hyperrogue/hyperpoint.cpp

// Hyperbolic Rogue
// This file contains hyperbolic points and matrices.
// Copyright (C) 2011-2018 Zeno Rogue, see 'hyper.cpp' for details

namespace hr {

eGeometry geometry;
eVariation variation;

// hyperbolic points and matrices

// basic functions and types
//===========================

#ifdef SINHCOSH
// ld sinh(ld alpha) { return (exp(alpha) - exp(-alpha)) / 2; }
// ld cosh(ld alpha) { return (exp(alpha) + exp(-alpha)) / 2; }

/* ld inverse_sinh(ld z) {
  return log(z+sqrt(1+z*z));
  }
 
double inverse_cos(double c) {
  double s = sqrt(1-c*c);
  double r = atan(s/c);
  if(r < 0) r = -r;
  return r;
  }

// ld tanh(ld x) { return sinh(x) / cosh(x); }
ld inverse_tanh(ld x) { return log((1+x)/(1-x)) / 2; } */

#endif
#ifndef M_PI
#define M_PI 3.14159265358979
#endif

ld squar(ld x) { return x*x; }

int sig(int z) { return (sphere || z<DIM)?1:-1; }

int curvature() {
  switch(cgclass) {
    case gcEuclid: return 0;
    case gcHyperbolic: return -1;
    case gcSphere: return 1;
    default: return 0;
    }
  }

ld sin_auto(ld x) {
  switch(cgclass) {
    case gcEuclid: return x;
    case gcHyperbolic: return sinh(x);
    case gcSphere: return sin(x);
    default: return x;
    }
  }

ld asin_auto(ld x) {
  switch(cgclass) {
    case gcEuclid: return x;
    case gcHyperbolic: return asinh(x);
    case gcSphere: return asin(x);
    default: return x;
    }
  }

ld asin_auto_clamp(ld x) {
  switch(cgclass) {
    case gcEuclid: return x;
    case gcHyperbolic: return asinh(x);
    case gcSphere: return x>1 ? M_PI/2 : x<-1 ? -M_PI/2 : std::isnan(x) ? 0 : asin(x);
    default: return x;
    }
  }

ld cos_auto(ld x) {
  switch(cgclass) {
    case gcEuclid: return 1;
    case gcHyperbolic: return cosh(x);
    case gcSphere: return cos(x);
    default: return 1;
    }
  }

ld tan_auto(ld x) {
  switch(cgclass) {
    case gcEuclid: return x;
    case gcHyperbolic: return tanh(x);
    case gcSphere: return tan(x);
    default: return 1;
    }
  }

ld atan_auto(ld x) {
  switch(cgclass) {
    case gcEuclid: return x;
    case gcHyperbolic: return atanh(x);
    case gcSphere: return atan(x);
    default: return 1;
    }
  }

ld atan2_auto(ld y, ld x) {
  switch(cgclass) {
    case gcEuclid: return y/x;
    case gcHyperbolic: return atanh(y/x);
    case gcSphere: return atan2(y, x);
    default: return 1;
    }
  }

// hyperbolic point:
//===================

// we represent the points on the hyperbolic plane
// by points in 3D space (Minkowski space) such that x^2+y^2-z^2 == -1, z > 0
// (this is analogous to representing a sphere with points such that x^2+y^2+z^2 == 1)

hyperpoint hpxy(ld x, ld y DC(, ld z)) { 
  return hpxyz(x,y,DC(z,) euclid ? 1 : sphere ? sqrt(1-x*x-y*y D3(-z*z)) : sqrt(1+x*x+y*y D3(+z*z)));
  }

// center of the pseudosphere
const hyperpoint Hypc(0,0,0 DC(,0));

// origin of the hyperbolic plane
const hyperpoint C0(0,0 DC(,0) ,1);

// a point (I hope this number needs no comments ;) )
const hyperpoint Cx1(1,0 DC(,0) ,1.41421356237);

// this function returns approximate square of distance between two points
// (in the spherical analogy, this would be the distance in the 3D space,
// through the interior, not on the surface)
// also used to verify whether a point h1 is on the hyperbolic plane by using Hypc for h2

bool zero2(hyperpoint h) { return h[0] == 0 && h[1] == 0 D3(&& h[2] == 0); }

bool zero3(hyperpoint h) { return h[0] == 0 && h[1] == 0 && h[2] == 0 D3(&& h[3] == 0); }

ld intval(const hyperpoint &h1, const hyperpoint &h2) {
  ld res = 0;
  for(int i=0; i<MDIM; i++) res += squar(h1[i] - h2[i]) * sig(i);
  if(elliptic) {
    ld res2 = 0;
    for(int i=0; i<MDIM; i++) res2 += squar(h1[i] + h2[i]) * sig(i);
    return min(res, res2);
    }
  return res;
  }

ld intvalxy(const hyperpoint &h1, const hyperpoint &h2) {
  return squar(h1[0]-h2[0]) + squar(h1[1]-h2[1]) D3(+ squar(h1[2]-h2[2]));
  }

ld intvalxyz(const hyperpoint &h1, const hyperpoint &h2) {
  return squar(h1[0]-h2[0]) + squar(h1[1]-h2[1]) + squar(h1[2]-h2[2]) D3(+ squar(h1[3]-h2[3]));
  }

ld sqhypot2(const hyperpoint& h) {
  return h[0]*h[0]+h[1]*h[1] D3(+h[2]*h[2]);
  }
  
ld sqhypot3(const hyperpoint& h) {
  return h[0]*h[0]+h[1]*h[1]+h[2]*h[2] D3(+h[3]*h[3]);
  }
  
ld hypot2(const hyperpoint& h) {
  return sqrt(sqhypot2(h));
  }
  
ld hypot3(const hyperpoint& h) {
  return sqrt(sqhypot3(h));
  }
  
ld zlevel(const hyperpoint &h) {
  if(euclid) return h[DIM];
  else if(sphere) return sqrt(intval(h, Hypc));
  else return (h[DIM] < 0 ? -1 : 1) * sqrt(-intval(h, Hypc));
  }

ld hypot_auto(ld x, ld y) {
  switch(cgclass) {
    case gcEuclid:
      return hypot(x, y);
    case gcHyperbolic:
      return acosh(cosh(x) * cosh(y));
    case gcSphere:
      return acos(cos(x) * cos(y));
    default:
      return hypot(x, y);
    }
  }

// move H back to the sphere/hyperboloid/plane
hyperpoint normalize(hyperpoint H) {
  ld Z = zlevel(H);
  for(int c=0; c<MDIM; c++) H[c] /= Z;
  return H;
  }

// get the center of the line segment from H1 to H2
hyperpoint mid(const hyperpoint& H1, const hyperpoint& H2) {
  using namespace hyperpoint_vec;
  return normalize(H1 + H2);
  }

// like mid, but take 3D into account
hyperpoint midz(const hyperpoint& H1, const hyperpoint& H2) {
  using namespace hyperpoint_vec;
  hyperpoint H3 = H1 + H2;
  
  ld Z = 2;
  
  if(!euclid) Z = zlevel(H3) * 2 / (zlevel(H1) + zlevel(H2));
  for(int c=0; c<MDIM; c++) H3[c] /= Z;
  
  return H3;
  }

// matrices
//==========

// matrices represent isometries of the hyperbolic plane
// (just like isometries of the sphere are represented by rotation matrices)

// rotate by alpha degrees
transmatrix cspin(int a, int b, ld alpha) {
  transmatrix T = Id;
  T[a][a] = +cos(alpha); T[a][b] = +sin(alpha);
  T[b][a] = -sin(alpha); T[b][b] = +cos(alpha);
  return T;
  }

transmatrix spin(ld alpha) { return cspin(0, 1, alpha); }

transmatrix eupush(ld x, ld y DC(, ld z)) {
  transmatrix T = Id;
  T[0][DIM] = x;
  T[1][DIM] = y;
  D3( T[2][DIM] = z );
  return T;
  }

transmatrix eupush(hyperpoint h) {
  transmatrix T = Id;
  for(int i=0; i<DIM; i++) T[i][DIM] = h[i];
  return T;
  }

transmatrix euscalezoom(hyperpoint h) {
  transmatrix T = Id;
  T[0][0] = h[0];
  T[0][1] = -h[1];
  T[1][0] = h[1];
  T[1][1] = h[0];
  return T;
  }

transmatrix euaffine(hyperpoint h) {
  transmatrix T = Id;
  T[0][1] = h[0];
  T[1][1] = exp(h[1]);
  return T;
  }

transmatrix cpush(int cid, ld alpha) {
  transmatrix T = Id;
  T[DIM][DIM] = T[cid][cid] = cos_auto(alpha);
  T[cid][DIM] = sin_auto(alpha);
  T[DIM][cid] = -curvature() * sin_auto(alpha);
  return T;
  }

// push alpha units to the right
transmatrix xpush(ld alpha) { return cpush(0, alpha); }

inline hyperpoint cpush0(int c, ld x) { 
  hyperpoint h = Hypc;
  h[DIM] = cos_auto(x);
  h[c] = sin_auto(x);
  return h;
  }

inline hyperpoint xpush0(ld x) { return cpush0(0, x); }

inline hyperpoint ypush0(ld x) { return cpush0(1, x); }

inline hyperpoint xspinpush0(ld alpha, ld x) { 
  hyperpoint h = Hypc;
  h[DIM] = cos_auto(x);
  h[0] = sin_auto(x) * cos(alpha);
  h[1] = sin_auto(x) * -sin(alpha);
  return h;
  }

// push alpha units vertically
transmatrix ypush(ld alpha) { return cpush(1, alpha); }

transmatrix parabolic1(ld u DC(, ld v)) {
  if(euclid)
    return ypush(u);
  else {
    ld diag = (u*u D3(+v*v))/2;
    #if DIM==2
    return transmatrix {{{-diag+1, u, diag}, {-u, 1, u}, {-diag, u, diag+1}}};
    #else
    return transmatrix {{{-diag+1, u, v, diag}, {-u, 1, 0, u}, {-v, 0, 1, v}, {-diag, u, v, diag+1}}};
    #endif
    }
  }

transmatrix spintoc(const hyperpoint& H, int t, int f) {
  transmatrix T = Id;
  ld R = hypot(H[f], H[t]);
  if(R >= 1e-12) {
    T[t][t] = +H[t]/R; T[t][f] = +H[f]/R;
    T[f][t] = -H[f]/R; T[f][f] = +H[t]/R;
    }
  return T;
  }

transmatrix rspintoc(const hyperpoint& H, int t, int f) {
  transmatrix T = Id;
  ld R = hypot(H[f], H[t]);
  if(R >= 1e-12) {
    T[t][t] = +H[t]/R; T[t][f] = -H[f]/R;
    T[f][t] = +H[f]/R; T[f][f] = +H[t]/R;
    }
  return T;
  }

// rotate the hyperbolic plane around C0 such that H[1] == 0 and H[0] >= 0
transmatrix spintox(const hyperpoint& H) { 
  #if DIM==2
  return spintoc(H, 0, 1); 
  #else
  transmatrix T1 = spintoc(H, 0, 1);
  return spintoc(T1*H, 0, 2) * T1;
  #endif
  }

void set_column(transmatrix& T, int i, const hyperpoint& H) {
  for(int j=0; j<MDIM; j++)
    T[j][i] = H[j];
  }

transmatrix build_matrix(hyperpoint h1, hyperpoint h2, hyperpoint h3) {
  transmatrix T;
  for(int i=0; i<MDIM; i++)
    T[i][0] = h1[i],
    T[i][1] = h2[i],
    T[i][2] = h3[i] 
    DC(, T[i][3] = 0);
  return T;
  }

// reverse of spintox(H)
transmatrix rspintox(const hyperpoint& H) { 
  #if DIM==2
  return rspintoc(H, 0, 1); 
  #else
  transmatrix T1 = spintoc(H, 0, 1);
  return rspintoc(H, 0, 1) * rspintoc(T1*H, 0, 2);
  #endif
  }

// for H such that H[1] == 0, this matrix pushes H to C0
transmatrix pushxto0(const hyperpoint& H) {
  transmatrix T = Id;
  T[0][0] = +H[DIM]; T[0][DIM] = -H[0];
  T[DIM][0] = curvature() * H[0]; T[DIM][DIM] = +H[DIM];
  return T;
  }

// reverse of pushxto0(H)
transmatrix rpushxto0(const hyperpoint& H) {
  transmatrix T = Id;
  T[0][0] = +H[DIM]; T[0][DIM] = H[0];
  T[DIM][0] = -curvature() * H[0]; T[DIM][DIM] = +H[DIM];
  return T;
  }

transmatrix ggpushxto0(const hyperpoint& H, ld co) {
  if(euclid) return eupush(co * H[0], co * H[1] DC(, co * H[2]));
  transmatrix res = Id;
  if(sqhypot2(H) < 1e-12) return res;
  ld fac = (H[DIM]-1) / sqhypot2(H);
  for(int i=0; i<DIM; i++)
  for(int j=0; j<DIM; j++)
    res[i][j] += H[i] * H[j] * fac;
    
  for(int d=0; d<DIM; d++)
    res[d][DIM] = co * H[d],
    res[DIM][d] = -curvature() * co * H[d];
  res[DIM][DIM] = H[DIM];
  
  return res;
  }

// generalization: H[1] can be non-zero
transmatrix gpushxto0(const hyperpoint& H) {
  return ggpushxto0(H, -1);
  }

transmatrix rgpushxto0(const hyperpoint& H) {
  return ggpushxto0(H, 1);
  }

// fix the matrix T so that it is indeed an isometry
// (without using this, imprecision could accumulate)

void fixmatrix(transmatrix& T) {
  if(euclid) {
    for(int x=0; x<DIM; x++) for(int y=0; y<=x; y++) {
      ld dp = 0;
      for(int z=0; z<DIM; z++) dp += T[z][x] * T[z][y];
      
      if(y == x) dp = 1 - sqrt(1/dp);
      
      for(int z=0; z<DIM; z++) T[z][x] -= dp * T[z][y];
      }
    for(int x=0; x<DIM; x++) T[2][x] = 0;
    T[2][2] = 1;
    }
  else for(int x=0; x<MDIM; x++) for(int y=0; y<=x; y++) {
    ld dp = 0;
    for(int z=0; z<MDIM; z++) dp += T[z][x] * T[z][y] * sig(z);
    
    if(y == x) dp = 1 - sqrt(sig(x)/dp);
    
    for(int z=0; z<MDIM; z++) T[z][x] -= dp * T[z][y];
    }
  }

// show the matrix on screen

ld det(const transmatrix& T) {
  #if DIM == 3
  ld det = 0;
  for(int i=0; i<3; i++) 
    det += T[0][i] * T[1][(i+1)%3] * T[2][(i+2)%3];
  for(int i=0; i<3; i++) 
    det -= T[0][i] * T[1][(i+2)%3] * T[2][(i+1)%3];
  #else
  ld det = 1;
  transmatrix M = T;
  for(int a=0; a<MDIM; a++) {
    for(int b=a; b<=DIM; b++)
      if(M[b][a]) {
        if(b != a)
          for(int c=a; c<MDIM; c++) tie(M[b][c], M[a][c]) = make_pair(-M[a][c], M[b][c]);
        break;
        }
    if(!M[a][a]) return 0;
    for(int b=a+1; b<=DIM; b++) {
      ld co = -M[b][a] / M[a][a];
      for(int c=a; c<MDIM; c++) M[b][c] += M[a][c] * co;
      }
    det *= M[a][a];
    }
  #endif
  return det;
  }

void inverse_error(const transmatrix& T) {
  println(hlog, "Warning: inverting a singular matrix: ", T);
  }

transmatrix inverse(const transmatrix& T) {
  profile_start(7);
  
  #if DIM == 3
  ld d = det(T);
  transmatrix T2;
  if(d == 0) {
    inverse_error(T); 
    return Id;
    }
  
  for(int i=0; i<3; i++) 
  for(int j=0; j<3; j++) 
    T2[j][i] = (T[(i+1)%3][(j+1)%3] * T[(i+2)%3][(j+2)%3] - T[(i+1)%3][(j+2)%3] * T[(i+2)%3][(j+1)%3]) / d;
  
  #else
  transmatrix T1 = T;
  transmatrix T2 = Id;

  for(int a=0; a<MDIM; a++) {
    for(int b=a; b<=DIM; b++)
      if(T1[b][a]) {
        if(b != a)
          for(int c=0; c<MDIM; c++) 
            tie(T1[b][c], T1[a][c]) = make_pair(T1[a][c], T1[b][c]),
            tie(T2[b][c], T2[a][c]) = make_pair(T2[a][c], T2[b][c]);            
        break;
        }
    if(!T1[a][a]) { inverse_error(T); return Id; }
    for(int b=a+1; b<=DIM; b++) {
      ld co = -T1[b][a] / T1[a][a];
      for(int c=0; c<MDIM; c++) T1[b][c] += T1[a][c] * co, T2[b][c] += T2[a][c] * co;
      }
    }
  
  for(int a=MDIM-1; a>=0; a--) {
    for(int b=0; b<a; b++) {
      ld co = -T1[b][a] / T1[a][a];
      for(int c=0; c<MDIM; c++) T1[b][c] += T1[a][c] * co, T2[b][c] += T2[a][c] * co;
      }
    ld co = 1 / T1[a][a];
    for(int c=0; c<MDIM; c++) T1[a][c] *= co, T2[a][c] *= co;
    }
  #endif

  profile_stop(7);
  return T2;
  }

// distance between mh and 0
double hdist0(const hyperpoint& mh) {
  switch(cgclass) {
    case gcHyperbolic:
      if(mh[DIM] < 1) return 0;
      return acosh(mh[DIM]);
    case gcEuclid: {
      ld d = sqhypot2(mh);
      return d;
      }
    case gcSphere: {
      ld res = mh[DIM] >= 1 ? 0 : mh[DIM] <= -1 ? M_PI : acos(mh[DIM]);
      if(elliptic && res > M_PI/2) res = M_PI-res;
      return res;
      }
    default:
      return 0;
    }
  }

ld circlelength(ld r) {
  switch(cgclass) {
    case gcEuclid:
      return 2 * M_PI * r;
    case gcHyperbolic:
      return 2 * M_PI * sinh(r);
    case gcSphere:
      return 2 * M_PI * sin(r);
    default:
      return 0;
    }
  }

// distance between two points
double hdist(const hyperpoint& h1, const hyperpoint& h2) {
  return hdist0(gpushxto0(h1) * h2);
  ld iv = intval(h1, h2);
  switch(cgclass) {
    case gcEuclid:
      return sqrt(iv);
    case gcHyperbolic:
      return 2 * asinh(sqrt(iv) / 2);
    case gcSphere:
      return 2 * asin_auto_clamp(sqrt(iv) / 2);
    default:
      return 0;
    }
  }

hyperpoint mscale(const hyperpoint& t, double fac) {
  hyperpoint res;
  for(int i=0; i<MDIM; i++) 
    res[i] = t[i] * fac;
  return res;
  }

transmatrix mscale(const transmatrix& t, double fac) {
  transmatrix res;
  for(int i=0; i<MDIM; i++) for(int j=0; j<MDIM; j++)
    res[i][j] = t[i][j] * fac;
  return res;
  }

transmatrix xyscale(const transmatrix& t, double fac) {
  transmatrix res;
  for(int i=0; i<MDIM; i++) for(int j=0; j<DIM; j++)
    res[i][j] = t[i][j] * fac;
  return res;
  }

transmatrix xyzscale(const transmatrix& t, double fac, double facz) {
  transmatrix res;
  for(int i=0; i<MDIM; i++) for(int j=0; j<DIM; j++)
    res[i][j] = t[i][j] * fac;
  for(int i=0; i<MDIM; i++) 
    res[i][DIM] = t[i][DIM] * facz;
  return res;
  }

// double downspin_zivory;

transmatrix mzscale(const transmatrix& t, double fac) {
  // take only the spin
  transmatrix tcentered = gpushxto0(tC0(t)) * t;
  // tcentered = tcentered * spin(downspin_zivory);
  fac -= 1;
  transmatrix res = t * inverse(tcentered) * ypush(-fac) * tcentered;
  fac *= .2;
  fac += 1;
  for(int i=0; i<MDIM; i++) for(int j=0; j<MDIM; j++)
    res[i][j] = res[i][j] * fac;
  return res;
  }

transmatrix pushone() { return euclid ? eupush(1, 0) : xpush(sphere?.5 : 1); }

bool operator == (hyperpoint h1, hyperpoint h2) {
  for(int i=0; i<MDIM; i++) if(h1[i] != h2[i]) return false;
  return true;
  }

// rotation matrix in R^3

transmatrix rotmatrix(double rotation, int c0, int c1) {
  transmatrix t = Id;
  t[c0][c0] = cos(rotation);
  t[c1][c1] = cos(rotation);
  t[c0][c1] = sin(rotation);
  t[c1][c0] = -sin(rotation);
  return t;
  }

hyperpoint mid3(hyperpoint h1, hyperpoint h2, hyperpoint h3) {
  using namespace hyperpoint_vec;
  return mid(h1+h2+h3, h1+h2+h3);
  }

hyperpoint mid_at(hyperpoint h1, hyperpoint h2, ld v) {
  using namespace hyperpoint_vec;
  hyperpoint h = h1 * (1-v) + h2 * v;
  return mid(h, h);
  }  

hyperpoint mid_at_actual(hyperpoint h, ld v) {
  using namespace hyperpoint_vec;
  return rspintox(h) * xpush0(hdist0(h) * v);
  }

}