k)), -u);
      }
  
    else {
      ld r = atan_auto(rparam);      
      ld cr = cos_auto(r);
      ld sr = sin_auto(r);
    
      ld z = cr * (K - 1/SV/SV);
      
      ld a = len / hypot(sr, cr/SV);
    
      ld k = K*a;
      ld u = z*a;
      
      ld xy = sr * sin(k);  
      ld zw = cr * sin(k);    

      if(sl2)
        return shiftpoint(hyperpoint(K*xy * cos(beta), K*xy * sin(beta), zw, cos(k)), -u + get_shift_cycles(k));

      return shiftpoint(hyperpoint(K*xy * cos(u+beta), K*xy * sin(u+beta), zw * cos(u) - cos(k) * sin(u), zw * sin(u) + cos(k)*cos(u)), 0);
      }
    }

#endif
EX }

/** stretched rotation space (S3 or SLR) */
EX namespace stretch {

  EX ld factor;
  
  EX bool mstretch;
  
  EX transmatrix m_itoa, m_atoi, m_pd;
  EX ld ms_christoffel[3][3][3];
  
  EX transmatrix mstretch_matrix;
  
  EX void enable_mstretch() {
    mstretch = true;

    for(int a=0; a<4; a++)
    for(int b=0; b<4; b++)
      if(a==3 || b==3) m_atoi[a][b] = (a==b);

    m_itoa = inverse3(m_atoi);
    
    for(int a=0; a<4; a++)
    for(int b=0; b<4; b++)
      if(a==3 || b==3)
        m_itoa[a][b] = m_atoi[a][b] = 0;

    for(int j=0; j<3; j++)
    for(int k=0; k<3; k++) {
      m_pd[j][k] = 0;
      for(int i=0; i<3; i++)
        m_pd[j][k] += m_atoi[i][j] * m_atoi[i][k];
      }

    auto& c = ms_christoffel;

    ld A00 = m_pd[0][0];
    ld A11 = m_pd[1][1];
    ld A22 = m_pd[2][2];
    ld A01 = m_pd[0][1] + m_pd[1][0];
    ld A02 = m_pd[0][2] + m_pd[2][0];
    ld A12 = m_pd[2][1] + m_pd[1][2];
    ld B01 = A01 * A01;
    ld B02 = A02 * A02;
    ld B12 = A12 * A12;
    ld B00 = A00 * A00;
    ld B11 = A11 * A11;
    ld B22 = A22 * A22;
    
    ld den = (-4*A00*A11*A22 + A00*B12 + B01*A22 - A01*A02*A12 + B02*A11);
    
    if(sl2) {
      c[ 0 ][ 0 ][ 0 ] =  (A01*(A01*A12 - 2*A02*A11) - A02*(2*A01*A22 - A02*A12))/den;
      c[ 0 ][ 0 ][ 1 ] =  (A00*A01*A12 - 2*A00*A02*A11 - A01*A11*A12 + A01*A12*A22 + 2*A02*B11 + 2*A02*A11*A22 - A02*B12)/-den ;
      c[ 0 ][ 0 ][ 2 ] =  (-A01*(4*A11*A22 - B12)/2 + A12*(A01*A12 - 2*A02*A11)/2 - (A00 + A22)*(2*A01*A22 - A02*A12))/den;
      c[ 0 ][ 1 ][ 0 ] =  (A00*A01*A12 - 2*A00*A02*A11 - A01*A11*A12 + A01*A12*A22 + 2*A02*B11 + 2*A02*A11*A22 - A02*B12)/-den ;
      c[ 0 ][ 1 ][ 1 ] =  -(A01*(A01*A12 - 2*A02*A11) + A12*(4*A11*A22 - B12))/den;
      c[ 0 ][ 1 ][ 2 ] =  (B01*A22 - B02*A11 + 4*B11*A22 - A11*B12 + 4*A11*B22 - B12*A22)/-den ;
      c[ 0 ][ 2 ][ 0 ] =  (-A01*(4*A11*A22 - B12)/2 + A12*(A01*A12 - 2*A02*A11)/2 - (A00 + A22)*(2*A01*A22 - A02*A12))/den;
      c[ 0 ][ 2 ][ 1 ] =  (B01*A22 - B02*A11 + 4*B11*A22 - A11*B12 + 4*A11*B22 - B12*A22)/-den ;
      c[ 0 ][ 2 ][ 2 ] =  -(A02*(2*A01*A22 - A02*A12) + A12*(4*A11*A22 - B12))/den;
      c[ 1 ][ 0 ][ 0 ] =  (-A01*(2*A00*A12 - A01*A02) + A02*(4*A00*A22 - B02))/den;
      c[ 1 ][ 0 ][ 1 ] =  (A02*(2*A01*A22 - A02*A12)/2 + A12*(4*A00*A22 - B02)/2 + (A00 - A11)*(2*A00*A12 - A01*A02))/den;
      c[ 1 ][ 0 ][ 2 ] =  (-4*B00*A22 + A00*B02 + A00*B12 - 4*A00*B22 - B01*A22 + B02*A22)/-den ;
      c[ 1 ][ 1 ][ 0 ] =  (A02*(2*A01*A22 - A02*A12)/2 + A12*(4*A00*A22 - B02)/2 + (A00 - A11)*(2*A00*A12 - A01*A02))/den;
      c[ 1 ][ 1 ][ 1 ] =  (A01*(2*A00*A12 - A01*A02) + A12*(2*A01*A22 - A02*A12))/den;
      c[ 1 ][ 1 ][ 2 ] =  (A01*(4*A00*A22 - B02)/2 + A02*(2*A00*A12 - A01*A02)/2 + (A11 + A22)*(2*A01*A22 - A02*A12))/den;
      c[ 1 ][ 2 ][ 0 ] =  (-4*B00*A22 + A00*B02 + A00*B12 - 4*A00*B22 - B01*A22 + B02*A22)/-den ;
      c[ 1 ][ 2 ][ 1 ] =  (A01*(4*A00*A22 - B02)/2 + A02*(2*A00*A12 - A01*A02)/2 + (A11 + A22)*(2*A01*A22 - A02*A12))/den;
      c[ 1 ][ 2 ][ 2 ] =  (A02*(4*A00*A22 - B02) + A12*(2*A01*A22 - A02*A12))/den;
      c[ 2 ][ 0 ][ 0 ] =  (A01*(4*A00*A11 - B01) - A02*(2*A00*A12 - A01*A02))/den;
      c[ 2 ][ 0 ][ 1 ] =  (4*B00*A11 - A00*B01 - 4*A00*B11 + A00*B12 + B01*A11 - B02*A11)/-den ;
      c[ 2 ][ 0 ][ 2 ] =  (-A01*(A01*A12 - 2*A02*A11)/2 + A12*(4*A00*A11 - B01)/2 - (A00 + A22)*(2*A00*A12 - A01*A02))/den;
      c[ 2 ][ 1 ][ 0 ] =  (4*B00*A11 - A00*B01 - 4*A00*B11 + A00*B12 + B01*A11 - B02*A11)/-den ;
      c[ 2 ][ 1 ][ 1 ] =  -(A01*(4*A00*A11 - B01) + A12*(A01*A12 - 2*A02*A11))/den;
      c[ 2 ][ 1 ][ 2 ] =  (A00*A01*A12 + 2*A00*A02*A11 - B01*A02 + A01*A11*A12 + A01*A12*A22 - 2*A02*B11 - 2*A02*A11*A22)/-den ;
      c[ 2 ][ 2 ][ 0 ] =  (-A01*(A01*A12 - 2*A02*A11)/2 + A12*(4*A00*A11 - B01)/2 - (A00 + A22)*(2*A00*A12 - A01*A02))/den;
      c[ 2 ][ 2 ][ 1 ] =  (A00*A01*A12 + 2*A00*A02*A11 - B01*A02 + A01*A11*A12 + A01*A12*A22 - 2*A02*B11 - 2*A02*A11*A22)/-den ;
      c[ 2 ][ 2 ][ 2 ] =  -(A02*(2*A00*A12 - A01*A02) + A12*(A01*A12 - 2*A02*A11))/den;
      }
    else {
      c[ 0 ][ 0 ][ 0 ] =  (A01*(A01*A12 - 2*A02*A11) + A02*(2*A01*A22 - A02*A12))/den ;
      c[ 0 ][ 0 ][ 1 ] =  (A02*(4*A11*A22 - B12)/2 + A12*(2*A01*A22 - A02*A12)/2 - (A00 - A11)*(A01*A12 - 2*A02*A11))/den ;
      c[ 0 ][ 0 ][ 2 ] =  (-A01*(4*A11*A22 - B12)/2 + A12*(A01*A12 - 2*A02*A11)/2 - (A00 - A22)*(2*A01*A22 - A02*A12))/den ;
      c[ 0 ][ 1 ][ 0 ] =  (A02*(4*A11*A22 - B12)/2 + A12*(2*A01*A22 - A02*A12)/2 - (A00 - A11)*(A01*A12 - 2*A02*A11))/den ;
      c[ 0 ][ 1 ][ 1 ] =  (-A01*(A01*A12 - 2*A02*A11) + A12*(4*A11*A22 - B12))/den ;
      c[ 0 ][ 1 ][ 2 ] =  (B01*A22 - B02*A11 + 4*B11*A22 - A11*B12 - 4*A11*B22 + B12*A22)/(4*A00*A11*A22 - A00*B12 - B01*A22 + A01*A02*A12 - B02*A11) ;
      c[ 0 ][ 2 ][ 0 ] =  (-A01*(4*A11*A22 - B12)/2 + A12*(A01*A12 - 2*A02*A11)/2 - (A00 - A22)*(2*A01*A22 - A02*A12))/den ;
      c[ 0 ][ 2 ][ 1 ] =  (B01*A22 - B02*A11 + 4*B11*A22 - A11*B12 - 4*A11*B22 + B12*A22)/(4*A00*A11*A22 - A00*B12 - B01*A22 + A01*A02*A12 - B02*A11) ;
      c[ 0 ][ 2 ][ 2 ] =  -(A02*(2*A01*A22 - A02*A12) + A12*(4*A11*A22 - B12))/den ;
      c[ 1 ][ 0 ][ 0 ] =  -(A01*(2*A00*A12 - A01*A02) + A02*(4*A00*A22 - B02))/den ;
      c[ 1 ][ 0 ][ 1 ] =  (-A02*(2*A01*A22 - A02*A12)/2 - A12*(4*A00*A22 - B02)/2 + (A00 - A11)*(2*A00*A12 - A01*A02))/den ;
      c[ 1 ][ 0 ][ 2 ] =  (-4*B00*A22 + A00*B02 + A00*B12 + 4*A00*B22 - B01*A22 - B02*A22)/(4*A00*A11*A22 - A00*B12 - B01*A22 + A01*A02*A12 - B02*A11) ;
      c[ 1 ][ 1 ][ 0 ] =  (-A02*(2*A01*A22 - A02*A12)/2 - A12*(4*A00*A22 - B02)/2 + (A00 - A11)*(2*A00*A12 - A01*A02))/den ;
      c[ 1 ][ 1 ][ 1 ] =  (A01*(2*A00*A12 - A01*A02) - A12*(2*A01*A22 - A02*A12))/den ;
      c[ 1 ][ 1 ][ 2 ] =  (A01*(4*A00*A22 - B02)/2 + A02*(2*A00*A12 - A01*A02)/2 + (A11 - A22)*(2*A01*A22 - A02*A12))/den ;
      c[ 1 ][ 2 ][ 0 ] =  (-4*B00*A22 + A00*B02 + A00*B12 + 4*A00*B22 - B01*A22 - B02*A22)/(4*A00*A11*A22 - A00*B12 - B01*A22 + A01*A02*A12 - B02*A11) ;
      c[ 1 ][ 2 ][ 1 ] =  (A01*(4*A00*A22 - B02)/2 + A02*(2*A00*A12 - A01*A02)/2 + (A11 - A22)*(2*A01*A22 - A02*A12))/den ;
      c[ 1 ][ 2 ][ 2 ] =  (A02*(4*A00*A22 - B02) + A12*(2*A01*A22 - A02*A12))/den ;
      c[ 2 ][ 0 ][ 0 ] =  (A01*(4*A00*A11 - B01) + A02*(2*A00*A12 - A01*A02))/den ;
      c[ 2 ][ 0 ][ 1 ] =  (4*B00*A11 - A00*B01 - 4*A00*B11 - A00*B12 + B01*A11 + B02*A11)/(4*A00*A11*A22 - A00*B12 - B01*A22 + A01*A02*A12 - B02*A11) ;
      c[ 2 ][ 0 ][ 2 ] =  (-A01*(A01*A12 - 2*A02*A11)/2 + A12*(4*A00*A11 - B01)/2 - (A00 - A22)*(2*A00*A12 - A01*A02))/den ;
      c[ 2 ][ 1 ][ 0 ] =  (4*B00*A11 - A00*B01 - 4*A00*B11 - A00*B12 + B01*A11 + B02*A11)/(4*A00*A11*A22 - A00*B12 - B01*A22 + A01*A02*A12 - B02*A11) ;
      c[ 2 ][ 1 ][ 1 ] =  (-A01*(4*A00*A11 - B01) + A12*(A01*A12 - 2*A02*A11))/den ;
      c[ 2 ][ 1 ][ 2 ] =  (A00*A01*A12 + 2*A00*A02*A11 - B01*A02 + A01*A11*A12 - A01*A12*A22 - 2*A02*B11 + 2*A02*A11*A22)/(4*A00*A11*A22 - A00*B12 - B01*A22 + A01*A02*A12 - B02*A11) ;
      c[ 2 ][ 2 ][ 0 ] =  (-A01*(A01*A12 - 2*A02*A11)/2 + A12*(4*A00*A11 - B01)/2 - (A00 - A22)*(2*A00*A12 - A01*A02))/den ;
      c[ 2 ][ 2 ][ 1 ] =  (A00*A01*A12 + 2*A00*A02*A11 - B01*A02 + A01*A11*A12 - A01*A12*A22 - 2*A02*B11 + 2*A02*A11*A22)/(4*A00*A11*A22 - A00*B12 - B01*A22 + A01*A02*A12 - B02*A11) ;
      c[ 2 ][ 2 ][ 2 ] =  -(A02*(2*A00*A12 - A01*A02) + A12*(A01*A12 - 2*A02*A11))/den ;
      }
    
    for(int i=0; i<3; i++)
    for(int j=0; j<3; j++)
    for(int k=0; k<3; k++)
      if(c[i][j][k])
        println(hlog, tie(i,j,k), " : ", c[i][j][k]);
      
    
    println(hlog, "ATOI = ", m_atoi);
    println(hlog, "ITOA = ", m_itoa, " vs ", 1/not_squared());
    println(hlog, "PD   = ", m_pd, " vs ", factor);

    ray::reset_raycaster();
    }

  EX bool applicable() {
    return (mtwisted && !nil) || (cgflags & qSTRETCHABLE);
    }

  EX bool in() {
    return (factor || mstretch) && applicable();
    }

  EX transmatrix translate(hyperpoint h) {
    if(!sphere) return slr::translate(h);
    return matrix4(
      h[3], -h[2],  h[1],  h[0],
      h[2],  h[3], -h[0],  h[1],
     -h[1],  h[0],  h[3],  h[2],
     -h[0], -h[1], -h[2],  h[3]
      );
    }
  
  EX transmatrix itranslate(hyperpoint h) {
    h[0] = -h[0];
    h[1] = -h[1];
    h[2] = -h[2];
    if(!sphere) return slr::translate(h);
    return translate(h);
    }

  hyperpoint mulz(const hyperpoint at, const hyperpoint velocity, ld zf) {
    auto vel = itranslate(at) * velocity;
    vel[2] *= zf;
    return translate(at) * vel;
    }
  
  EX ld squared() {
    return abs(1 + factor);
    }
  
  EX ld not_squared() {
    return sqrt(squared());
    }
  
  EX hyperpoint isometric_to_actual(const hyperpoint at, const hyperpoint velocity) {
    if(mstretch)
      return translate(at) * m_itoa * itranslate(at) * velocity;
    else
      return mulz(at, velocity, 1/not_squared());
    }
 
  EX hyperpoint actual_to_isometric(const hyperpoint at, const hyperpoint velocity) {
    if(mstretch)
      return translate(at) * m_atoi * itranslate(at) * velocity;
    else
      return mulz(at, velocity, not_squared());
    }
  
  EX hyperpoint christoffel(const hyperpoint at, const hyperpoint velocity, const hyperpoint transported) {
  
    auto vel = itranslate(at) * velocity;
    auto tra = itranslate(at) * transported;  
    
    hyperpoint c;
    
    if(mstretch) {    
      c = Hypc;
      for(int i=0; i<3; i++)
      for(int j=0; j<3; j++)
      for(int k=0; k<3; k++)
        c[i] += vel[j] * tra[k] * ms_christoffel[i][j][k];
      }
    
    else {    
      auto K = factor;    
      c[0] = (sphere ? -K : K+2) * (vel[1] * tra[2] + vel[2] * tra[1]);
      c[1] = (sphere ? K : -(K+2)) * (vel[0] * tra[2] + vel[2] * tra[0]);
      c[2] = 0;
      c[3] = 0;
      }

    return translate(at) * c;
    }  

  EX ld sqnorm(hyperpoint at, hyperpoint h) {
    if(sphere)
      return sqhypot_d(4, h);
    h = itranslate(at) * h;    
    return h[0] * h[0] + h[1] * h[1] + h[2] * h[2];
    }

  EX vector<hyperpoint> inverse_exp_all(hyperpoint h, int generations) {
  
    vector<hyperpoint> res;
    
    ld SV = stretch::not_squared();

    if(stretch::factor == 0) {
      ld d = hypot_d(3, h);
      if(h[3] >= 1 || h[3] <= -1|| d == 0) return res;
      ld a = acos(h[3]);
      
      res.push_back(point31(h[0] * a / d, h[1] * a / d, h[2] * a / d));
      
      a = a - TAU;

      res.push_back(point31(h[0] * a / d, h[1] * a / d, h[2] * a / d));
      
      return res;
      }
    
    if(h[0] == 0 && h[1] == 0) {
      ld a = atan2(h[2], h[3]);
      
      for(int it=-generations; it<generations; it++) {
        res.push_back(point31(0, 0, (a + TAU * it) * SV));
        }
      
      return res;
      }

    ld xy = hypot_d(2, h);
  
    ld base_min_a = asin(xy);
    ld base_max_a = M_PI - base_min_a;
  
    ld seek = 90._deg - atan2(h[3], h[2]);
  
    auto ang = [&] (ld a) {
      ld rp = xy / sin(a);
      ld co = abs(rp) >= 1 ? 0 : sqrt(1-rp*rp);
      
      return atan2(co * sin(a), cos(a)) - co * (1 - 1/SV/SV) * a;
      };
    
    for(int shift=-generations; shift<generations; shift++) {
      ld min_a = base_min_a + M_PI * shift;
      ld max_a = base_max_a + M_PI * shift;
  
      ld ang_min = ang(min_a);
      ld ang_max = ang(max_a);
      
      for(int mi=0; mi<2; mi++) {
        // 0 : minimum, 1 : maximum
        ld tl = min_a, tr = max_a;
        for(int it=0; it<20; it++) {
          ld t1 = tl * .51 + tr * .49;
          ld t2 = tl * .49 + tr * .51;
          if((ang(t1) < ang(t2)) == mi) 
            tr = t1;
          else
            tl = t2;
          }
        ld extreme = (tl + tr) / 2;
        ld ang_extreme = ang(extreme);
        for(int t=0; t<2; t++) {
          ld mmin = t == 0 ? min_a : extreme;
          ld mmax = t == 0 ? extreme : max_a;
          ld vmin = t == 0 ? ang_min : ang_extreme;
          ld vmax = t == 0 ? ang_extreme : ang_max;
          
          // make it increasing
          if(t != mi) swap(mmin, mmax), swap(vmin, vmax);
          
          // println(hlog, "*** ", mi, t, " ** ", tie(min_a, ang_min), tie(extreme, ang_extreme), tie(max_a, ang_max), " -> ", vmin, " to ", vmax);
          
          int cmin = ceil((vmin - seek) / TAU);
          int cmax = floor((vmax - seek) / TAU);
          for(int c = cmin; c <= cmax; c++) {
            ld cseek = seek + c * TAU;
  
            for(int it=0; it<40; it++) {
            
              ld a = (mmin + mmax) / 2;
              
              ld cros = ang(a);
              if(cros > cseek) mmax = a; else mmin = a;
              }
            
            ld a = (mmin + mmax) / 2;
            
            ld r = asin_clamp( xy / sin(a) );
          
            ld z_part = 1;
            ld x_part = SV * tan(r);
            
            ld db = hypot(x_part, z_part);
            x_part /= db;
            z_part /= db;
            
            ld alpha = atan2(-h[1], h[0]);
          
            ld z = cos(r) * (1 - 1/SV/SV);
            ld u = z * a;
          
            ld r_angle = alpha + u;
             
            ld len = a * hypot(sin_auto(r), cos_auto(r)/SV);
      
            auto answer = point3(cos(r_angle) * x_part * len, -sin(r_angle) * x_part * len, z_part * len);
            
            // int id = (shift << 10) + (mi << 9) + (t << 8) + c;
            
            /*
            auto f = formula_exp(answer);
            
            ld err = sqhypot_d(4, f - h);
            
            println(hlog, "************************* ", answer, ": error = ", err, " id = ", id, " params = ", tie(shift, mi, t, c));
            */
  
            res.emplace_back(answer);
            }
          }
        }    
      }
    
    return res;
    }


EX }

EX namespace nisot {

  EX hyperpoint christoffel(const hyperpoint at, const hyperpoint velocity, const hyperpoint transported) {
    if(nil) return nilv::christoffel(at, velocity, transported);
    #if CAP_SOLV
    else if(sn::in()) return sn::christoffel(at, velocity, transported);
    #endif
    else if(stretch::in() || sl2) return stretch::christoffel(at, velocity, transported);
    else return point3(0, 0, 0);
    }

  EX bool in_table_range(hyperpoint h) {
    #if CAP_SOLV
    if(sol) return sn::in_table_range(h);
    #endif
    return true;
    }

  EX hyperpoint get_acceleration(const hyperpoint& at, const hyperpoint& vel) {
    return christoffel(at, vel, vel);
    }
  
  EX void geodesic_step(hyperpoint& at, hyperpoint& vel) {
    /* RK4 method */
    auto acc1 = get_acceleration(at, vel);
    auto acc2 = get_acceleration(at + vel/2, vel + acc1/2);
    auto acc3 = get_acceleration(at + vel/2 + acc1/4, vel + acc2/2);
    auto acc4 = get_acceleration(at + vel + acc2/2, vel + acc3);
    
    at += vel + (acc1+acc2+acc3)/6;
    vel += (acc1+2*acc2+2*acc3+acc4)/6;
    }
  
  EX int rk_steps = 20;

  EX hyperpoint numerical_exp(hyperpoint v) {
    hyperpoint at = point31(0, 0, 0);
    v /= rk_steps;
    v[3] = 0;
    for(int i=0; i<rk_steps; i++) geodesic_step(at, v);
    return at;
    }

  EX transmatrix parallel_transport_bare(transmatrix Pos, hyperpoint h) {
  
    bool stretch = stretch::in() || sl2;

    h[3] = 0;
  
    if(stretch::in() && stretch::mstretch)
      Pos = stretch::mstretch_matrix * Pos;
    
    auto tPos = transpose(Pos);
    
    h = Pos * h;

    int steps = rk_steps;
    h /= steps;

    auto& at = tPos[3];
    auto& vel = h;
    
    array<ld, 4> ms;
    
    if(stretch) {
      for(int i=0; i<3; i++) {
        ms[i] = stretch::sqnorm(at, tPos[i]);
        tPos[i] = stretch::isometric_to_actual(at, tPos[i]);
        }
      ms[3] = stretch::sqnorm(at, vel);
      if(!ms[3]) return Pos;
      vel = stretch::isometric_to_actual(at, vel);
      }

    for(int i=0; i<steps; i++) {
      auto acc1 = get_acceleration(at, vel);
      auto at1 = at + vel/2; auto vel1 = vel + acc1/2;
      auto acc2 = get_acceleration(at1, vel1);
      auto at2 = at1 + acc1/4; auto vel2 = vel + acc2/2;
      auto acc3 = get_acceleration(at2, vel2);
      auto at3 = at + vel + acc2/2; auto vel3 = vel + acc3;
      auto acc4 = get_acceleration(at3, vel3);

      for(int j=0; j<3; j++) {
        auto& tra = tPos[j];
        
        auto tacc1 = christoffel(at, vel, tra);
        auto tacc2 = christoffel(at1, vel1, tra + tacc1/2);
        auto tacc3 = christoffel(at2, vel2, tra + tacc2/2);
        auto tacc4 = christoffel(at3, vel3, tra + tacc3);
        
        tra += (tacc1+tacc2*2+tacc3*2+tacc4) / 6;
        }

      at += vel + (acc1+acc2+acc3)/6;
      vel += (acc1+2*acc2+2*acc3+acc4)/6;
      
      if(stretch) {
        at = normalize(at);
        
        auto fix = [&] (hyperpoint& h, ld& m) {
          h = stretch::itranslate(at) * h;
          h[3] = 0;
          ld m1;
          if(stretch::mstretch) {
            m1 = 0;
            for(int i=0; i<3; i++) for(int j=0; j<3; j++)
              m1 += h[i] * stretch::m_pd[i][j] * h[j];
            }
          else
            m1 = h[0] * h[0] + h[1] * h[1] + h[2] * h[2] * stretch::squared();
          h /= sqrt(m1/m);
          h = stretch::translate(at) * h;
          };

        for(int i=0; i<3; i++) fix(tPos[i], ms[i]);
        fix(vel, ms[3]); 
        }

      }

    if(stretch) {
      vel = stretch::actual_to_isometric(at, vel);
      for(int i=0; i<3; i++) tPos[i] = stretch::actual_to_isometric(at, tPos[i]);
      }

    Pos = transpose(tPos);
    
    if(stretch::in() && stretch::mstretch)
      Pos = inverse(stretch::mstretch_matrix) * Pos;
    
    return Pos;
    }

  EX void fixmatrix(transmatrix& T) {
    if(sphere) return hr::fixmatrix(T);
    transmatrix push = eupush( tC0(T) );
    transmatrix push_back = eupush(tC0(T), -1);
    transmatrix gtl = push_back * T;
    fix_rotation(gtl);
    T = push * gtl;
    }

  EX transmatrix parallel_transport(const transmatrix Position, const hyperpoint direction) {
    auto P = Position;
    nisot::fixmatrix(P);  
    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 = unshift(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);
    transmatrix push_back = translate(at, -1);
    hyperpoint back_goal = push_back * goal;
    back_goal = inverse_exp(shiftless(back_goal), prec);
    
    transmatrix back_Position = push_back * Position;

    return rspintox(inverse(back_Position) * back_goal);
    }
  
  EX hrmap *new_map() { 
    #if CAP_SOLV
    if(sn::in()) return new sn::hrmap_solnih;
    #endif
    if(mproduct) return new product::hrmap_product;
    #if MAXMDIM >= 4
    if(mhybrid) return new twist::hrmap_twisted;
    if(nil) return new nilv::hrmap_nil;
    #endif
    return NULL;
    }

  #if CAP_COMMANDLINE
  auto config = addHook(hooks_args, 0, [] () {
    using namespace arg;
    #if CAP_SOLV
    if(argis("-solrange")) {
      shift_arg_formula(sn::solrange_xy);
      shift_arg_formula(sn::solrange_z);
      return 0;
      }
    #endif
    if(argis("-slrange")) {
      shift_arg_formula(slr::range_xy);
      shift_arg_formula(slr::range_z);
      return 0;
      }
    #if CAP_SOLV
    else if(argis("-fsol")) {
      shift(); sn::solt.fname = args();
      return 0;
      }
    else if(argis("-nihsol")) {
      shift(); sn::niht.fname = args();
      return 0;
      }
    #endif
    else if(argis("-product")) {
      PHASEFROM(2);
      set_geometry(gProduct);
      return 0;
      }
    else if(argis("-prodlevel")) {
      /* specify an exact value for cgi.plevel */
      shift(); hybrid::set_plevel(argf());
      return 0;
      }
    else if(argis("-s2xe")) {
      PHASEFROM(2);
      shift(); s2xe::qrings = argi();
      return 0;
      }
    else if(argis("-twisted-product")) {
      PHASEFROM(2);
      bool quo = closed_manifold;
      set_geometry(gTwistedProduct);
      if(quo) hybrid::fixup_csteps();
      return 0;
      }
    else if(argis("-rotspace")) {
      PHASEFROM(2);
      hybrid::enable_rotspace();
      return 0;
      }
    else if(argis("-rot_uscale")) {
      PHASEFROM(2);
      shift_arg_formula(hybrid::underlying_scale);
      return 0;
      }
    else if(argis("-nilperiod")) {
      PHASEFROM(2);
      if(nil) stop_game();
      for(int a=0; a<3; a++) { shift(); nilv::nilperiod[a] = argi(); }
      nilv::set_flags();
      return 0;
      }
    else if(argis("-nilwidth")) {
      PHASEFROM(2);
      shift_arg_formula(nilv::nilwidth);
      return 0;
      }
    else if(argis("-nilsi")) {
      PHASEFROM(2);
      stop_game();
      shift(); 
      nilv::nil_structure_index = argi();
      nilv::set_flags();
      start_game();
      return 0;
      }
    else if(argis("-rk-steps")) {
      PHASEFROM(2);
      shift(); rk_steps = argi();
      return 0;
      }      
    else if(argis("-nilv")) {
      PHASEFROM(2);
      if(nil) stop_game();
      shift();
      ginf[gNil].sides = argi();
      return 0;
      }
    #if CAP_SOLV
    else if(argis("-catperiod")) {
      PHASEFROM(2);
      if(sol) stop_game();
      shift(); asonov::period_xy = argi();
      shift(); asonov::period_z = argi();
      asonov::set_flags();
      return 0;
      }
    #endif
    else if(argis("-prodperiod")) {
      PHASEFROM(2);
      if(mproduct) stop_game();
      shift(); hybrid::csteps = argi();
      hybrid::reconfigure();
      return 0;
      }
    else if(argis("-rot-stretch")) {
      PHASEFROM(2);
      shift_arg_formula(stretch::factor, ray::reset_raycaster);
      return 0;
      }
    else if(argis("-mstretch")) {
      PHASEFROM(2);
      auto& M = stretch::m_atoi;
      M = Id;
      stretch::enable_mstretch();
      while(true) {
        shift();
        string s = args();
        if(isize(s) == 2 && among(s[0], 'a', 'b','c') && among(s[1], 'a', 'b', 'c'))
          shift_arg_formula(M[s[0]-'a'][s[1]-'a'], stretch::enable_mstretch);
        else break;
        }
      // shift_arg_formula(stretch::yfactor, ray::reset_raycaster);
      return 0;
      }
    else if(argis("-mstretch1")) {
      PHASEFROM(2);
      auto& M = stretch::m_atoi;
      M = Id;
      M[2][2] = stretch::not_squared();
      stretch::enable_mstretch();
      // shift_arg_formula(stretch::yfactor, ray::reset_raycaster);
      return 0;
      }
    else if(argis("-prodturn")) {
      PHASEFROM(2);
      if(mproduct) stop_game();
      shift(); product::cspin = argi();
      shift(); product::cmirror = argi();
      return 0;
      }
    else if(argis("-nil-model")) {
      shift(); nilv::model_used = argf();
      return 0;
      }
    return 1;
    });
  #endif  
  }

}