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PSL(2,R)

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Zeno Rogue
2019-08-24 11:55:45 +02:00
parent e0852419fc
commit 136b931609
21 changed files with 525 additions and 42 deletions
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385
nonisotropic.cpp
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@@ -10,6 +10,7 @@ namespace hr {
EX namespace nisot {
typedef array<float, 3> ptlow;
EX int current_view_level;
EX transmatrix local_perspective;
#if HDR
inline bool local_perspective_used() { return nonisotropic || prod; }
@@ -18,6 +19,8 @@ EX namespace nisot {
EX bool geodesic_movement = true;
EX transmatrix translate(hyperpoint h) {
if(sl2)
return slr::translate(h);
transmatrix T = Id;
for(int i=0; i<GDIM; i++) T[i][LDIM] = h[i];
if(sol) {
@@ -564,8 +567,6 @@ EX namespace product {
ginf[gProduct].g.graphical_dimension++;
}
EX int current_view_level;
hrmap *pmap;
geometry_information *pcgip;
@@ -634,7 +635,7 @@ EX namespace product {
if(sphere) gmatrix[c] = V; /* some computations need gmatrix0 for underlying geometry */
bool s = sphere;
in_actual([&] {
cell *c0 = get_at(c, current_view_level);
cell *c0 = get_at(c, nisot::current_view_level);
cwall_offset = wall_offset(c0);
if(s) cwall_mask = (1<<c->type) - 1;
else {
@@ -652,7 +653,7 @@ EX namespace product {
for(int z=-max_z; z<=max_z; z++) {
if(z == 0) cwall_mask ^= (2<<c->type);
if(z == 1) cwall_mask ^= (1<<c->type);
cell *c1 = get_at(c, current_view_level+z);
cell *c1 = get_at(c, nisot::current_view_level+z);
setdist(c1, 7, NULL);
drawcell(c1, V * mscale(Id, cgi.plevel * z), spinv, mirrored);
}
@@ -769,11 +770,360 @@ EX namespace product {
EX }
EX namespace slr {
/* This implementation is based on:
// https://pdfs.semanticscholar.org/bf46/824df892593a1b6d1c84a5f99e90eece7c54.pdf
// However, to make it consistent with the conventions in HyperRogue,
// coordinates 0<->2 and 1<->3 are swapped,
// then coordinates 2<->3 are swapped
*/
EX hyperpoint from_phigans(hyperpoint h) {
ld r = asinh(hypot_d(2, h));
ld x = h[0];
ld y = h[1];
ld z = h[2];
return hyperpoint(x * cos(z) + y * sin(z), y * cos(z) - x * sin(z), cosh(r) * sin(z), cosh(r) * cos(z));
}
EX hyperpoint to_phigans(hyperpoint h) {
ld z = atan2(h[2], h[3]);
ld x = h[0];
ld y = h[1];
return point31(x * cos(z) - y * sin(z), y * cos(z) + x * sin(z), z);
}
/** in the 'phigans' model */
hyperpoint christoffel(const hyperpoint Position, const hyperpoint Velocity, const hyperpoint Transported) {
ld x = Position[0];
ld y = Position[1];
ld s = x*x + y*y + 1;
ld x2 = x * x;
ld y2 = y * y;
ld x4 = x2 * x2;
ld y4 = y2 * y2;
return point3(
+ Velocity[ 0 ] * Transported[ 0 ] * (x*(4*x2*y2 + 4*y4 + 9*y2 + 1))
+ Velocity[ 0 ] * Transported[ 1 ] * (-y*(4*x4 + 4*x2*y2 + 9*x2 + 2))
+ Velocity[ 0 ] * Transported[ 2 ] * (-x*y*(x2 + y2 + 2))
+ Velocity[ 1 ] * Transported[ 0 ] * (-y*(4*x4 + 4*x2*y2 + 9*x2 + 2))
+ Velocity[ 1 ] * Transported[ 1 ] * (x*(4*x4 + 4*x2*y2 + 9*x2 + 5))
+ Velocity[ 1 ] * Transported[ 2 ] * (x4 + x2*y2 + 2*x2 + 1)
+ Velocity[ 2 ] * Transported[ 0 ] * (-x*y*(x2 + y2 + 2))
+ Velocity[ 2 ] * Transported[ 1 ] * (x4 + x2*y2 + 2*x2 + 1),
+ Velocity[ 0 ] * Transported[ 0 ] * (y*(4*x2*y2 + 4*y4 + 9*y2 + 5))
+ Velocity[ 0 ] * Transported[ 1 ] * (-x*(4*x2*y2 + 4*y4 + 9*y2 + 2))
+ Velocity[ 0 ] * Transported[ 2 ] * (-x2*y2 - y4 - 2*y2 - 1)
+ Velocity[ 1 ] * Transported[ 0 ] * (-x*(4*x2*y2 + 4*y4 + 9*y2 + 2))
+ Velocity[ 1 ] * Transported[ 1 ] * (y*(4*x4 + 4*x2*y2 + 9*x2 + 1))
+ Velocity[ 1 ] * Transported[ 2 ] * (x*y*(x2 + y2 + 2))
+ Velocity[ 2 ] * Transported[ 0 ] * (-x2*y2 - y4 - 2*y2 - 1)
+ Velocity[ 2 ] * Transported[ 1 ] * (x*y*(x2 + y2 + 2)),
+ Velocity[ 0 ] * Transported[ 0 ] * (-4*x*y)
+ Velocity[ 0 ] * Transported[ 1 ] * (2*x2 - 2*y2)
+ Velocity[ 0 ] * Transported[ 2 ] * x
+ Velocity[ 1 ] * Transported[ 0 ] * (2*x2 - 2*y2)
+ Velocity[ 1 ] * Transported[ 1 ] * 4*x*y
+ Velocity[ 1 ] * Transported[ 2 ] * y
+ Velocity[ 2 ] * Transported[ 0 ] * x
+ Velocity[ 2 ] * Transported[ 1 ] * y
) / s;
}
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));
}
ld rootsin(ld square, ld s) {
if(square > 0) return sinh(sqrt(square) * s) / sqrt(square);
else if(square < 0) return sin(sqrt(-square) * s) / sqrt(-square);
else return s;
}
/** it==0 is standard asin, it==1 is the next solution (PI-asin) */
ld asin_it(ld z, int it) {
auto ans = asin(z);
if(it & 1) ans = M_PI - ans;
return ans;
}
ld arootsin(ld square, ld v, int it) {
if(square > 0) return asinh(v * sqrt(square)) / sqrt(square);
else if(square < 0) return asin_it(v * sqrt(-square), it) / sqrt(-square);
else return v;
}
ld roottan(ld square, ld s) {
if(square > 0) return tanh(sqrt(square) * s) / sqrt(square);
else if(square < 0) return tan(sqrt(-square) * s) / sqrt(-square);
else return s;
}
hyperpoint geodesic_polar(ld alpha, ld beta, ld s) {
auto c = cos(2*alpha);
ld t;
if(c > 0)
t = atan(sin(alpha) * tanh(sqrt(c) * s) / sqrt(c));
else if(c < 0) {
/* the formula in the paper is roughly atan(k*tan(s))
* however, atan is not always to be taken in [-PI/2,PI/2]:
* if s is in [kPI-PI/2, kPI+PI/2], we should also increase the result by kPI
*/
ld x = sqrt(-c) * s;
ld steps = floor(x/M_PI + 0.5);
t = atan(sin(alpha) * tan(sqrt(-c) * s) / sqrt(-c)) + M_PI * steps;
}
else t = atan(sin(alpha) * s);
return polar(
asinh(cos(alpha) * rootsin(c, s)),
beta - t,
2*sin(alpha)*s - t
);
}
EX hyperpoint formula_exp(hyperpoint h) {
ld s = hypot_d(3, h);
ld beta = atan2(h[1], h[0]);
ld alpha = asin(h[2] / s);
return geodesic_polar(alpha, beta, s);
}
void find_alpha(ld phi, ld r, ld theta, ld &alpha, ld &s, ld &beta) {
if(phi < 0) { find_alpha(-phi, r, -theta, alpha, s, beta); alpha = -alpha; beta = -beta; return; }
ld mina = 0, maxa = M_PI/2;
bool next_nan;
ld c;
for(int it=0; it<40; it++) {
alpha = (mina + maxa) / 2;
c = cos(2 * alpha);
s = arootsin(c, sinh(r) / cos(alpha), 0);
if(isnan(s)) { next_nan = true, maxa = alpha; continue; }
ld got_phi = 2*sin(alpha)*s - atan(sin(alpha) * roottan(c, s));
if(got_phi > phi) next_nan = false, maxa = alpha;
else mina = alpha;
}
if(next_nan) {
mina = M_PI/4;
for(int it=0; it<40; it++) {
alpha = (mina + maxa) / 2;
c = cos(2 * alpha);
s = arootsin(c, sinh(r) / cos(alpha), 1);
ld got_phi = 2*sin(alpha)*s - atan(sin(alpha) * roottan(c, s)) - M_PI;
if(got_phi < phi) maxa = alpha;
else mina = alpha;
}
beta = theta + atan(sin(alpha) * roottan(c, s)) + M_PI;
}
else beta = theta + atan(sin(alpha) * roottan(c, s));
}
EX hyperpoint get_inverse_exp(hyperpoint h) {
if(sqhypot_d(2, h) < 1e-12) return point3(0, 0, atan2(h[2], h[3]));
ld r = asinh(hypot_d(2, h));
ld phi = atan2(h[2], h[3]);
ld theta = atan2(h[1], h[0]) + phi;
ld alpha, s, beta;
find_alpha(phi, r, theta, alpha, s, beta);
return point3(s * cos(beta) * cos(alpha), s * sin(beta) * cos(alpha), s * sin(alpha));
}
EX string slshader =
"float atan2(float y, float x) {"
" if(x == 0.) return y > 0. ? PI/2. : -PI/2.;"
" if(x > 0.) return atan(y / x);"
" if(y >= 0.) return atan(y / x) + PI;"
" if(y < 0.) return atan(y / x) - PI;"
" }"
"vec4 inverse_exp(vec4 h) {"
"if(h[0]*h[0] + h[1] * h[1] < 1e-6) return vec4(0, 0, atan(h[2], h[3]), 1);"
"float r = asinh(sqrt(h[0] * h[0] + h[1] * h[1]));"
"float phi = atan2(h[2], h[3]);"
"float theta = atan2(h[1], h[0]) + phi;"
"float alpha;"
"float s;"
"float beta;"
"float sgn = 1.;"
"if(phi < 0.) { phi = -phi; theta = -theta; sgn = -1.; }"
"float c;"
"s = sinh(r) / cos(PI/4.);"
"float gphi = 2.*sin(PI/4.)*s - atan(sin(PI/4.) * s);"
"if(gphi > phi) {"
" float mina = 0.;"
" float maxa = PI/4.;"
" for(int it=0; it<40; it++) {"
" alpha = (mina + maxa) / 2.;"
" c = sqrt(cos(2. * alpha));"
" s = asinh(sinh(r) / cos(alpha) * c) / c;"
" gphi = 2.*sin(alpha)*s - atan(sin(alpha) * tanh(c * s) / c);"
" if(gphi > phi) maxa = alpha;"
" else mina = alpha;"
" }"
" beta = theta + atan(sin(alpha) * tanh(c * s) / c);"
" }"
"else {"
" int next_nan = 1;"
" float mina = PI/4.;"
" float maxa = PI/2.;"
" for(int it=0; it<40; it++) {"
" alpha = (mina + maxa) / 2.;"
" c = sqrt(-cos(2. * alpha));"
" if(sinh(r) * c > cos(alpha)) { next_nan = 1; maxa = alpha; continue; }"
" s = asin(sinh(r) * c / cos(alpha)) / c;"
" gphi = 2.*sin(alpha)*s - atan(sin(alpha) * tan(c*s) / c);"
" if(gphi > phi) { next_nan = 0; maxa = alpha; }"
" else mina = alpha;"
" }"
" if(next_nan != 0) {"
" mina = PI/4.;"
" for(int it=0; it<10; it++) {"
" alpha = (mina + maxa) / 2.;"
" c = sqrt(-cos(2. * alpha));"
" s = PI - asin(sinh(r) * c / cos(alpha)) / c;"
" gphi = 2.*sin(alpha)*s - atan(sin(alpha) * tan(c*s) / c) - PI;"
" if(gphi < phi) maxa = alpha; else mina = alpha;"
" }"
" beta = theta + atan(sin(alpha) * tanh(c * s) / c) + PI;"
" }"
" else beta = theta + atan(sin(alpha) * tanh(c * s) / c);"
" }"
"alpha = alpha * sgn; beta = beta * sgn;"
"return vec4(s * cos(beta) * cos(alpha), s * sin(beta) * cos(alpha), s * sin(alpha), 1);"
"}";
EX transmatrix adjmatrix(int i, int j) {
ld zs = 2 * M_PI / 28;
if(i == 7) return zpush(-zs) * spin(-2*zs);
if(i == 8) return zpush(+zs) * spin(+2*zs);
return spin(2 * M_PI * i / 7) * xpush(tessf7/2) * spin(M_PI - 2 * M_PI * j / 7);
}
struct hrmap_psl2 : hrmap {
hrmap *underlying_map;
map<pair<cell*, int>, cell*> at;
map<cell*, pair<cell*, int>> where;
heptagon *getOrigin() override { return underlying_map->getOrigin(); }
template<class T> auto in_underlying(const T& t) -> decltype(t()) {
dynamicval<eGeometry> g(geometry, gNormal);
dynamicval<hrmap*> gu(currentmap, underlying_map);
return t();
}
cell *getCell(cell *u, int h) {
h = gmod(h, 14);
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); }
virtual transmatrix relative_matrix(cell *c2, cell *c1, const struct hyperpoint& point_hint) override {
for(int i=0; i<c1->type; i++) if(c1->move(i) == c2) return adjmatrix(i, c1->c.spin(i));
if(gmatrix0.count(c2) && gmatrix0.count(c1))
return inverse(gmatrix0[c1]) * gmatrix0[c2];
return Id; // not implemented yet
}
void draw() override {
set<cell*> visited;
cell* start = viewcenter();
vector<pair<cell*, transmatrix>> dq;
visited.insert(start);
dq.emplace_back(start, cview());
for(int i=0; i<isize(dq); i++) {
cell *c = dq[i].first;
transmatrix V = dq[i].second;
if(V[3][3] < 0) V = centralsym * V;
if(!do_draw(c, V)) continue;
drawcell(c, V, 0, false);
for(int i=0; i<9; i++) {
// note: need do cmove before c.spin
cell *c1 = c->cmove(i);
if(visited.count(c1)) continue;
visited.insert(c1);
dq.emplace_back(c1, V * adjmatrix(i, c->c.spin(i)));
}
}
}
hrmap_psl2() {
in_underlying([this] { initcells(); underlying_map = currentmap; });
for(hrmap*& m: allmaps) if(m == underlying_map) m = NULL;
}
~hrmap_psl2() {
in_underlying([this] { delete currentmap; });
for(auto& p: at) tailored_delete(p.second);
}
};
EX cell *get_at(cell *base, int level) {
return ((hrmap_psl2*)currentmap)->getCell(base, level);
}
EX pair<cell*, int> get_where(cell *c) { return ((hrmap_psl2*)currentmap)->where[c]; }
void find_cell_connection(cell *c, int d) {
auto m = (hrmap_psl2*)currentmap;
if(d >= c->type - 2) {
cell *c1 = get_at(m->where[c].first, m->where[c].second + (d == c->type-1 ? 1 : -1));
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);
cell *c1 = get_at(cu1, m->where[c].second - d1*2 + d*2 + 7);
c->c.connect(d, c1, d1, false);
}
}
EX }
EX namespace nisot {
EX hyperpoint christoffel(const hyperpoint at, const hyperpoint velocity, const hyperpoint transported) {
if(sol) return solv::christoffel(at, velocity, transported);
else if(nil) return nilv::christoffel(at, velocity, transported);
else if(sl2) return slr::christoffel(at, velocity, transported);
else return point3(0, 0, 0);
}
@@ -789,6 +1139,7 @@ EX namespace nisot {
EX hyperpoint inverse_exp(const hyperpoint h, iePrecision p, bool just_direction IS(true)) {
if(sol) return solv::get_inverse_exp(h, p == iLazy, just_direction);
if(nil) return nilv::get_inverse_exp(h, p == iLazy ? 5 : 20);
if(sl2) return slr::get_inverse_exp(h);
if(prod) return product::inverse_exp(h);
return point3(h[0], h[1], h[2]);
}
@@ -822,6 +1173,7 @@ EX namespace nisot {
EX hyperpoint get_exp(hyperpoint v, int steps) {
if(sol) return direct_exp(v, steps);
if(nil) return nilv::formula_exp(v);
if(sl2) return slr::formula_exp(v);
if(prod) return product::direct_exp(v);
return v;
}
@@ -831,19 +1183,35 @@ EX namespace nisot {
hyperpoint h = tC0(T);
h[3] = 0;
h = Pos * h;
auto tPos = transpose(Pos);
const ld eps = 1e-4;
if(sl2) {
hyperpoint p = slr::to_phigans(tPos[3]);
for(int i=0; i<3; i++)
tPos[i] = (slr::to_phigans(tPos[3] + tPos[i] * eps) - p) / eps;
tPos[3] = p;
h = transpose(tPos) * h;
}
else h = Pos * h;
int steps = 100;
h /= steps;
auto tPos = transpose(Pos);
for(int i=0; i<steps; i++) {
for(int j=0; j<3; j++)
tPos[j] += christoffel(tPos[3], h, tPos[j]);
geodesic_step(tPos[3], h);
}
if(sl2) {
hyperpoint p = slr::from_phigans(tPos[3]);
for(int i=0; i<3; i++)
tPos[i] = (slr::from_phigans(tPos[3] + tPos[i] * eps) - p) / eps;
tPos[3] = p;
}
return transpose(tPos);
}
@@ -895,6 +1263,7 @@ EX namespace nisot {
EX hrmap *new_map() {
if(sol) return new solv::hrmap_sol;
if(nil) return new nilv::hrmap_nil;
if(sl2) return new slr::hrmap_psl2;
if(geometry == gProduct) return new product::hrmap_product;
return NULL;
}