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1398 lines (1259 loc) · 49.7 KB
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// Hyperbolic Rogue -- embeddings
// Copyright (C) 2011-2019 Zeno Rogue, see 'hyper.cpp' for details
/** \file embeddings.cpp
* \brief Embedding 2D geometries into 3D
*
* This file handles primarily embedding 2D geometries into 3D.
*
* The following coordinate systems are used for embedding of 2D geometries into 3D:
*
* - *base* coordinates are simply the coordinate in the underlying 2D geometry. They support only two dimensions.
* - *logical* coordinates: X and Y are in the Beltrami-Klein or gnomonic model, or in horocyclic coordinates for binary-like tilings. Z coordinate is the altitude above the plane.
* - *logical_scaled* coordinates: X and Y are scaled (and possibly rotated in the XY plane) in order to match the scale and orientation of the ambient 3D geometry. They are a linear transformation of logical.
* - *intermediate* coordinates: they use the same assignment of coordinates as actual, but they are a linear transformation of logical scaled.
* - *actual* coordinates: final coordinates in the ambient 3D geometry.
*
*/
#include "hyper.h"
namespace hr {
EX namespace geom3 {
#if HDR
enum eSpatialEmbedding {
seNone,
seDefault,
seLowerCurvature,
seMuchLowerCurvature,
seProduct,
seNil,
seSol,
seNIH,
seSolN,
seCliffordTorus,
seProductH,
seProductS,
seSL2,
seCylinderE,
seCylinderH,
seCylinderHE,
seCylinderNil,
seCylinderHoro,
seCylinderSL2
};
#endif
EX vector<pair<string, string>> spatial_embedding_options = {
{"2D engine", "Use HyperRogue's 2D engine to simulate same curvature. Works well in top-down and third-person perspective. The Hypersian Rug mode can be used to project this to a surface."},
{"same curvature", "Embed as an equidistant surface in the 3D version of the same geometry."},
{"lower curvature", "Embed as a surface in a space of lower curvature."},
{"much lower curvature", "Embed sphere as a sphere in hyperbolic space."},
{"product", "Add one extra dimension in the Euclidean way."},
{"Nil", "Embed Euclidean plane into Nil."},
{"Sol", "Embed Euclidean or hyperbolic plane into Sol."},
{"stretched hyperbolic", "Embed Euclidean or hyperbolic plane into stretched hyperbolic geometry."},
{"stretched Sol", "Embed Euclidean or hyperbolic plane into stretched Sol geometry."},
{"Clifford Torus", "Embed Euclidean rectangular torus into S3."},
{"hyperbolic product", "Embed Euclidean or hyperbolic plane in the H2xR product space."},
{"spherical product", "Embed Euclidean cylinder or spherical plane in the H2xR product space."},
{"SL(2,R)", "Embed Euclidean plane in twisted product geometry."},
{"cylinder", "Embed Euclidean cylinder in Euclidean space."},
{"hyperbolic cylinder", "Embed Euclidean cylinder in hyperbolic space."},
{"product cylinder", "Embed Euclidean cylinder in H2xR space."},
{"Nil cylinder", "Embed Euclidean cylinder in Nil."},
{"horocylinder", "Embed Euclidean as a horocylinder in H2xR space."},
{"SL2 cylinder", "Embed Euclidean as a cylinder in twisted product geometry."},
};
EX bool clifford_torus_valid() {
#if CAP_RUG
rug::clifford_torus ct;
ld h = ct.xh | ct.yh;
return !(sqhypot_d(2, ct.xh) < 1e-3 || sqhypot_d(2, ct.yh) < 1e-3 || abs(h) > 1e-3);
#else
return false;
#endif
}
EX string why_wrong(eSpatialEmbedding sp) {
string ans = "";
if(among(sp, seNil, seCliffordTorus, seProductH, seProductS, seSL2) || any_cylinder(sp)) {
if(!PURE)
ans += " pure";
if(!meuclid)
ans += " E";
if((sp == seProductS || any_cylinder(sp)) && !quotient)
ans += " cyl";
if(sp == seCliffordTorus && !clifford_torus_valid())
ans += " torus";
}
if(among(sp, seSol, seNIH, seSolN)) {
if((meuclid && !PURE) || !bt::in()) ans += " pure or bin";
}
return ans;
}
EX eSpatialEmbedding spatial_embedding = seDefault;
EX eSpatialEmbedding want_spatial_embedding = seNone;
EX ld euclid_embed_scale = 1;
EX ld euclid_embed_scale_y = 1;
EX ld euclid_embed_rotate = 0;
EX bool auto_configure = true;
EX bool flat_embedding = false;
EX bool inverted_embedding = false;
EX bool apply_break_cylinder = true;
EX ld euclid_embed_scale_mean() { return euclid_embed_scale * sqrt(euclid_embed_scale_y); }
EX void set_euclid_embed_scale(ld x) { euclid_embed_scale = x; euclid_embed_scale_y = 1; euclid_embed_rotate = 0; }
EX bool supports_flat() { return among(spatial_embedding, seDefault, seProductH, seProductS); }
EX bool supports_invert() { return among(spatial_embedding, seDefault, seLowerCurvature, seMuchLowerCurvature, seNil, seSol, seNIH, seSolN, seProductH, seProductS) || any_cylinder(spatial_embedding); }
EX vector<geometryinfo> ginf_backup;
EX eGeometryClass mgclass() {
return (embedded_plane ? ginf_backup : ginf)[geometry].g.kind;
}
EX eGeometryClass ggclass() {
return (flipped ? ginf_backup : ginf)[geometry].g.kind;
}
EX bool any_cylinder(eSpatialEmbedding e) {
return among(e, seCylinderE, seCylinderH, seCylinderHE, seCylinderHoro, seCylinderNil, seCylinderSL2);
}
EX bool in_product() {
return ggclass() == gcProduct;
}
EX bool flipped;
EX geometry_information* unflipped;
EX void light_flip_geom() {
indenter ind(2);
swap(ginf[geometry].g, geom3::ginf_backup[geometry].g);
swap(ginf[geometry].flags, geom3::ginf_backup[geometry].flags);
if(fake::in()) {
// println(hlog, "warning: flipping while still in fake");
FPIU(light_flip_geom());
}
// hyperbolic arcm needs gNormal for cdata. Also swap gSphere and gEuclid for compute_geometry
else if(arcm::in()) {
dynamicval<eGeometry> g(geometry, gNormal); light_flip_geom();
geometry = gSphere; light_flip_geom();
geometry = gEuclid; light_flip_geom();
}
}
EX void light_flip(bool f) {
if(f != flipped) {
if(!flipped) cgip->use_count++;
if(!flipped) unflipped = cgip;
light_flip_geom();
flipped = f;
if(!flipped) cgip = unflipped;
if(!flipped) cgip->use_count--;
}
}
#if HDR
template<class T> auto in_flipped(const T& f) -> decltype(f()) {
light_flip(true);
finalizer ff([] { light_flip(false); });
return f();
}
template<class T> auto in_not_flipped(const T& f) -> decltype(f()) {
light_flip(false);
finalizer ff([] { light_flip(true); });
return f();
}
#define IPF(x) geom3::in_flipped([&] { return (x); })
#endif
EX void apply_always3() {
if(!vid.always3 && !ginf_backup.empty()) {
ginf = ginf_backup;
ginf_backup.clear();
}
if(vid.always3 && ginf_backup.empty()) {
ginf_backup = ginf;
for(geometryinfo& gi: ginf)
apply_always3_to(gi);
}
}
EX void apply_always3_to(geometryinfo& gi) {
auto &g = gi.g;
if(vid.always3 && g.gameplay_dimension == 2 && g.graphical_dimension == 2) {
/* same-in-same by default */
auto og = g;
g.graphical_dimension++;
g.homogeneous_dimension++;
g.sig[3] = g.sig[2];
g.sig[2] = g.sig[1];
bool ieuclid = g.kind == gcEuclid;
bool isphere = g.kind == gcSphere;
bool ieuc_or_binary = ieuclid || (gi.flags & qBINARY);
if(spatial_embedding == seProduct && !ieuclid) g = giProduct, g.sig[2] = og.sig[2];
if(spatial_embedding == seProductH && ieuclid) g = giProductH;
if(spatial_embedding == seProductS && ieuclid) g = giProductS;
if(spatial_embedding == seLowerCurvature) g = (isphere ? giEuclid3 : giHyperb3);
if(spatial_embedding == seMuchLowerCurvature) g = giHyperb3;
if(spatial_embedding == seNil && ieuclid) g = giNil;
if(spatial_embedding == seCliffordTorus && ieuclid) g = giSphere3;
if(spatial_embedding == seSol && ieuc_or_binary) g = giSol;
if(spatial_embedding == seNIH && ieuc_or_binary) g = giNIH;
if(spatial_embedding == seSolN && ieuc_or_binary) g = giSolN;
if(spatial_embedding == seSL2 && ieuclid) g = giSL2;
if(spatial_embedding == seCylinderH && ieuclid) g = giHyperb3;
if(spatial_embedding == seCylinderHE && ieuclid) g = giProductH;
if(spatial_embedding == seCylinderHoro && ieuclid) g = giProductH;
if(spatial_embedding == seCylinderNil && ieuclid) g = giNil;
if(spatial_embedding == seCylinderSL2 && ieuclid) g = giSL2;
g.gameplay_dimension = 2;
}
}
EX void configure_clifford_torus() {
#if CAP_RUG
dynamicval<ld> dtessf(cgi.tessf, 1);
rug::clifford_torus ct;
if(hypot_d(2, ct.xh) < 1e-6 || hypot_d(2, ct.yh) < 1e-6) {
euclid_embed_scale = TAU / 20.;
euclid_embed_scale_y = 1;
euclid_embed_rotate = 0;
vid.depth = 45._deg - 1;
vid.wall_height = 0.2;
vid.eye = vid.wall_height / 2 - vid.depth;
return;
}
euclid_embed_scale = TAU / hypot_d(2, ct.xh);
euclid_embed_scale_y = TAU / hypot_d(2, ct.yh) / euclid_embed_scale;
euclid_embed_rotate = atan2(ct.xh[1], ct.xh[0]) / degree;
ld alpha = atan2(ct.xfactor, ct.yfactor);
vid.depth = alpha - 1;
vid.wall_height = min(1 / euclid_embed_scale_mean(), (90._deg - alpha) * 0.9);
vid.eye = vid.wall_height / 2 - vid.depth;
#endif
}
EX void configure_cylinder() {
#if CAP_RUG
dynamicval<ld> dtessf(cgi.tessf, 1);
rug::clifford_torus ct;
hyperpoint vec;
if(sqhypot_d(2, ct.yh) > 1e-6) vec = ct.yh;
else if(sqhypot_d(2, ct.xh) > 1e-6) vec = ct.xh;
else vec = hyperpoint(10, 0, 0, 0);
euclid_embed_scale = TAU / hypot_d(2, vec);
euclid_embed_scale_y = 1;
euclid_embed_rotate = atan2(vec[1], vec[0]) / degree;
#endif
}
EX }
#if HDR
struct embedding_method {
virtual ld center_z() { return 0; }
virtual hyperpoint tile_center();
virtual transmatrix intermediate_to_actual_translation(hyperpoint i) = 0;
virtual hyperpoint intermediate_to_actual(hyperpoint i) { return intermediate_to_actual_translation(i) * tile_center(); }
virtual hyperpoint actual_to_intermediate(hyperpoint a) = 0;
virtual hyperpoint orthogonal_move(const hyperpoint& a, ld z);
virtual transmatrix map_relative_push(hyperpoint h);
virtual ld get_logical_z(hyperpoint a) { return (intermediate_to_logical_scaled * actual_to_intermediate(a))[2]; }
virtual hyperpoint logical_to_actual(hyperpoint l) { return intermediate_to_actual(logical_to_intermediate * l); }
virtual hyperpoint actual_to_logical(hyperpoint a) { return intermediate_to_logical * actual_to_intermediate(a); }
virtual hyperpoint base_to_actual(hyperpoint h) = 0;
virtual transmatrix base_to_actual(const transmatrix &T) = 0;
virtual hyperpoint actual_to_base(hyperpoint h) = 0;
virtual transmatrix actual_to_base(const transmatrix &T) = 0;
virtual hyperpoint normalize_flat(hyperpoint a) { return flatten(normalize(a)); }
virtual hyperpoint flatten(hyperpoint a);
virtual void set_radar_transform();
virtual transmatrix get_lsti() { return Id; }
virtual transmatrix get_lti() { return logical_scaled_to_intermediate; }
virtual hyperpoint base_to_logical(hyperpoint h) = 0;
virtual hyperpoint logical_to_base(hyperpoint h) = 0;
virtual ld anim_center_z() { return center_z(); }
virtual hyperpoint anim_tile_center();
virtual void logical_fix(transmatrix&) = 0;
virtual ld height_limit(ld sign);
virtual bool is_euc_in_product() { return false; }
virtual bool is_product_embedding() { return false; }
virtual bool is_euc_in_sl2() { return false; }
virtual bool is_same_in_same() { return false; }
virtual bool is_sph_in_low() { return false; }
virtual bool is_hyp_in_solnih() { return false; }
virtual bool is_euc_scalable() { return false; }
virtual bool is_euc_in_hyp() { return false; }
virtual bool is_euc_in_sph() { return false; }
virtual bool is_euc_in_nil() { return false; }
virtual bool is_euc_in_noniso() { return false; }
virtual bool is_in_noniso() { return false; }
virtual bool is_cylinder() { return false; }
virtual bool no_spin() { return false; }
/* convert the tangent space in logical coordinates to actual coordinates */
transmatrix logical_to_intermediate;
/* convert the tangent space in actual coordinates to logical coordinates */
transmatrix intermediate_to_logical;
/* convert the tangent space in logical coordinates to actual coordinates */
transmatrix logical_scaled_to_intermediate;
/* convert the tangent space in actual coordinates to logical coordinates */
transmatrix intermediate_to_logical_scaled;
void prepare_lta();
void auto_configure();
virtual ~embedding_method() {}
/* should we break cylinder between M1 and M2 */
virtual bool break_cylinder(const shiftmatrix& M1, const shiftmatrix& M2) { return false; }
};
#endif
EX geometry_information *swapper;
ld embedding_method::height_limit(ld sign) {
if(sign > 0) {
if(sol || nih) return 2.5;
if(hyperbolic || sl2 || in_h2xe()) return 5;
if(sphere || nil || in_s2xe()) return M_PI/2;
return 100;
}
if(sign < 0) {
if(center_z()) return -center_z();
if(sol || nih) return -2.5;
if(hyperbolic || sl2 || in_h2xe()) return -5;
if(sphere || nil || in_s2xe()) return -M_PI/2;
return -100;
}
return 0;
}
hyperpoint embedding_method::tile_center() {
ld z = center_z();
if(z == 0) return C0;
return lzpush(z) * C0;
}
hyperpoint embedding_method::anim_tile_center() {
ld z = anim_center_z();
if(z == 0) return C0;
return lzpush(z) * C0;
}
transmatrix embedding_method::map_relative_push(hyperpoint a) {
auto i = actual_to_intermediate(a);
return intermediate_to_actual_translation(i);
}
hyperpoint embedding_method::orthogonal_move(const hyperpoint& a, ld z) {
auto i = actual_to_intermediate(a);
auto l = intermediate_to_logical_scaled * i;
l[2] += z;
i = logical_scaled_to_intermediate * l;
return intermediate_to_actual(i);
}
hyperpoint embedding_method::flatten(hyperpoint a) {
auto i = actual_to_intermediate(a);
auto l = intermediate_to_logical * i;
l[2] = 0; i = logical_to_intermediate * l;
return intermediate_to_actual(i);
}
/** dummy 'embedding method' used when no embedding is used (2D engine or 3D map) */
struct emb_none : embedding_method {
hyperpoint actual_to_intermediate(hyperpoint a) override {
if(mhybrid) return base_to_logical(a);
return a;
}
hyperpoint intermediate_to_actual(hyperpoint i) override {
if(mhybrid) return logical_to_base(i);
return i;
}
transmatrix intermediate_to_actual_translation(hyperpoint i) override {
if(gproduct) i = intermediate_to_actual(i);
return rgpushxto0(i);
}
hyperpoint flatten(hyperpoint a) override {
if(gproduct || GDIM == 2) return a / exp(zlevel(a));
return embedding_method::flatten(a);
}
hyperpoint normalize_flat(hyperpoint a) override { return gproduct ? flatten(a) : normalize(a); }
transmatrix base_to_actual(const transmatrix& T) override { return T; }
hyperpoint base_to_actual(hyperpoint h) override { return h; }
transmatrix actual_to_base(const transmatrix& T) override { return T; }
hyperpoint actual_to_base(hyperpoint h) override { return h; }
hyperpoint orthogonal_move(const hyperpoint& h, ld z) override {
if(GDIM == 2) return scale_point(h, geom3::scale_at_lev(z));
if(gproduct) return scale_point(h, exp(z));
if(sl2) return slr::translate(h) * cpush0(2, z);
if(nil) return nisot::translate(h) * cpush0(2, z);
if(translatable) return hpxy3(h[0], h[1], h[2] + z);
/* copied from emb_same_in_same */
ld u = 1;
if(h[2]) z += asin_auto(h[2]), u /= cos_auto(asin_auto(h[2]));
u *= cos_auto(z);
return hpxy3(h[0] * u, h[1] * u, sinh(z));
}
hyperpoint base_to_logical(hyperpoint h) override {
if(sn::in() || !bt::in())
return h;
#if CAP_BT
if(bt::in() && !mproduct) return bt::minkowski_to_bt(h);
#endif
if(mproduct) {
ld z = zlevel(h);
h /= h[2];
h[2] = z;
}
if(sl2) {
ld z = atan2(h[2], h[3]);
h = slr::translate(h) * zpush0(-atan2(h[2], h[3]));
h[0] = h[0] / h[3]; h[1] = h[1] / h[3]; h[2] = z;
return h;
}
return h;
}
hyperpoint logical_to_base(hyperpoint h) override {
if(sn::in() || !bt::in())
return ultra_normalize(h);
#if CAP_BT
if(bt::in() && !mproduct)
return bt::bt_to_minkowski(h);
#endif
if(mproduct) {
ld z = h[2];
h[2] = 1;
flatten(h);
h *= exp(z);
}
if(sl2) {
ld z = h[2];
h[2] = 0; h[3] = 1; normalize(h);
h = slr::translate(h) * zpush0(z);
return h;
}
return h;
}
void logical_fix(transmatrix& T) override {
if(nonisotropic) {
hyperpoint h = tC0(T);
transmatrix rot = gpushxto0(h) * T;
fix_rotation(rot);
T = rgpushxto0(h) * rot;
}
else fixmatrix(T);
fixelliptic(T);
}
};
/** embeddings methods that are not emb_none */
struct emb_actual : embedding_method {
hyperpoint base_to_logical(hyperpoint h) override {
#if CAP_BT
if(bt::in()) {
auto h1 = bt::inverse_horopoint(h);
h1[2] = 0; h1[3] = 1;
return h1;
}
#endif
h /= h[2];
h[2] = 0; h[3] = 1;
return h;
}
hyperpoint logical_to_base(hyperpoint h) override {
#if CAP_BT
if(bt::in()) {
auto h1 = bt::get_horopoint(h);
h1[3] = 1;
return h1;
}
#endif
h[2] = 1; h = normalize(h);
h[3] = 1;
return h;
}
void logical_fix(transmatrix& T) override {
hyperpoint a = T * tile_center();
hyperpoint i0 = actual_to_intermediate(a);
auto l0 = intermediate_to_logical * i0;
auto l = l0; l[2] = 0;
auto i = logical_to_intermediate * l;
auto rot0= inverse(intermediate_to_actual_translation(i0)) * T ;
auto rot = intermediate_to_logical_scaled * rot0 * logical_scaled_to_intermediate;
ld alpha = atan2(rot[0][1], rot[0][0]);
T = intermediate_to_actual_translation(i) * spin(alpha);
fixelliptic(T);
}
};
/** embed in the 3D variant of the same geometry */
struct emb_same_in_same : emb_actual {
bool is_same_in_same() override { return true; }
transmatrix intermediate_to_actual_translation(hyperpoint i) override { return rgpushxto0(logical_to_actual(i)); }
hyperpoint actual_to_intermediate(hyperpoint a) override { return actual_to_logical(a); }
hyperpoint orthogonal_move(const hyperpoint& h, ld z) override {
if(euclid) { hyperpoint h1 = h; h1[2] += z; return h1; }
ld u = 1;
if(h[2]) z += asin_auto(h[2]), u /= cos_auto(asin_auto(h[2]));
u *= cos_auto(z);
return hpxy3(h[0] * u, h[1] * u, sinh(z));
}
transmatrix base_to_actual(const transmatrix &T0) override {
auto T = T0;
for(int i=0; i<4; i++) T[i][3] = T[i][2], T[i][2] = 0;
for(int i=0; i<4; i++) T[3][i] = T[2][i], T[i][2] = 0;
for(int i=0; i<4; i++) T[i][2] = T[2][i] = 0;
T[2][2] = 1;
return T;
}
transmatrix actual_to_base(const transmatrix &T0) override {
auto T = T0;
for(int i=0; i<4; i++) T[i][2] = T[i][3], T[i][3] = 0;
for(int i=0; i<4; i++) T[2][i] = T[3][i], T[3][i] = 0;
T[3][3] = 1;
if(MDIM == 3) fixmatrix(T); else IPF(fixmatrix(T));
for(int i=0; i<MDIM; i++) for(int j=0; j<MDIM; j++) if(isnan(T[i][j])) return Id;
return T;
}
hyperpoint base_to_actual(hyperpoint h) override {
h[3] = h[2]; h[2] = 0;
return h;
}
hyperpoint actual_to_base(hyperpoint h) override {
h[2] = h[3]; h[3] = 0;
return h;
}
transmatrix map_relative_push(hyperpoint h) override {
ld z = asin_auto(h[2]);
ld u = 1 / cos_auto(z);
auto h1 = hpxy3(h[0] * u, h[1] * u, 0);
return rgpushxto0(h1) * zpush(z);
}
hyperpoint actual_to_logical(hyperpoint h) override {
if(euclid) { h[3] = 1; return h; }
ld z = asin_auto(h[2]);
ld u = 1 / cos_auto(z);
auto h1 = hpxy3(h[0] * u, h[1] * u, 0);
h1[2] = h1[3];
geom3::light_flip(true);
h1 = base_to_logical(h1);
geom3::light_flip(false);
h1[2] = z;
return h1;
}
hyperpoint logical_to_actual(hyperpoint h) override {
if(euclid) { h[3] = 1; return h; }
geom3::light_flip(true);
auto b = logical_to_base(h);
geom3::light_flip(false);
b[3] = b[2]; b[2] = 0;
return orthogonal_move(b, h[2]);
}
hyperpoint flatten(hyperpoint h) override {
ld z = asin_auto(h[2]);
ld u = 1 / cos_auto(z);
return hpxy3(h[0] * u, h[1] * u, 0);
}
void logical_fix(transmatrix& T) override {
// optimization
for(int i=0; i<4; i++) T[i][2] = T[2][i] = i == 2;
fixmatrix(T);
fixelliptic(T);
}
};
/** embed in the product geometry */
struct emb_product_embedding : emb_actual {
bool is_product_embedding() override { return true; }
transmatrix intermediate_to_actual_translation(hyperpoint i) override { return rgpushxto0(logical_to_actual(i)); }
hyperpoint actual_to_intermediate(hyperpoint a) override { return actual_to_logical(a); }
hyperpoint flatten(hyperpoint h) override { h /= exp(zlevel(h)); return h; }
hyperpoint orthogonal_move(const hyperpoint& h, ld z) override { return h * exp(z); }
transmatrix base_to_actual(const transmatrix &T) override { return T; }
transmatrix actual_to_base(const transmatrix &T0) override {
auto T = T0; fixmatrix(T);
for(int i=0; i<MDIM; i++) for(int j=0; j<MDIM; j++) if(isnan(T[i][j])) return Id;
return T;
}
hyperpoint base_to_actual(hyperpoint h) override { return h; }
hyperpoint actual_to_base(hyperpoint h) override { return flatten(h); }
transmatrix map_relative_push(hyperpoint h) override { return rgpushxto0(h); }
hyperpoint actual_to_logical(hyperpoint h) override {
ld z = zlevel(h);
h /= exp(z);
h = base_to_logical(h);
h[2] = z;
return h;
}
hyperpoint logical_to_actual(hyperpoint h) override {
return logical_to_base(h) * exp(h[2]);
}
};
struct emb_euc_scalable : emb_actual {
bool is_euc_scalable() override { return true; }
transmatrix get_lti() override {
transmatrix lti = Id;
lti[0][0] *= geom3::euclid_embed_scale;
lti[1][1] *= geom3::euclid_embed_scale * geom3::euclid_embed_scale_y;
return logical_scaled_to_intermediate * cspin(0, 1, geom3::euclid_embed_rotate * degree) * lti;
}
};
/** embed Euclidean plane as horosphere */
struct emb_euc_in_hyp : emb_euc_scalable {
bool is_euc_in_hyp() override { return true; }
hyperpoint actual_to_intermediate(hyperpoint a) override { return deparabolic13(a); }
transmatrix intermediate_to_actual_translation(hyperpoint i) override { return parabolic13_at(i); }
transmatrix base_to_actual(const transmatrix &T) override {
geom3::light_flip(true);
hyperpoint mov = T * C02;
transmatrix U = gpushxto0(mov) * T;
mov = logical_to_intermediate * mov;
geom3::light_flip(false);
for(int i=0; i<4; i++) U[i][3] = U[3][i] = i == 3;
return parabolic13(mov[0], mov[1]) * U;
}
hyperpoint base_to_actual(hyperpoint h) override {
h = logical_to_intermediate * h;
h[3] = h[2]; h[2] = 0; return parabolic13(h[0], h[1]) * C0;
}
hyperpoint actual_to_base(hyperpoint h) override {
hyperpoint h1 = deparabolic13(h); h1[2] = 1;
return intermediate_to_logical * h1;
}
transmatrix actual_to_base(const transmatrix& T) override { hyperpoint h = deparabolic13(T * C0); return eupush(h[0], h[1]); }
ld anim_center_z() override { return vid.depth; }
};
/** sphere into a isotropic space of higher curvature */
struct emb_sphere_in_low : emb_actual {
bool is_sph_in_low() override { return true; }
transmatrix intermediate_to_actual_translation(hyperpoint i) override {
return map_relative_push(logical_to_actual(i)) * zpush(-1);
}
hyperpoint actual_to_intermediate(hyperpoint a) override { return actual_to_logical(a); }
ld center_z() override { return 1; }
transmatrix map_relative_push(hyperpoint a) override {
ld z = hdist0(a);
geom3::light_flip(true);
auto h1 = normalize(a);
transmatrix T = rgpushxto0(h1);
geom3::light_flip(false);
return T * zpush(z);
}
transmatrix base_to_actual(const transmatrix &T0) override {
auto T = T0;
for(int i=0; i<4; i++) T[i][3] = T[3][i] = i == 3;
return T;
}
hyperpoint base_to_actual(hyperpoint h) override {
if(euclid) h[3] = 1;
else h *= sinh(1), h[3] = cosh(1);
return h;
}
hyperpoint actual_to_base(hyperpoint h) override { return h; }
transmatrix actual_to_base(const transmatrix& T) override { return T; }
ld get_logical_z(hyperpoint a) override { return hdist0(a) - 1; }
hyperpoint flatten(hyperpoint a) override {
ld d = hdist0(a);
if(d == 0) return a;
a *= sin_auto(1) / sin_auto(d);
a[3] = cos_auto(1);
return a;
}
hyperpoint orthogonal_move(const hyperpoint& h, ld z) override {
ld z0 = hdist0(h);
ld f = sin_auto(z0 + z) / sin_auto(z0);
hyperpoint hf = h * f;
hf[3] = cos_auto(z0 + z);
return hf;
}
hyperpoint logical_to_actual(hyperpoint h) override {
auto z = h[2];
h[2] = 1;
geom3::light_flip(true);
h = normalize(h);
geom3::light_flip(false);
h *= sin_auto(1 + z);
h[3] = cos_auto(1 + z);
return h;
}
hyperpoint actual_to_logical(hyperpoint h) override {
ld z = get_logical_z(h);
geom3::light_flip(true);
h = kleinize(h);
geom3::light_flip(false);
h[2] = z; h[3] = 1;
return h;
}
void logical_fix(transmatrix& T) override {
fix4(T);
fixmatrix(T);
fixelliptic(T);
}
};
/** abstract class for embeddings of Euclidean plane; these embeddings are not isotropic */
struct emb_euclid_noniso : emb_euc_scalable {
bool is_euc_in_noniso() override { return true; }
bool is_in_noniso() override { return true; }
transmatrix base_to_actual(const transmatrix &T) override {
auto T0 = T;
hyperpoint h = get_column(T0, 2);
h[2] = 0; h[3] = 1;
return intermediate_to_actual_translation( logical_to_intermediate * h);
}
hyperpoint base_to_actual(hyperpoint h) override {
h[2] = 0; h[3] = 1;
return intermediate_to_actual_translation( logical_to_intermediate * h ) * tile_center();
}
hyperpoint actual_to_base(hyperpoint h) override {
hyperpoint h1 = intermediate_to_logical * actual_to_intermediate(h);
h1[2] = 1; h1[3] = 0;
return h1;
}
transmatrix actual_to_base(const transmatrix& T) override { hyperpoint h = actual_to_base(T * tile_center()); return eupush(h[0], h[1]); }
};
struct emb_euc_in_product : emb_euclid_noniso {
bool is_euc_in_product() override { return true; }
bool no_spin() override { return true; }
hyperpoint actual_to_intermediate(hyperpoint a) override {
ld bz = zlevel(a);
auto h1 = a / exp(bz);
ld by = asin_auto(h1[1]);
ld bx = atan2_auto(h1[0], h1[2]);
return hyperpoint(bx, by, bz, 1);
}
transmatrix get_lsti() override { return cspin90(2, 1); }
transmatrix intermediate_to_actual_translation(hyperpoint i) override {
return zpush(i[2]) * xpush(i[0]) * ypush(i[1]);
}
};
struct emb_euc_in_sl2 : emb_euclid_noniso {
transmatrix esl2_zpush(ld z) { return cspin(2, 3, z) * cspin(0, 1, z); }
hyperpoint intermediate_to_actual(hyperpoint i) override {
return esl2_zpush(i[2]) * xpush(i[0]) * ypush0(i[1]);
}
transmatrix intermediate_to_actual_translation(hyperpoint i) override {
return esl2_zpush(i[2]) * xpush(i[0]) * ypush(i[1]);
}
hyperpoint actual_to_intermediate(hyperpoint h) override {
ld a1 = (h[0] * h[3] - h[1] * h[2]) / (-h[2] * h[2] - h[1] * h[1] -h[0] * h[0] - h[3] * h[3]);
// a1 is S*sqrt(1+S*S) / (1+2*S*S), where S = sinh(-x) and C = cosh(-x); U is S*S
ld a = a1 * a1;
ld b = 4 * a - 1;
ld U = sqrt(.25 - a/b) - .5;
ld S = sqrt(U) * (a1 > 0 ? 1 : -1);
ld x = -asinh(S);
h = lorentz(0, 3, -x) * lorentz(1, 2, x) * h;
ld y = h[3]*h[3] > h[2]*h[2] ? atanh(h[1] / h[3]) : atanh(h[0] / h[2]);
h = lorentz(0, 2, -y) * lorentz(1, 3, -y) * h;
ld z = atan2(h[2], h[3]);
return hyperpoint(x, y, z, 0);
}
bool is_euc_in_sl2() override { return true; }
bool no_spin() override { return true; }
transmatrix get_lsti() override { return cspin90(2, 1); }
};
bool break_dims(const shiftmatrix& M1, const shiftmatrix& M2, int i, int j) {
transmatrix uM1 = current_display->radar_transform * unshift(M1);
transmatrix uM2 = current_display->radar_transform * unshift(M2);
return uM1[j][j] < 0 && uM2[j][j] < 0 && uM1[i][j] * uM2[i][j] < 0;
}
/* for both seCylinderH and seCylinderE. Possibly actually works for CliffordTorus too */
struct emb_euc_cylinder : emb_euclid_noniso {
bool is_cylinder() override { return true; }
ld center_z() override { return 1; }
transmatrix get_lsti() override { return cspin90(0, 1); }
hyperpoint actual_to_intermediate(hyperpoint a) override {
ld z0 = asin_auto(hypot(a[1], a[2]));
ld x0 = a[0];
if(z0 == 0) return hyperpoint(x0, 0, 0, 1);
x0 = asin_auto(x0 / cos_auto(z0));
ld y0 = z0 ? atan2(a[1], a[2]) : 0;
return point31(x0, y0, z0-1);
}
transmatrix intermediate_to_actual_translation(hyperpoint i) override {
return xpush(i[0]) * cspin(1, 2, i[1]) * zpush(i[2]);
}
bool break_cylinder(const shiftmatrix& M1, const shiftmatrix& M2) override { return break_dims(M1, M2, 1, 2); }
};
struct emb_euc_cylinder_he : emb_euc_cylinder {
bool no_spin() override { return true; }
transmatrix get_lsti() override { return cspin90(0, 2) * cspin90(0, 1); }
hyperpoint actual_to_intermediate(hyperpoint a) override {
ld z0 = zlevel(a);
a /= exp(z0);
ld y0 = atan2(a[1], a[0]);
ld x0 = asin_auto(hypot(a[0], a[1]));
return hyperpoint(x0-1, y0, z0, 1);
}
transmatrix intermediate_to_actual_translation(hyperpoint i) override {
return zpush(i[2]) * cspin(1, 0, i[1]) * xpush(i[0]);
}
bool break_cylinder(const shiftmatrix& M1, const shiftmatrix& M2) override { return break_dims(M1, M2, 1, 0); }
};
struct emb_euc_cylinder_twisted : emb_euc_cylinder {
transmatrix get_lsti() override { return cspin90(0, 2) * cspin90(0, 1); }
transmatrix get_lti() override {
ld depth = 0; // for now?
ld alpha = nil ? (1 + depth) / 2. : sinh(1 + depth) / 2.;
ld c = pow(1 + alpha * alpha, -0.5);
transmatrix U = Id;
U[1][1] = (alpha*alpha+1) * c;
U[0][1] = alpha * c;
return logical_scaled_to_intermediate * U * intermediate_to_logical_scaled * emb_euc_cylinder::get_lti();
}
};
struct emb_euc_cylinder_nil : emb_euc_cylinder_twisted {
hyperpoint actual_to_intermediate(hyperpoint a) override {
ld y0 = atan2(a[1], a[0]);
ld x0 = hypot(a[0], a[1]);
return hyperpoint(x0-1, y0, a[2], 1);
}
transmatrix intermediate_to_actual_translation(hyperpoint i) override {
return zpush(i[2]) * cspin(1, 0, i[1]) * xpush(i[0]);
}
bool break_cylinder(const shiftmatrix& M1, const shiftmatrix& M2) override { return break_dims(M1, M2, 1, 0); }
};
struct emb_euc_cylinder_horo : emb_euc_cylinder {
ld center_z() override { return 0; }
bool no_spin() override { return true; }
hyperpoint actual_to_intermediate(hyperpoint a) override {
ld z0 = zlevel(a);
a /= exp(z0);
auto hy = deparabolic13(a);
hy[2] = z0;
return hy;
}
transmatrix intermediate_to_actual_translation(hyperpoint i) override {
return zpush(i[2]) * parabolic1(i[1]) * xpush(i[0]);
}
transmatrix get_lsti() override {
return cspin90(0, 2);
}
bool break_cylinder(const shiftmatrix& M1, const shiftmatrix& M2) override { return false; }
};
struct emb_euc_cylinder_sl2 : emb_euc_cylinder_twisted {
bool no_spin() override { return true; }
hyperpoint actual_to_intermediate(hyperpoint a) override {
hyperpoint i = point31(0, 0, 0);
i[2] = atan2(a[2], a[3]);
a = cspin(1, 0, i[2]) * cspin(3, 2, i[2]) * a;
i[1] = (a[0] || a[1]) ? -atan2(a[1], a[0]) : 0;
a = cspin(1, 0, i[1]) * a;
i[0] = asinh(a[0])-1;
return i;
}
transmatrix intermediate_to_actual_translation(hyperpoint i) override {
return cspin(2, 3, i[2]) * cspin(0, 1, i[2] + i[1]) * xpush(i[0]);
}
bool break_cylinder(const shiftmatrix& M1, const shiftmatrix& M2) override { return break_dims(M1, M2, 0, 1); }
};
/** Clifford torus */
struct emb_euc_in_sph : emb_euclid_noniso {
bool is_euc_in_sph() override { return true; }
ld center_z() override { return 1; }
// ld height_limit(ld sign) override { return sign < 0 ? 0 : 90._deg; }
hyperpoint actual_to_intermediate(hyperpoint a) override {
ld tx = hypot(a[0], a[2]);
ld ty = hypot(a[1], a[3]);
ld x0 = atan2(a[0], a[2]);
ld y0 = atan2(a[1], a[3]);
ld z0 = atan2(tx, ty);
return hyperpoint(x0, y0, z0-1, 1);
}
transmatrix intermediate_to_actual_translation(hyperpoint i) override {
return cspin(0, 2, i[0]) * cspin(1, 3, i[1]) * cspin(2, 3, i[2]);
}
bool break_cylinder(const shiftmatrix& M1, const shiftmatrix& M2) override { return break_dims(M1, M2, 0, 2) || break_dims(M1, M2, 1, 3); }
};
/* todo model change */
struct emb_euc_in_nil : emb_euclid_noniso {
bool is_euc_in_nil() override { return true; }
hyperpoint actual_to_intermediate(hyperpoint a) override { a[2] -= a[0] * a[1] / 2; return a; }
transmatrix intermediate_to_actual_translation(hyperpoint i) override { i[2] += i[0] * i[1] / 2; return rgpushxto0(i); }
transmatrix get_lsti() override { return cspin90(2, 1); }
};
struct emb_euc_in_solnih : emb_euclid_noniso {
hyperpoint actual_to_intermediate(hyperpoint a) override { return a; }
transmatrix intermediate_to_actual_translation(hyperpoint i) override { return rgpushxto0(i); }
};
struct emb_hyp_in_solnih : emb_actual {
bool is_hyp_in_solnih() override { return true; }
bool is_in_noniso() override { return true; }
transmatrix intermediate_to_actual_translation(hyperpoint i) override {
if(cgclass == gcSol) i[0] *= exp(-i[2]);
if(cgclass == gcSolN) i[0] *= pow(2, -i[2]);
if(cgclass == gcNIH) i[0] *= pow(2, i[2]);
return rgpushxto0(i);
}
hyperpoint actual_to_intermediate(hyperpoint a) override {
if(cgclass == gcSol) a[0] *= exp(a[2]);
if(cgclass == gcSolN) a[0] *= pow(2, a[2]);
if(cgclass == gcNIH) a[0] *= pow(2, -a[2]);
return a;
}
transmatrix base_to_actual(const transmatrix &T) override {
auto T1 = T;
auto h = get_column(T1, 2);
return rgpushxto0(base_to_actual(h));
}
hyperpoint base_to_actual(hyperpoint h) override {
// copied from deparabolic13
h /= (1 + h[2]);
h[0] -= 1;
h /= sqhypot_d(2, h);
h[0] += .5;
ld hx = log(2) + log(-h[0]);
if(cgclass == gcNIH) hx /= log(3);
if(cgclass == gcSolN) hx /= log(3);
ld hy = h[1] * 2;
return point31(0, -hy, hx);
}
transmatrix actual_to_base(const transmatrix& T) override {
hyperpoint h = T * C0;
auto f = geom3::flipped;
geom3::light_flip(true);
transmatrix b = parabolic1(h[1]) * xpush(h[2]);
geom3::light_flip(f);
return b;
}
hyperpoint actual_to_base(hyperpoint h) override {
auto f = geom3::flipped;
geom3::light_flip(true);
hyperpoint b = parabolic1(h[1]) * xpush0(h[2]);
geom3::light_flip(f);
return b;
}
transmatrix get_lsti() override { return cspin90(0, 1) * cspin90(1, 2) * cspin90(0, 1); }
hyperpoint orthogonal_move(const hyperpoint& a, ld z) override { return nisot::translate(a) * cpush0(0, z); }
};
/* the remaining methods */
/*=======================*/
void embedding_method::prepare_lta() {
bool b = geom3::flipped;
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