Twisted S2xE
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Zeno Rogue
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3d3bde6e29
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6c4ca2254c
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#include "../hyper.h"
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// Twisted S2xE.
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// We use a model with coordinates (r,phi,z), where (r,phi) are the polar coordinates in S2.
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// Metric: ds^2 = (dr)^2 + (sin r * dphi)^2 + (dz + K * (1-cos(r)) dphi)^2
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// See https://youtu.be/lZCkEuud6aU and https://youtu.be/rfu6m_xGxWY
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namespace hr {
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EX namespace ts2 {
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eGeometry ts2 = eGeometry(-1);
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EX bool in() { return geometry == gTS; }
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EX ld K = -.9;
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EX hyperpoint at = point3(.5, 0, 0);
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EX transmatrix camera;
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void init() {
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ld r = .5;
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at = point3(r, 0, 0);
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camera = build_matrix(
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point3(1, 0, 0),
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point3(0, 1/sin(r), K/sin(r)*(cos(r)-1)),
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point3(0, 0, 1),
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point31(0,0,0)
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);
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camera = camera * cspin(1, 2, 90*degree);
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}
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// a dummy map that does nothing
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struct hrmap_ts2 : hrmap {
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heptagon *origin;
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heptagon *getOrigin() override { return origin; }
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struct transmatrix relative_matrix(heptagon *h2, heptagon *h1, const hyperpoint& hint) override {
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return Id;
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}
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hrmap_ts2() {
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init();
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heptagon*& h =origin;
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h = tailored_alloc<heptagon> (S7);
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h->c7 = newCell(S7, h);
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h->distance = 0;
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h->dm4 = 0;
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h->fieldval = 0;
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h->cdata = NULL;
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h->alt = NULL;
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}
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heptagon *create_step(heptagon *parent, int d) override {
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parent->c.connect(d, parent, d, false);
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return parent;
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}
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void draw() override {
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println(hlog, "at = ", at);
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for(int i=0; i<3; i++)
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println(hlog, i, ": ", camera * point3(i==0, i==1, i==2));
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}
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};
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EX hrmap* new_map() { return new hrmap_ts2; };
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EX hyperpoint christoffel(const hyperpoint at, const hyperpoint velocity, const hyperpoint transported) {
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const ld r = at[0];
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hyperpoint c = point3(0,0,0);
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// const ld r2 = r * r;
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const ld K2 = K * K;
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const ld sr = sin(r);
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// const ld sr2 = sr * sr;
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const ld cr = cos(r) - 1;
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const ld cr2 = cr * cr;
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c[ 0 ] = 0
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+ velocity[ 1 ] * transported[ 1 ] * (-K2*(cr) + cos(r))*sr
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+ velocity[ 1 ] * transported[ 2 ] * K*sr/2
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+ velocity[ 2 ] * transported[ 1 ] * K*sr/2;
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c[ 1 ] = 0
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+ velocity[ 0 ] * transported[ 1 ] * (K2*cos(r) - K2 - 2*cos(r))/(2*sr)
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+ velocity[ 0 ] * transported[ 2 ] * -K/(2*sr)
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+ velocity[ 1 ] * transported[ 0 ] * (K2*cos(r) - K2 - 2*cos(r))/(2*sr)
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+ velocity[ 2 ] * transported[ 0 ] * -K/(2*sr);
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c[ 2 ] = 0
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+ velocity[ 0 ] * transported[ 1 ] * K*(K2 - 1)*cr2/(2*sr)
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+ velocity[ 0 ] * transported[ 2 ] * K2*(1 - cos(r))/(2*sr)
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+ velocity[ 1 ] * transported[ 0 ] * K*(K2 - 1)*cr2/(2*sr)
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+ velocity[ 2 ] * transported[ 0 ] * K2*(1 - cos(r))/(2*sr);
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return c;
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}
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void geodesic_step(hyperpoint& at, hyperpoint& velocity) {
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auto acc = ts2::christoffel(at, velocity, velocity);
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auto at2 = at + velocity / 2;
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auto velocity2 = velocity + acc / 2;
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auto acc2 = ts2::christoffel(at2, velocity2, velocity2);
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at = at + velocity + acc2 / 2;
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velocity = velocity + acc;
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}
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EX bool shift_view(hyperpoint dist) {
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if(!in()) return false;
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auto tPos = transpose(camera);
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hyperpoint h = camera * dist;
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int steps = 100;
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h /= steps;
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for(int i=0; i<steps; i++) {
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for(int j=0; j<3; j++)
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tPos[j] += christoffel(at, h, tPos[j]);
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geodesic_step(at, h);
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}
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camera = transpose(tPos);
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return true;
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}
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EX bool rotate_view(transmatrix T) {
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if(!in()) return false;
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camera = camera * inverse(T);
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return true;
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}
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EX void radar() {
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hyperpoint a = at;
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hyperpoint v = camera * point3(0,0,1) / 100.;
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int it;
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for(it=0; it<1000; it++) {
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geodesic_step(a, v);
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// if(a[0] < .1) break;
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if(a[0] > .5 && a[0] < .6 && cos(a[1]) > .9 && cos(a[2]) > .9) break;
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}
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println(hlog, "radar = ", it);
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}
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void twist() {
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for(K = -2; K<=2; K += .1)
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for(ld start: {0.5f, 1.f}) {
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hyperpoint at = point3(start, 0, 0);
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hyperpoint vel = point3(0, 1e-5, 0);
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// println(hlog, "simulating");
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hyperpoint at1 = at, at2 = at;
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int it = 0;
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for(; ; it++) {
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at2 = at1; at1 = at;
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// if(it % 1000 == 0) println(hlog, format("%6d. ", it), at, " vel = ", vel);
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geodesic_step(at, vel);
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if(at2[0] > at1[0] && at1[0] < at[0]) break;
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}
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println(hlog, format("%8d. ", it), lalign(40, kz(at)), " vel = ", lalign(40, kz(vel)), " K = ", K);
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}
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}
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int readArgs() {
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using namespace arg;
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if(0) ;
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else if(argis("-ts2")) {
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if(ts2 == (eGeometry)(-1)) {
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ts2 = (eGeometry) isize(ginf);
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ginf.push_back(geometryinfo{
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"TS", "none", "TS", "ts", 1, 1, qEXPERIMENTAL | qRAYONLY, giSphere3, 0x31400, {{7, 2}}, eVariation::pure
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});
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}
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set_geometry(ts2);
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}
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else if(argis("-twist")) {
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twist();
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exit(1);
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}
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else return 1;
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return 0;
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}
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auto fundamentalhook = addHook(hooks_args, 100, readArgs);
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EX string fragmentshader() {
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return
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"varying mediump vec4 at;\n"
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"uniform mediump vec4 uStart;\n"
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"uniform mediump mat4 uLP;\n"
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"const float K = float(" + fts(ts2::K) + ");\n"
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"const float maxdist = 20.;\n"
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"vec4 christoffel(vec4 at, vec4 vel) {\n"
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"float r = at.x;\n"
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"float K2 = K * K;\n"
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"float sr = sin(r);\n"
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"float cr = cos(r) - 1.;\n"
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"float cr2 = cr * cr;\n"
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"vec4 c;\n"
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"c[ 0 ] = 0.\n"
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"+ vel.y * vel.y * (-K2*(cr) + cos(r))*sr\n"
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"+ vel.y * vel.z * K*sr/2.\n"
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"+ vel.z * vel.y * K*sr/2.;\n"
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"c[ 1 ] = 0."
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"+ vel.x * vel.y * (K2*cos(r) - K2 - 2.*cos(r))/(2.*sr)\n"
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"+ vel.x * vel.z * -K/(2.*sr)\n"
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"+ vel.y * vel.x * (K2*cos(r) - K2 - 2.*cos(r))/(2.*sr)\n"
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"+ vel.z * vel.x * -K/(2.*sr);\n"
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"c[ 2 ] = 0."
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"+ vel.x * vel.y * K*(K2 - 1.)*cr2/(2.*sr)\n"
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"+ vel.x * vel.z * K2*(1. - cos(r))/(2.*sr)\n"
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"+ vel.y * vel.x * K*(K2 - 1.)*cr2/(2.*sr)\n"
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"+ vel.z * vel.x * K2*(1. - cos(r))/(2.*sr);\n"
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"return c;}\n"
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"void main() {\n"
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" mediump vec4 at0 = at;\n"
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" at0.y = -at.y;\n"
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" at0.xyz = at0.xyz / length(at0.xyz);\n"
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" mediump vec4 position = uStart;\n"
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" mediump vec4 tangent = uLP * at0;\n"
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" tangent = tangent;\n"
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" float dist = 0.;\n"
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" int iter = 0;"
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" while(dist < maxdist && iter < 10000) {"
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// we make smaller steps if we are close to the singularities at the poles
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" float step = sin(position.x) * .05;"
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" dist = dist + step;\n"
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" iter++;\n"
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" tangent = tangent * step;\n"
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" vec4 acc = christoffel(position, tangent);"
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" vec4 at2 = position + tangent / 2.;"
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" vec4 tangent2 = tangent + acc / 2.;"
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" vec4 acc2 = christoffel(at2, tangent2);"
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" position = position + tangent + acc2 / 2.;"
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" tangent = tangent + acc;\n"
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" tangent = tangent / step;\n"
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" if(position.x > .5 && position.x < .6 && cos(position.y) > .9 && cos(position.z/2./K) > .9) {\n"
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" float bri = float(1. - dist / maxdist);\n"
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" int e = 0;\n"
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" if(position.x < .51 || position.x > .59) e++;\n"
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" if(cos(position.y) < .91) e++;\n"
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" if(cos(position.z/2./K) < .91) e++;\n"
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" if(e >= 2) bri /= 2.;\n"
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" gl_FragColor = vec4(bri, bri, bri, 1.);\n"
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" return;"
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" }"
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" }\n"
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" gl_FragColor = vec4(0.,0.,0.,1.);\n"
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" }";
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}
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int ah2 = addHook(ray::hooks_rayshader, 100, [] (string &vsh, string &fsh) { fsh = ts2::fragmentshader(); })
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+ addHook(ray::hooks_rayset, 100, [] (shared_ptr<ray::raycaster> o) {
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if(!in()) return false;
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glUniformMatrix4fv(o->uLP, 1, 0, glhr::tmtogl_transpose3(ts2::camera).as_array());
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auto pg = glhr::pointtogl(ts2::at);
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glUniform4f(o->uStart, pg[0], pg[1], pg[2], pg[3]);
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return true;
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})
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+ addHook(hooks_newmap, 100, [] () { if(in()) return ts2::new_map(); return (hrmap*) nullptr; })
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+ addHook(hooks_rotate_view, 100, rotate_view)
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+ addHook(hooks_shift_view, 100, shift_view)
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;
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EX }
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}
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