embeddings:: clarified the meaning of various coordinate systems, both in a comment and in idea
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Zeno Rogue
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embeddings.cpp
163
embeddings.cpp
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@ -1,3 +1,21 @@
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// Hyperbolic Rogue -- embeddings
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// Copyright (C) 2011-2019 Zeno Rogue, see 'hyper.cpp' for details
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/** \file embeddings.cpp
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* \brief Embedding 2D geometries into 3D
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*
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* This file handles primarily embedding 2D geometries into 3D.
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*
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* The following coordinate systems are used for embedding of 2D geometries into 3D:
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*
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* - *base* coordinates are simply the coordinate in the underlying 2D geometry. They support only two dimensions.
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* - *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.
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* - *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.
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* - *intermediate* coordinates: they use the same assignment of coordinates as actual, but they are a linear transformation of logical scaled.
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* - *actual* coordinates: final coordinates in the ambient 3D geometry.
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*
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*/
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#include "hyper.h"
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namespace hr {
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@ -204,6 +222,7 @@ EX }
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virtual transmatrix map_relative_push(hyperpoint h);
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virtual ld get_logical_z(hyperpoint a) { return (intermediate_to_logical_scaled * actual_to_intermediate(a))[2]; }
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virtual hyperpoint logical_to_actual(hyperpoint l) { return intermediate_to_actual(logical_to_intermediate * l); }
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virtual hyperpoint actual_to_logical(hyperpoint a) { return intermediate_to_logical * actual_to_intermediate(a); }
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virtual hyperpoint base_to_actual(hyperpoint h) = 0;
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virtual transmatrix base_to_actual(const transmatrix &T) = 0;
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virtual hyperpoint actual_to_base(hyperpoint h) = 0;
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@ -213,6 +232,8 @@ EX }
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virtual transmatrix get_radar_transform(const transmatrix& V);
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virtual transmatrix get_lsti() { return Id; }
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virtual transmatrix get_lti() { return logical_scaled_to_intermediate; }
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virtual hyperpoint base_to_logical(hyperpoint h) = 0;
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virtual hyperpoint logical_to_base(hyperpoint h) = 0;
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virtual bool is_euc_in_product() { return false; }
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virtual bool is_product_embedding() { return false; }
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@ -249,8 +270,6 @@ EX }
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EX geometry_information *swapper;
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struct auaua : embedding_method { int z; };
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hyperpoint embedding_method::tile_center() {
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ld z = center_z();
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if(z == 0) return C0;
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@ -299,15 +318,58 @@ struct emb_none : embedding_method {
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u *= cos_auto(z);
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return hpxy3(h[0] * u, h[1] * u, sinh(z));
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}
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hyperpoint base_to_logical(hyperpoint h) override {
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if(sn::in() || !bt::in())
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return h;
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#if CAP_BT
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if(bt::in() && !mproduct) return bt::minkowski_to_bt(h);
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#endif
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return h;
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}
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hyperpoint logical_to_base(hyperpoint h) override {
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if(sn::in() || !bt::in())
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return ultra_normalize(h);
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#if CAP_BT
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if(bt::in() && !mproduct)
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return bt::bt_to_minkowski(h);
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#endif
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return h;
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}
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};
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/** embeddings methods that are not emb_none */
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struct emb_actual : embedding_method {
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hyperpoint base_to_logical(hyperpoint h) override {
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if(bt::in()) {
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auto h1 = bt::inverse_horopoint(h);
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h1[2] = 0; h1[3] = 1;
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return h1;
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}
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h /= h[2];
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h[2] = 0; h[3] = 1;
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return h;
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}
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hyperpoint logical_to_base(hyperpoint h) override {
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if(bt::in()) {
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auto h1 = bt::get_horopoint(h);
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h1[3] = 1;
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return h1;
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}
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h[2] = 1; h = normalize(h);
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h[3] = 1;
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return h;
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}
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};
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/** embed in the 3D variant of the same geometry */
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struct emb_same_in_same : auaua {
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struct emb_same_in_same : emb_actual {
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virtual bool is_same_in_same() { return true; }
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transmatrix intermediate_to_actual_translation(hyperpoint i) override { return rgpushxto0(i); }
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hyperpoint actual_to_intermediate(hyperpoint a) override { return a; }
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hyperpoint flatten(hyperpoint h) override { if(abs(h[2]) > 1e-6) println(hlog, "h2 = ", h[2]); return h; }
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hyperpoint orthogonal_move(const hyperpoint& h, ld z) override {
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ld u = 1;
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if(h[2]) z += asin_auto(h[2]), u /= cos_auto(asin_auto(h[2]));
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@ -340,16 +402,42 @@ struct emb_same_in_same : auaua {
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return h;
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}
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transmatrix map_relative_push(hyperpoint h) override {
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ld z = -asin_auto(h[2]);
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ld z = asin_auto(h[2]);
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ld u = 1 / cos_auto(z);
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auto h1 = hpxy3(h[0] * u, h[1] * u, 0);
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return rgpushxto0(h1) * zpush(-z);
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return rgpushxto0(h1) * zpush(z);
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}
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hyperpoint actual_to_logical(hyperpoint h) override {
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ld z = asin_auto(h[2]);
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ld u = 1 / cos_auto(z);
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auto h1 = hpxy3(h[0] * u, h[1] * u, 0);
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h1[2] = h1[3];
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geom3::light_flip(true);
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h1 = base_to_logical(h1);
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geom3::light_flip(false);
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return h1;
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}
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hyperpoint logical_to_actual(hyperpoint h) override {
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geom3::light_flip(true);
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auto b = logical_to_base(h);
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geom3::light_flip(false);
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b[3] = b[2]; b[2] = 0;
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return orthogonal_move(b, h[2]);
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}
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hyperpoint flatten(hyperpoint h) override {
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ld z = asin_auto(h[2]);
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ld u = 1 / cos_auto(z);
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return hpxy3(h[0] * u, h[1] * u, 0);
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}
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};
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/** embed in the product geometry */
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struct emb_product_embedding : auaua {
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struct emb_product_embedding : emb_actual {
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virtual bool is_product_embedding() { return true; }
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transmatrix intermediate_to_actual_translation(hyperpoint i) { return rgpushxto0(i); }
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hyperpoint actual_to_intermediate(hyperpoint a) { return a; }
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@ -364,11 +452,23 @@ struct emb_product_embedding : auaua {
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hyperpoint base_to_actual(hyperpoint h) override { return h; }
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hyperpoint actual_to_base(hyperpoint h) override { return flatten(h); }
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transmatrix map_relative_push(hyperpoint h) override { return rgpushxto0(h); }
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hyperpoint actual_to_logical(hyperpoint h) override {
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ld z = zlevel(h);
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h /= exp(z);
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h = base_to_logical(h);
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h[2] = z;
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return h;
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}
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hyperpoint logical_to_actual(hyperpoint h) override {
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return logical_to_base(h) * exp(h[2]);
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}
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};
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/** embed Euclidean plane as horosphere */
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struct emb_euc_in_hyp : embedding_method {
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struct emb_euc_in_hyp : emb_actual {
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bool is_euc_in_hyp() override { return true; }
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hyperpoint actual_to_intermediate(hyperpoint a) override { return deparabolic13(a); }
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transmatrix intermediate_to_actual_translation(hyperpoint i) override { return parabolic13_at(i); }
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@ -389,11 +489,15 @@ struct emb_euc_in_hyp : embedding_method {
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/** sphere into a isotropic space of higher curvature */
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struct emb_sphere_in_low : embedding_method {
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struct emb_sphere_in_low : emb_actual {
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bool is_sph_in_low() override { return true; }
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bool is_depth_limited() override { return true; }
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transmatrix intermediate_to_actual_translation(hyperpoint i) override { throw hr_exception("illegal function"); }
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hyperpoint actual_to_intermediate(hyperpoint a) override { throw hr_exception("illegal function"); }
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transmatrix intermediate_to_actual_translation(hyperpoint i) override {
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throw hr_exception("illegal function");
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}
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hyperpoint actual_to_intermediate(hyperpoint a) override {
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throw hr_exception("illegal function");
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}
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ld center_z() { return 1; }
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transmatrix map_relative_push(hyperpoint a) {
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ld z = hdist0(a);
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@ -430,11 +534,29 @@ struct emb_sphere_in_low : embedding_method {
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hf[3] = cos_auto(z0 + z);
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return hf;
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}
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hyperpoint logical_to_actual(hyperpoint h) {
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auto z = h[2];
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h[2] = 1;
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geom3::light_flip(true);
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h = normalize(h);
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geom3::light_flip(false);
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h *= (1 + z);
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h[3] = 1;
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return h;
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}
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hyperpoint actual_to_logical(hyperpoint h) {
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ld z = get_logical_z(h);
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geom3::light_flip(true);
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h = kleinize(h);
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geom3::light_flip(false);
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h[2] = z; h[3] = 1;
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return h;
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}
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};
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/** abstract class for embeddings of Euclidean plane; these embeddings are not isotropic */
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struct emb_euclid_noniso : embedding_method {
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struct emb_euclid_noniso : emb_actual {
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bool is_euc_in_noniso() override { return true; }
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bool is_in_noniso() override { return true; }
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transmatrix base_to_actual(const transmatrix &T) override {
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@ -602,7 +724,6 @@ struct emb_euc_in_nil : emb_euclid_noniso {
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hyperpoint actual_to_intermediate(hyperpoint a) override { return a; }
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transmatrix intermediate_to_actual_translation(hyperpoint i) override { return rgpushxto0(i); }
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transmatrix get_lsti() override { return cspin90(2, 1); }
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hyperpoint orthogonal_move(const hyperpoint& a, ld z) override { return nisot::translate(a) * cpush0(1, z); }
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};
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struct emb_euc_in_solnih : emb_euclid_noniso {
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@ -610,11 +731,21 @@ struct emb_euc_in_solnih : emb_euclid_noniso {
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transmatrix intermediate_to_actual_translation(hyperpoint i) override { return rgpushxto0(i); }
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};
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struct emb_hyp_in_solnih : embedding_method {
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struct emb_hyp_in_solnih : emb_actual {
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bool is_hyp_in_solnih() override { return true; }
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bool is_in_noniso() override { return true; }
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transmatrix intermediate_to_actual_translation(hyperpoint i) override { return rgpushxto0(i); }
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hyperpoint actual_to_intermediate(hyperpoint a) override { return a; }
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transmatrix intermediate_to_actual_translation(hyperpoint i) override {
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if(cgclass == gcSol) i[0] *= exp(-i[2]);
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if(cgclass == gcSolN) i[0] *= pow(2, -i[2]);
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if(cgclass == gcNIH) i[0] *= pow(2, i[2]);
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return rgpushxto0(i);
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}
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hyperpoint actual_to_intermediate(hyperpoint a) override {
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if(cgclass == gcSol) a[0] *= exp(a[2]);
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if(cgclass == gcSolN) a[0] *= pow(2, a[2]);
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if(cgclass == gcNIH) a[0] *= pow(2, -a[2]);
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return a;
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}
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transmatrix base_to_actual(const transmatrix &T) override {
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auto T1 = T;
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auto h = get_column(T1, 2);
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