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https://github.com/zenorogue/hyperrogue.git
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606 lines
14 KiB
C++
606 lines
14 KiB
C++
// Hyperbolic Rogue
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// This file contains hyperbolic points and matrices.
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// Copyright (C) 2011-2018 Zeno Rogue, see 'hyper.cpp' for details
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namespace hr {
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eGeometry geometry;
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eVariation variation;
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// hyperbolic points and matrices
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// basic functions and types
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//===========================
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#ifdef SINHCOSH
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// ld sinh(ld alpha) { return (exp(alpha) - exp(-alpha)) / 2; }
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// ld cosh(ld alpha) { return (exp(alpha) + exp(-alpha)) / 2; }
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/* ld inverse_sinh(ld z) {
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return log(z+sqrt(1+z*z));
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}
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double inverse_cos(double c) {
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double s = sqrt(1-c*c);
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double r = atan(s/c);
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if(r < 0) r = -r;
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return r;
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}
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// ld tanh(ld x) { return sinh(x) / cosh(x); }
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ld inverse_tanh(ld x) { return log((1+x)/(1-x)) / 2; } */
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#endif
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#ifndef M_PI
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#define M_PI 3.14159265358979
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#endif
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ld squar(ld x) { return x*x; }
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int sig(int z) { return (sphere || z<2)?1:-1; }
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int curvature() {
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switch(cgclass) {
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case gcEuclid: return 0;
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case gcHyperbolic: return -1;
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case gcSphere: return 1;
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default: return 0;
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}
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}
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ld sin_auto(ld x) {
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switch(cgclass) {
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case gcEuclid: return x;
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case gcHyperbolic: return sinh(x);
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case gcSphere: return sin(x);
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default: return x;
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}
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}
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ld asin_auto(ld x) {
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switch(cgclass) {
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case gcEuclid: return x;
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case gcHyperbolic: return asinh(x);
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case gcSphere: return asin(x);
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default: return x;
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}
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}
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ld asin_auto_clamp(ld x) {
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switch(cgclass) {
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case gcEuclid: return x;
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case gcHyperbolic: return asinh(x);
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case gcSphere: return x>1 ? M_PI/2 : x<-1 ? -M_PI/2 : std::isnan(x) ? 0 : asin(x);
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default: return x;
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}
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}
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ld cos_auto(ld x) {
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switch(cgclass) {
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case gcEuclid: return 1;
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case gcHyperbolic: return cosh(x);
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case gcSphere: return cos(x);
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default: return 1;
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}
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}
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ld tan_auto(ld x) {
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switch(cgclass) {
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case gcEuclid: return x;
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case gcHyperbolic: return tanh(x);
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case gcSphere: return tan(x);
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default: return 1;
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}
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}
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ld atan_auto(ld x) {
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switch(cgclass) {
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case gcEuclid: return x;
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case gcHyperbolic: return atanh(x);
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case gcSphere: return atan(x);
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default: return 1;
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}
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}
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ld atan2_auto(ld y, ld x) {
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switch(cgclass) {
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case gcEuclid: return y/x;
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case gcHyperbolic: return atanh(y/x);
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case gcSphere: return atan2(y, x);
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default: return 1;
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}
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}
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// hyperbolic point:
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//===================
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// we represent the points on the hyperbolic plane
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// by points in 3D space (Minkowski space) such that x^2+y^2-z^2 == -1, z > 0
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// (this is analogous to representing a sphere with points such that x^2+y^2+z^2 == 1)
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hyperpoint hpxy(ld x, ld y) {
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return hpxyz(x,y, euclid ? 1 : sphere ? sqrt(1-x*x-y*y) : sqrt(1+x*x+y*y));
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}
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// center of the pseudosphere
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const hyperpoint Hypc(0,0,0);
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// origin of the hyperbolic plane
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const hyperpoint C0(0,0,1);
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// a point (I hope this number needs no comments ;) )
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const hyperpoint Cx1(1,0,1.41421356237);
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// this function returns approximate square of distance between two points
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// (in the spherical analogy, this would be the distance in the 3D space,
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// through the interior, not on the surface)
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// also used to verify whether a point h1 is on the hyperbolic plane by using Hypc for h2
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bool zero2(hyperpoint h) { return h[0] == 0 && h[1] == 0; }
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bool zero3(hyperpoint h) { return h[0] == 0 && h[1] == 0 && h[2] == 0; }
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ld intval(const hyperpoint &h1, const hyperpoint &h2) {
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if(elliptic) {
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double d1 = squar(h1[0]-h2[0]) + squar(h1[1]-h2[1]) + squar(h1[2]-h2[2]);
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double d2 = squar(h1[0]+h2[0]) + squar(h1[1]+h2[1]) + squar(h1[2]+h2[2]);
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return min(d1, d2);
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}
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return squar(h1[0]-h2[0]) + squar(h1[1]-h2[1]) + (sphere?1:euclid?0:-1) * squar(h1[2]-h2[2]);
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}
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ld intvalxy(const hyperpoint &h1, const hyperpoint &h2) {
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return squar(h1[0]-h2[0]) + squar(h1[1]-h2[1]);
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}
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ld intvalxyz(const hyperpoint &h1, const hyperpoint &h2) {
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return squar(h1[0]-h2[0]) + squar(h1[1]-h2[1]) + squar(h1[2]-h2[2]);
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}
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ld hypot2(const hyperpoint& h) {
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return sqrt(h[0]*h[0]+h[1]*h[1]);
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}
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ld hypot3(const hyperpoint& h) {
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return sqrt(h[0]*h[0]+h[1]*h[1]+h[2]*h[2]);
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}
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ld sqhypot2(const hyperpoint& h) {
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return h[0]*h[0]+h[1]*h[1];
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}
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ld sqhypot3(const hyperpoint& h) {
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return h[0]*h[0]+h[1]*h[1]+h[2]*h[2];
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}
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ld zlevel(const hyperpoint &h) {
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if(euclid) return h[2];
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else if(sphere) return sqrt(intval(h, Hypc));
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else return sqrt(-intval(h, Hypc));
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}
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// display a hyperbolic point
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char *display(const hyperpoint& H) {
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static char buf[100];
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sprintf(buf, "%8.4f:%8.4f:%8.4f", double(H[0]), double(H[1]), double(H[2]));
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return buf;
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}
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ld hypot_auto(ld x, ld y) {
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switch(cgclass) {
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case gcEuclid:
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return hypot(x, y);
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case gcHyperbolic:
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return acosh(cosh(x) * cosh(y));
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case gcSphere:
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return acos(cos(x) * cos(y));
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default:
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return hypot(x, y);
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}
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}
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// move H back to the sphere/hyperboloid/plane
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hyperpoint normalize(hyperpoint H) {
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ld Z = zlevel(H);
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for(int c=0; c<3; c++) H[c] /= Z;
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return H;
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}
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// get the center of the line segment from H1 to H2
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hyperpoint mid(const hyperpoint& H1, const hyperpoint& H2) {
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using namespace hyperpoint_vec;
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return normalize(H1 + H2);
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}
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// like mid, but take 3D into account
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hyperpoint midz(const hyperpoint& H1, const hyperpoint& H2) {
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using namespace hyperpoint_vec;
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hyperpoint H3 = H1 + H2;
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ld Z = 2;
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if(!euclid) Z = zlevel(H3) * 2 / (zlevel(H1) + zlevel(H2));
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for(int c=0; c<3; c++) H3[c] /= Z;
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return H3;
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}
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// matrices
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//==========
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// matrices represent isometries of the hyperbolic plane
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// (just like isometries of the sphere are represented by rotation matrices)
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// rotate by alpha degrees
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transmatrix spin(ld alpha) {
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transmatrix T = Id;
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T[0][0] = +cos(alpha); T[0][1] = +sin(alpha);
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T[1][0] = -sin(alpha); T[1][1] = +cos(alpha);
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T[2][2] = 1;
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return T;
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}
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transmatrix eupush(ld x, ld y) {
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transmatrix T = Id;
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T[0][2] = x;
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T[1][2] = y;
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return T;
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}
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transmatrix eupush(hyperpoint h) {
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transmatrix T = Id;
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T[0][2] = h[0];
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T[1][2] = h[1];
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return T;
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}
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transmatrix euscalezoom(hyperpoint h) {
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transmatrix T = Id;
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T[0][0] = h[0];
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T[0][1] = -h[1];
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T[1][0] = h[1];
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T[1][1] = h[0];
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return T;
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}
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transmatrix euaffine(hyperpoint h) {
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transmatrix T = Id;
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T[0][1] = h[0];
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T[1][1] = exp(h[1]);
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return T;
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}
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// push alpha units to the right
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transmatrix xpush(ld alpha) {
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if(euclid) return eupush(alpha, 0);
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transmatrix T = Id;
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if(sphere) {
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T[0][0] = +cos(alpha); T[0][2] = +sin(alpha);
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T[2][0] = -sin(alpha); T[2][2] = +cos(alpha);
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}
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else {
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T[0][0] = +cosh(alpha); T[0][2] = +sinh(alpha);
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T[2][0] = +sinh(alpha); T[2][2] = +cosh(alpha);
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}
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return T;
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}
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inline hyperpoint xpush0(ld x) {
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hyperpoint h;
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if(euclid) return hpxy(x, 0);
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else if(sphere) h[0] = sin(x), h[1] = 0, h[2] = cos(x);
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else h[0] = sinh(x), h[1] = 0, h[2] = cosh(x);
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return h;
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}
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inline hyperpoint xspinpush0(ld alpha, ld x) {
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// return spin(alpha)*xpush0(x);
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ld s;
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hyperpoint h;
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if(euclid) return hpxy(x*cos(alpha), -x*sin(alpha));
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else if(sphere) s=sin(x), h[0] = s*cos(alpha), h[1] = -s*sin(alpha), h[2] = cos(x);
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else s=sinh(x), h[0] = s*cos(alpha), h[1] = -s*sin(alpha), h[2] = cosh(x);
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return h;
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}
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// push alpha units vertically
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transmatrix ypush(ld alpha) {
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if(euclid) return eupush(0, alpha);
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transmatrix T = Id;
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if(sphere) {
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T[1][1] = +cos(alpha); T[1][2] = +sin(alpha);
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T[2][1] = -sin(alpha); T[2][2] = +cos(alpha);
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}
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else {
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T[1][1] = +cosh(alpha); T[1][2] = +sinh(alpha);
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T[2][1] = +sinh(alpha); T[2][2] = +cosh(alpha);
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}
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return T;
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}
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// rotate the hyperbolic plane around C0 such that H[1] == 0 and H[0] >= 0
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transmatrix spintox(const hyperpoint& H) {
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transmatrix T = Id;
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ld R = sqrt(H[0] * H[0] + H[1] * H[1]);
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if(R >= 1e-12) {
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T[0][0] = +H[0]/R; T[0][1] = +H[1]/R;
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T[1][0] = -H[1]/R; T[1][1] = +H[0]/R;
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}
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return T;
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}
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void set_column(transmatrix& T, int i, const hyperpoint& H) {
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for(int j=0; j<3; j++)
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T[j][i] = H[j];
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}
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transmatrix build_matrix(hyperpoint h1, hyperpoint h2, hyperpoint h3) {
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transmatrix T;
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for(int i=0; i<3; i++)
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T[i][0] = h1[i],
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T[i][1] = h2[i],
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T[i][2] = h3[i];
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return T;
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}
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// reverse of spintox(H)
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transmatrix rspintox(const hyperpoint& H) {
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transmatrix T = Id;
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ld R = sqrt(H[0] * H[0] + H[1] * H[1]);
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if(R >= 1e-12) {
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T[0][0] = +H[0]/R; T[0][1] = -H[1]/R;
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T[1][0] = +H[1]/R; T[1][1] = +H[0]/R;
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}
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return T;
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}
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// for H such that H[1] == 0, this matrix pushes H to C0
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transmatrix pushxto0(const hyperpoint& H) {
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if(euclid) return eupush(-H[0], -H[1]);
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transmatrix T = Id;
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if(sphere) {
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T[0][0] = +H[2]; T[0][2] = -H[0];
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T[2][0] = +H[0]; T[2][2] = +H[2];
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}
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else {
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T[0][0] = +H[2]; T[0][2] = -H[0];
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T[2][0] = -H[0]; T[2][2] = +H[2];
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}
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return T;
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}
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// reverse of pushxto0(H)
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transmatrix rpushxto0(const hyperpoint& H) {
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if(euclid) return eupush(H[0], H[1]);
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transmatrix T = Id;
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if(sphere) {
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T[0][0] = +H[2]; T[0][2] = +H[0];
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T[2][0] = -H[0]; T[2][2] = +H[2];
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}
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else {
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T[0][0] = +H[2]; T[0][2] = +H[0];
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T[2][0] = +H[0]; T[2][2] = +H[2];
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}
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return T;
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}
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// generalization: H[1] can be non-zero
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transmatrix gpushxto0(const hyperpoint& H) {
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if(euclid) return eupush(-H[0], -H[1]);
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hyperpoint H2 = spintox(H) * H;
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return rspintox(H) * pushxto0(H2) * spintox(H);
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}
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transmatrix rgpushxto0(const hyperpoint& H) {
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if(euclid) return eupush(H[0], H[1]);
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hyperpoint H2 = spintox(H) * H;
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return rspintox(H) * rpushxto0(H2) * spintox(H);
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}
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// fix the matrix T so that it is indeed an isometry
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// (without using this, imprecision could accumulate)
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void fixmatrix(transmatrix& T) {
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if(euclid) {
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for(int x=0; x<2; x++) for(int y=0; y<=x; y++) {
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ld dp = 0;
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for(int z=0; z<2; z++) dp += T[z][x] * T[z][y];
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if(y == x) dp = 1 - sqrt(1/dp);
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for(int z=0; z<2; z++) T[z][x] -= dp * T[z][y];
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}
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for(int x=0; x<2; x++) T[2][x] = 0;
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T[2][2] = 1;
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}
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else for(int x=0; x<3; x++) for(int y=0; y<=x; y++) {
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ld dp = 0;
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for(int z=0; z<3; z++) dp += T[z][x] * T[z][y] * sig(z);
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if(y == x) dp = 1 - sqrt(sig(x)/dp);
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for(int z=0; z<3; z++) T[z][x] -= dp * T[z][y];
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}
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}
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// show the matrix on screen
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void display(const transmatrix& T) {
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for(int y=0; y<3; y++) {
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for(int x=0; x<3; x++) printf("%10.7f", double(T[y][x]));
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printf(" -> %10.7f\n", double(squar(T[y][0]) + squar(T[y][1]) + sig(2) * squar(T[y][2])));
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// printf("\n");
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}
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for(int x=0; x<3; x++) printf("%10.7f", double(squar(T[0][x]) + squar(T[1][x]) + sig(2) * squar(T[2][x])));
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printf("\n");
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for(int x=0; x<3; x++) {
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int y = (x+1) % 3;
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printf("%10.7f", double(T[0][x]*T[0][y] + T[1][x]*T[1][y] + sig(2) * T[2][x]*T[2][y]));
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}
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printf("\n\n");
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}
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ld det(const transmatrix& T) {
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ld det = 0;
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for(int i=0; i<3; i++)
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det += T[0][i] * T[1][(i+1)%3] * T[2][(i+2)%3];
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for(int i=0; i<3; i++)
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det -= T[0][i] * T[1][(i+2)%3] * T[2][(i+1)%3];
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return det;
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}
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void inverse_error(const transmatrix& T) {
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printf("Warning: inverting a singular matrix\n");
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display(T);
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}
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transmatrix inverse(const transmatrix& T) {
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profile_start(7);
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ld d = det(T);
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transmatrix T2;
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if(d == 0) {
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inverse_error(T);
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return Id;
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}
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for(int i=0; i<3; i++)
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for(int j=0; j<3; j++)
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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;
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profile_stop(7);
|
|
return T2;
|
|
}
|
|
|
|
// distance between mh and 0
|
|
double hdist0(const hyperpoint& mh) {
|
|
switch(cgclass) {
|
|
case gcHyperbolic:
|
|
if(mh[2] < 1) return 0;
|
|
return acosh(mh[2]);
|
|
case gcEuclid: {
|
|
ld d = sqrt(mh[0]*mh[0]+mh[1]*mh[1]);
|
|
return d;
|
|
}
|
|
case gcSphere: {
|
|
ld res = mh[2] >= 1 ? 0 : mh[2] <= -1 ? M_PI : acos(mh[2]);
|
|
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<3; i++)
|
|
res[i] = t[i] * fac;
|
|
return res;
|
|
}
|
|
|
|
transmatrix mscale(const transmatrix& t, double fac) {
|
|
transmatrix res;
|
|
for(int i=0; i<3; i++) for(int j=0; j<3; j++)
|
|
res[i][j] = t[i][j] * fac;
|
|
return res;
|
|
}
|
|
|
|
transmatrix xyscale(const transmatrix& t, double fac) {
|
|
transmatrix res;
|
|
for(int i=0; i<3; i++) for(int j=0; j<2; 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<3; i++) for(int j=0; j<2; j++)
|
|
res[i][j] = t[i][j] * fac;
|
|
for(int i=0; i<3; i++)
|
|
res[i][2] = t[i][2] * 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<3; i++) for(int j=0; j<3; 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) {
|
|
return h1[0] == h2[0] && h1[1] == h2[1] && h1[2] == h2[2];
|
|
}
|
|
|
|
// 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);
|
|
}
|
|
|
|
}
|