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hyperpoint.cpp
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2078 lines (1772 loc) · 57.7 KB
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// Hyperbolic Rogue -- basic computations in non-Euclidean geometry
// Copyright (C) 2011-2019 Zeno Rogue, see 'hyper.cpp' for details
/** \file hyperpoint.cpp
* \brief basic computations in non-Euclidean geometry
*
* This implements hyperpoint (a point in non-Euclidean space), transmatrix (a transformation matrix),
* and various basic routines related to them: rotations, translations, inverses and determinants, etc.
* For nonisotropic geometries, it rather refers to nonisotropic.cpp.
*/
#include "hyper.h"
namespace hr {
#if HDR
#ifndef M_PI
#define M_PI 3.14159265358979
#endif
static constexpr ld A_PI = M_PI;
static constexpr ld TAU = 2 * A_PI;
static constexpr ld degree = A_PI / 180;
static const ld golden_phi = (sqrt(5)+1)/2;
static const ld log_golden_phi = log(golden_phi);
constexpr ld operator""_deg(long double deg) { return deg * A_PI / 180; }
#endif
eGeometry geometry;
eVariation variation;
#if HDR
/** \brief A point in our continuous space
*
* Originally used for representing points in the hyperbolic plane.
* Currently used for all kinds of supported spaces, as well as
* for all vector spaces (up to 4 dimensions). We are using
* the normalized homogeneous coordinates, which allows us to work with most
* geometries in HyperRogue in a uniform way.
* In the hyperbolic plane, this is the Minkowski hyperboloid model:
* (x,y,z) such that x*x+y*y-z*z == -1 and z > 0.
*
* In spherical geometry, we have x*x+y*y+z*z == 1.
*
* In Euclidean geometry, we have z = 1.
*
* In isotropic 3D geometries an extra coordinate is added.
*
* In nonisotropic coordinates h[3] == 1.
*
* In product geometries the 'z' coordinate is modelled by multiplying all
* three coordinates with exp(z).
*
*/
struct hyperpoint : array<ld, MAXMDIM> {
hyperpoint() {}
#if MAXMDIM == 4
constexpr hyperpoint(ld x, ld y, ld z, ld w) : array<ld, MAXMDIM> {{x,y,z,w}} {}
#else
constexpr hyperpoint(ld x, ld y, ld z, ld w) : array<ld, MAXMDIM> {{x,y,z}} {}
#endif
inline hyperpoint& operator *= (ld d) {
for(int i=0; i<MXDIM; i++) self[i] *= d;
return self;
}
inline hyperpoint& operator /= (ld d) {
for(int i=0; i<MXDIM; i++) self[i] /= d;
return self;
}
inline hyperpoint& operator += (const hyperpoint h2) {
for(int i=0; i<MXDIM; i++) self[i] += h2[i];
return self;
}
inline hyperpoint& operator -= (const hyperpoint h2) {
for(int i=0; i<MXDIM; i++) self[i] -= h2[i];
return self;
}
inline friend hyperpoint operator * (ld d, hyperpoint h) { return h *= d; }
inline friend hyperpoint operator * (hyperpoint h, ld d) { return h *= d; }
inline friend hyperpoint operator / (hyperpoint h, ld d) { return h /= d; }
inline friend hyperpoint operator + (hyperpoint h, hyperpoint h2) { return h += h2; }
inline friend hyperpoint operator - (hyperpoint h, hyperpoint h2) { return h -= h2; }
inline friend hyperpoint operator - (hyperpoint h) { return h * -1; }
// cross product
inline friend hyperpoint operator ^ (hyperpoint h1, hyperpoint h2) {
return hyperpoint(
h1[1] * h2[2] - h1[2] * h2[1],
h1[2] * h2[0] - h1[0] * h2[2],
h1[0] * h2[1] - h1[1] * h2[0],
0
);
}
friend ld dot_d(int c, hyperpoint h1, hyperpoint h2) {
ld sum = 0;
for(int i=0; i<c; i++) sum += h1[i] * h2[i];
return sum;
}
// Euclidean inner product
inline friend ld operator | (hyperpoint h1, hyperpoint h2) {
return dot_d(MXDIM, h1, h2);
}
};
/** \brief A matrix acting on hr::hyperpoint
*
* Since we are using homogeneous coordinates for hr::hyperpoint,
* rotations and translations can be represented
* as matrix multiplications. Other applications of matrices in HyperRogue
* (in dimension up to 4) are also implemented using transmatrix.
*/
struct transmatrix {
ld tab[MAXMDIM][MAXMDIM];
hyperpoint& operator [] (int i) { return (hyperpoint&)tab[i][0]; }
const hyperpoint& operator [] (int i) const { return (const hyperpoint&)tab[i]; }
inline friend hyperpoint operator * (const transmatrix& T, const hyperpoint& H) {
hyperpoint z;
for(int i=0; i<MXDIM; i++) {
z[i] = 0;
for(int j=0; j<MXDIM; j++) z[i] += T[i][j] * H[j];
}
return z;
}
inline friend transmatrix operator * (const transmatrix& T, const transmatrix& U) {
transmatrix R;
for(int i=0; i<MXDIM; i++) for(int j=0; j<MXDIM; j++) {
R[i][j] = 0;
for(int k=0; k<MXDIM; k++)
R[i][j] += T[i][k] * U[k][j];
}
return R;
}
};
/** @brief hyperpoint with shift
* shift has two uses:
* (1) in the 'universal cover of SL' geometry, shift is used for the extra angular coordinate
* (2) in band models, shift is used to draw faraway points correctly
*/
struct shiftpoint {
hyperpoint h;
ld shift;
shiftpoint() {}
shiftpoint(hyperpoint _h, ld _shift) : h(_h), shift(_shift) {}
ld& operator [] (int i) { return h[i]; }
const ld& operator [] (int i) const { return h[i]; }
inline friend shiftpoint operator + (const shiftpoint& h, const hyperpoint& h2) {
return shiftpoint{h.h+h2, h.shift};
}
inline friend shiftpoint operator - (const shiftpoint& h, const hyperpoint& h2) {
return shiftpoint{h.h-h2, h.shift};
}
};
inline shiftpoint shiftless(const hyperpoint& h, ld shift = 0) {
shiftpoint res; res.h = h; res.shift = shift; return res;
}
struct shiftmatrix {
transmatrix T;
ld shift;
shiftmatrix() {}
shiftmatrix(const transmatrix& _h, ld _shift) : T(_h), shift(_shift) {}
hyperpoint& operator [] (int i) { return T[i]; }
const hyperpoint& operator [] (int i) const { return T[i]; }
inline friend shiftpoint operator * (const shiftmatrix& T, const hyperpoint& h) {
return shiftpoint{T.T*h, T.shift};
}
inline friend shiftmatrix operator * (const shiftmatrix& T, const transmatrix& U) {
return shiftmatrix{T.T*U, T.shift};
}
};
struct shiftmatrix_or_null : shiftmatrix {
bool is_null;
shiftmatrix_or_null& operator = (const shiftmatrix& T) { ((shiftmatrix&) self) = T; is_null = false; return self; }
shiftmatrix_or_null() { is_null = true; }
};
inline shiftmatrix shiftless(const transmatrix& T, ld shift = 0) {
shiftmatrix res; res.T = T; res.shift = shift; return res;
}
/** returns a diagonal matrix */
constexpr transmatrix diag(ld a, ld b, ld c, ld d) {
#if MAXMDIM==3
return transmatrix{{{a,0,0}, {0,b,0}, {0,0,c}}};
#else
return transmatrix{{{a,0,0,0}, {0,b,0,0}, {0,0,c,0}, {0,0,0,d}}};
#endif
}
constexpr hyperpoint Hypc = hyperpoint(0, 0, 0, 0);
/** identity matrix */
constexpr transmatrix Id = diag(1,1,1,1);
/** zero matrix */
constexpr transmatrix Zero = diag(0,0,0,0);
/** a transmatrix with 2D and 3D version, useful for configuration */
struct trans23 {
transmatrix v2, v3;
transmatrix& get() { return MDIM == 3 ? v2 : v3; }
const transmatrix& get() const { return MDIM == 3 ? v2 : v3; }
trans23() { v2 = Id; v3 = Id; }
trans23(const transmatrix& T) { v2 = T; v3 = T; }
trans23(const transmatrix& T2, const transmatrix& T3) { v2 = T2; v3 = T3; }
bool operator == (const trans23& b) const;
bool operator != (const trans23& b) const { return !(self == b); }
trans23 operator * (trans23 T) {
trans23 t;
auto& dim = cginf.g.homogeneous_dimension;
dynamicval<int> d1(dim, dim);
dim = 3; t.v2 = v2 * T.v2;
dim = 4; t.v3 = v3 * T.v3;
return t;
}
friend trans23 operator * (transmatrix M, trans23 T) {
trans23 t(M);
return t * T;
}
};
/** mirror image */
constexpr transmatrix Mirror = diag(1,-1,1,1);
/** mirror image: flip in the Y coordinate */
constexpr transmatrix MirrorY = diag(1,-1,1,1);
/** mirror image: flip in the X coordinate */
constexpr transmatrix MirrorX = diag(-1,1,1,1);
/** mirror image: flip in the Z coordinate */
constexpr transmatrix MirrorZ = diag(1,1,-1,1);
/** rotate by PI in the XY plane */
constexpr transmatrix pispin = diag(-1,-1,1,1);
/** central symmetry matrix */
constexpr transmatrix centralsym = diag(-1,-1,-1,-1);
inline hyperpoint hpxyz(ld x, ld y, ld z) { return MDIM == 3 ? hyperpoint(x,y,z,0) : hyperpoint(x,y,0,z); }
inline hyperpoint hpxyz3(ld x, ld y, ld z, ld w) { return MDIM == 3 ? hyperpoint(x,y,w,0) : hyperpoint(x,y,z,w); }
constexpr hyperpoint point3(ld x, ld y, ld z) { return hyperpoint(x,y,z,0); }
constexpr hyperpoint point30(ld x, ld y, ld z) { return hyperpoint(x,y,z,0); }
constexpr hyperpoint point31(ld x, ld y, ld z) { return hyperpoint(x,y,z,1); }
constexpr hyperpoint point2(ld x, ld y) { return hyperpoint(x,y,0,0); }
constexpr hyperpoint C02 = hyperpoint(0,0,1,0);
constexpr hyperpoint C03 = hyperpoint(0,0,0,1);
/** C0 is the origin in our space */
#define C0 (MDIM == 3 ? C02 : C03)
#endif
bool trans23::operator == (const trans23& b) const { return eqmatrix(v2, b.v2) && eqmatrix(v3, b.v3); }
// basic functions and types
//===========================
EX ld squar(ld x) { return x*x; }
EX int sig(int z) { return ginf[geometry].g.sig[z]; }
EX int curvature() {
switch(cgclass) {
case gcEuclid: return 0;
case gcHyperbolic: return -1;
case gcSphere: return 1;
case gcProduct: return PIU(curvature());
default: return 0;
}
}
EX ld sin_auto(ld x) {
switch(cgclass) {
case gcEuclid: return x;
case gcHyperbolic: return sinh(x);
case gcSphere: return sin(x);
case gcProduct: return PIU(sin_auto(x));
case gcSL2: return sinh(x);
default: return x;
}
}
EX ld asin_auto(ld x) {
switch(cgclass) {
case gcEuclid: return x;
case gcHyperbolic: return asinh(x);
case gcSphere: return asin(x);
case gcProduct: return PIU(asin_auto(x));
case gcSL2: return asinh(x);
default: return x;
}
}
EX ld acos_auto(ld x) {
switch(cgclass) {
case gcHyperbolic: return acosh(x);
case gcSphere: return acos(x);
case gcProduct: return PIU(acos_auto(x));
case gcSL2: return acosh(x);
default: return x;
}
}
/** \brief volume of a three-dimensional ball of radius r in the current isotropic geometry */
EX ld volume_auto(ld r) {
switch(cgclass) {
case gcEuclid: return r * r * r * 240._deg;
case gcHyperbolic: return M_PI * (sinh(2*r) - 2 * r);
case gcSphere: return M_PI * (2 * r - sin(2*r));
default: return 0;
}
}
/** \brief area of a circle of radius r in the current isotropic geometry */
EX ld area_auto(ld r) {
switch(cgclass) {
case gcEuclid: return r * r * M_PI;
case gcHyperbolic: return TAU * (cosh(r) - 1);
case gcSphere: return TAU * (1 - cos(r));
default: return 0;
}
}
EX ld inverse_area_auto(ld vol) {
switch(cgclass) {
case gcEuclid: return sqrt(vol / M_PI);
case gcHyperbolic: return acosh_clamp(vol / TAU + 1);
case gcSphere: return acos_clamp(1 - vol / TAU);
default: return 0;
}
}
EX ld inverse_volume_euc(ld vol) {
return pow(vol * 3/4/M_PI, 1/3.);
}
EX ld inverse_volume_auto(ld vol) {
switch(cgclass) {
case gcEuclid: return inverse_volume_euc(vol);
case gcHyperbolic: {
ld r = vol < 10 ? inverse_volume_euc(vol) : asinh(vol/M_PI)/2;
if(r>0) for(int i=0; i<5; i++) r += (vol - volume_auto(r)) / (4 * M_PI * pow(sinh(r), 2));
return r;
}
case gcSphere: {
if(vol <= 0) return 0;
if(vol >= TAU * M_PI) return M_PI;
ld r = vol < M_PI ? inverse_volume_euc(vol) : vol > TAU * M_PI - M_PI ? M_PI - inverse_volume_euc(TAU*M_PI-vol) : vol / TAU;
if(sin(r)) for(int i=0; i<5; i++) r += (vol - volume_auto(r)) / (4 * M_PI * pow(sin(r), 2));
return r;
}
default: return 0;
}
}
/** \brief volume in 3D, area in 2D */
EX ld wvolarea_auto(ld r) {
if(WDIM == 3) return volume_auto(r);
else return area_auto(r);
}
/** \brief volume in 3D, area in 2D -- inverse */
EX ld inverse_wvolarea_auto(ld r) {
if(WDIM == 3) return inverse_volume_auto(r);
else return inverse_area_auto(r);
}
EX ld asin_clamp(ld x) { return x>1 ? 90._deg : x<-1 ? -90._deg : std::isnan(x) ? 0 : asin(x); }
EX ld acos_clamp(ld x) { return x>1 ? 0 : x<-1 ? M_PI : std::isnan(x) ? 0 : acos(x); }
EX ld asin_auto_clamp(ld x) {
switch(cgclass) {
case gcEuclid: return x;
case gcHyperbolic: return asinh(x);
case gcSL2: return asinh(x);
case gcSphere: return asin_clamp(x);
case gcProduct: return PIU(asin_auto_clamp(x));
default: return x;
}
}
EX ld acosh_clamp(ld x) { return x < 1 ? 0 : acosh(x); }
EX ld acos_auto_clamp(ld x) {
switch(cgclass) {
case gcHyperbolic: return acosh_clamp(x);
case gcSL2: return acosh_clamp(x);
case gcSphere: return acos_clamp(x);
case gcProduct: return PIU(acos_auto_clamp(x));
default: return x;
}
}
EX ld cos_auto(ld x) {
switch(cgclass) {
case gcEuclid: return 1;
case gcHyperbolic: return cosh(x);
case gcSL2: return cosh(x);
case gcSphere: return cos(x);
case gcProduct: return PIU(cos_auto(x));
default: return 1;
}
}
EX ld tan_auto(ld x) {
switch(cgclass) {
case gcEuclid: return x;
case gcHyperbolic: return tanh(x);
case gcSphere: return tan(x);
case gcProduct: return PIU(tan_auto(x));
case gcSL2: return tanh(x);
default: return 1;
}
}
EX ld atan_auto(ld x) {
switch(cgclass) {
case gcEuclid: return x;
case gcHyperbolic: return atanh(x);
case gcSphere: return atan(x);
case gcProduct: return PIU(atan_auto(x));
case gcSL2: return atanh(x);
default: return x;
}
}
EX ld atan2_auto(ld y, ld x) {
switch(cgclass) {
case gcEuclid: return y/x;
case gcHyperbolic: return atanh(y/x);
case gcSL2: return atanh(y/x);
case gcSphere: return atan2(y, x);
case gcProduct: return PIU(atan2_auto(y, x));
default: return y/x;
}
}
/** This function returns the length of the edge opposite the angle alpha in
* a triangle with angles alpha, beta, gamma. This is called the cosine rule,
* and of course works only in non-Euclidean geometry. */
EX ld edge_of_triangle_with_angles(ld alpha, ld beta, ld gamma) {
return acos_auto((cos(alpha) + cos(beta) * cos(gamma)) / (sin(beta) * sin(gamma)));
}
EX hyperpoint hpxy(ld x, ld y) {
if(embedded_plane) {
geom3::light_flip(true);
hyperpoint h = hpxy(x, y);
geom3::light_flip(false);
return cgi.emb->base_to_actual(h);
}
if(sl2) return hyperpoint(x, y, 0, sqrt(1+x*x+y*y));
if(mtwisted) return hyperpoint(x, y, 0, sqrt(1-x*x-y*y));
return PIU(hpxyz(x,y, translatable ? 1 : sphere ? sqrt(1-x*x-y*y) : sqrt(1+x*x+y*y)));
}
EX hyperpoint hpxy3(ld x, ld y, ld z) {
return hpxyz3(x,y,z, sl2 ? sqrt(1+x*x+y*y-z*z) :translatable ? 1 : sphere ? sqrt(1-x*x-y*y-z*z) : sqrt(1+x*x+y*y+z*z));
}
#if HDR
// a point (I hope this number needs no comments ;) )
constexpr hyperpoint Cx12 = hyperpoint(1,0,1.41421356237,0);
constexpr hyperpoint Cx13 = hyperpoint(1,0,0,1.41421356237);
#define Cx1 (GDIM==2?Cx12:Cx13)
#endif
EX bool zero_d(int d, hyperpoint h) {
for(int i=0; i<d; i++) if(h[i]) return false;
return true;
}
/** inner product in the current geometry */
EX ld geo_inner(const hyperpoint &h1, const hyperpoint &h2) {
ld res = 0;
for(int i=0; i<MDIM; i++) res += h1[i] * h2[i] * sig(i);
return res;
}
/** this function returns approximate square of distance between two points
* (in the spherical analogy, this would be the distance in the 3D space,
* through the interior, not on the surface)
* also used to verify whether a point h1 is on the hyperbolic plane by using Hypc for h2
*/
EX ld intval(const hyperpoint &h1, const hyperpoint &h2) {
ld res = 0;
for(int i=0; i<MDIM; i++) res += squar(h1[i] - h2[i]) * sig(i);
if(elliptic) {
ld res2 = 0;
for(int i=0; i<MDIM; i++) res2 += squar(h1[i] + h2[i]) * sig(i);
return min(res, res2);
}
return res;
}
EX ld quickdist(const hyperpoint &h1, const hyperpoint &h2) {
if(gproduct) return hdist(h1, h2);
return intval(h1, h2);
}
/** square Euclidean hypotenuse in the first d dimensions */
EX ld sqhypot_d(int d, const hyperpoint& h) {
ld sum = 0;
for(int i=0; i<d; i++) sum += h[i]*h[i];
return sum;
}
/** Euclidean hypotenuse in the first d dimensions */
EX ld hypot_d(int d, const hyperpoint& h) {
return sqrt(sqhypot_d(d, h));
}
/** @brief h1 and h2 define a line; to_other_side(h1,h2)*x is x moved orthogonally to this line, by double the distance from C0
* (I suppose it could be done better)
*/
EX transmatrix to_other_side(hyperpoint h1, hyperpoint h2) {
if(cgi.emb->is_sph_in_low() && !geom3::flipped) {
geom3::light_flip(true);
h1 = normalize(h1);
h2 = normalize(h2);
transmatrix T = to_other_side(h1, h2);
fix4(T);
geom3::light_flip(false);
return T;
}
if(sol && meuclid) {
/* works in 4x4... */
return gpushxto0(h1) * gpushxto0(h2);
}
ld d = hdist(h1, h2);
hyperpoint v;
if(euclid)
v = (h2 - h1) / d;
else
v = (h1 * cos_auto(d) - h2) / sin_auto(d);
ld d1;
if(euclid)
d1 = -(v|h1) / (v|v);
else
d1 = atan_auto(-v[LDIM] / h1[LDIM]);
hyperpoint hm = h1 * cos_auto(d1) + (sphere ? -1 : 1) * v * sin_auto(d1);
return rspintox(hm) * xpush(-hdist0(hm) * 2) * spintox(hm);
}
/** @brief positive for a material vertex, 0 for ideal vertex, negative for ultra-ideal vertex */
EX ld material(const hyperpoint& h) {
if(sphere || in_s2xe()) return intval(h, Hypc);
else if(hyperbolic || in_h2xe()) return -intval(h, Hypc);
#if MAXMDIM >= 4
else if(sl2) return h[2]*h[2] + h[3]*h[3] - h[0]*h[0] - h[1]*h[1];
#endif
else return h[LDIM];
}
EX int safe_classify_ideals(hyperpoint h) {
if(hyperbolic || in_h2xe()) {
h /= h[LDIM];
ld x = MDIM == 3 ? 1 - (h[0] * h[0] + h[1] * h[1]) : 1 - (h[0] * h[0] + h[1] * h[1] + h[2] * h[2]);
if(x > 1e-6) return 1;
if(x < -1e-6) return -1;
return 0;
}
return 1;
}
EX ld ideal_limit = 10;
EX ld ideal_each = degree;
EX hyperpoint safe_approximation_of_ideal(hyperpoint h) {
return towards_inf(C0, h, ideal_limit);
}
/** the point on the line ab which is closest to zero. Might not be normalized. Works even if a and b are (ultra)ideal */
EX hyperpoint closest_to_zero(hyperpoint a, hyperpoint b) {
if(sqhypot_d(MDIM, a-b) < 1e-9) return a;
if(isnan(a[0])) return a;
a /= a[LDIM];
b /= b[LDIM];
ld mul_a = 0, mul_b = 0;
for(int i=0; i<LDIM; i++) {
ld z = a[i] - b[i];
mul_a += a[i] * z;
mul_b -= b[i] * z;
}
return (mul_b * a + mul_a * b) / (mul_a + mul_b);
}
/** should be called get_lof */
EX ld zlevel(const hyperpoint &h) {
if(sl2) return sqrt(-intval(h, Hypc));
else if(translatable) return h[LDIM];
else if(sphere) return sqrt(intval(h, Hypc));
else if(in_e2xe()) return log(h[2]);
else if(gproduct) return log(sqrt(abs(intval(h, Hypc)))); /* abs works with both underlying spherical and hyperbolic */
else return (h[LDIM] < 0 ? -1 : 1) * sqrt(-intval(h, Hypc));
}
EX ld hypot_auto(ld x, ld y) {
switch(cgclass) {
case gcEuclid:
return hypot(x, y);
case gcHyperbolic:
return acosh(cosh(x) * cosh(y));
case gcSphere:
return acos(cos(x) * cos(y));
default:
return hypot(x, y);
}
}
/** normalize the homogeneous coordinates */
EX hyperpoint normalize(hyperpoint H) {
if(gproduct) return H;
ld Z = zlevel(H);
for(int c=0; c<MXDIM; c++) H[c] /= Z;
return H;
}
/** like normalize but makes (ultra)ideal points material */
EX hyperpoint ultra_normalize(hyperpoint H) {
if(material(H) <= 0) {
H[LDIM] = hypot_d(LDIM, H) + 1e-10;
}
return normalize(H);
}
/** get the center of the line segment from H1 to H2 */
EX hyperpoint mid(const hyperpoint& H1, const hyperpoint& H2) {
if(gproduct) {
auto d1 = product_decompose(H1);
auto d2 = product_decompose(H2);
hyperpoint res1 = PIU( mid(d1.second, d2.second) );
hyperpoint res = res1 * exp((d1.first + d2.first) / 2);
return res;
}
return normalize(H1 + H2);
}
EX shiftpoint mid(const shiftpoint& H1, const shiftpoint& H2) {
return shiftless(mid(H1.h, H2.h), (H1.shift + H2.shift)/2);
}
/** like mid, but take 3D into account */
EX hyperpoint midz(const hyperpoint& H1, const hyperpoint& H2) {
if(gproduct) return mid(H1, H2);
hyperpoint H3 = H1 + H2;
ld Z = 2;
if(!euclid) Z = zlevel(H3) * 2 / (zlevel(H1) + zlevel(H2));
for(int c=0; c<MXDIM; c++) H3[c] /= Z;
return H3;
}
// matrices
//==========
/** rotate by alpha degrees in the coordinates a, b */
EX transmatrix cspin(int a, int b, ld alpha) {
transmatrix T = Id;
T[a][a] = +cos(alpha); T[a][b] = +sin(alpha);
T[b][a] = -sin(alpha); T[b][b] = +cos(alpha);
return T;
}
EX transmatrix lorentz(int a, int b, ld v) {
transmatrix T = Id;
T[a][a] = T[b][b] = cosh(v);
T[a][b] = T[b][a] = sinh(v);
return T;
}
/** rotate by 90 degrees in the coordinates a, b */
EX transmatrix cspin90(int a, int b) {
transmatrix T = Id;
T[a][a] = 0; T[a][b] = 1;
T[b][a] = -1; T[b][b] = 0;
return T;
}
/** rotate by 180 degrees in the coordinates a, b */
EX transmatrix cspin180(int a, int b) {
transmatrix T = Id;
T[a][a] = T[b][b] = -1;
return T;
}
EX transmatrix random_spin3() {
ld alpha2 = asin(randd() * 2 - 1);
ld alpha = randd() * TAU;
ld alpha3 = randd() * TAU;
return cspin(0, 1, alpha) * cspin(0, 2, alpha2) * cspin(1, 2, alpha3);
}
EX transmatrix random_spin() {
if(WDIM == 2) return spin(randd() * TAU);
else return random_spin3();
}
EX transmatrix eupush(ld x, ld y) {
transmatrix T = Id;
T[0][LDIM] = x;
T[1][LDIM] = y;
return T;
}
EX transmatrix euclidean_translate(ld x, ld y, ld z) {
transmatrix T = Id;
T[0][LDIM] = x;
T[1][LDIM] = y;
T[2][LDIM] = z;
return T;
}
EX transmatrix euscalexx(ld x) {
transmatrix T = Id;
T[0][0] = x;
T[1][1] = x;
return T;
}
EX transmatrix euscale(ld x, ld y) {
transmatrix T = Id;
T[0][0] = x;
T[1][1] = y;
return T;
}
EX transmatrix euscale3(ld x, ld y, ld z) {
transmatrix T = Id;
T[0][0] = x;
T[1][1] = y;
T[2][2] = z;
return T;
}
EX hyperpoint eupoint(ld x, ld y) {
return hyperpoint(x, y, MDIM == 3 ? 1 : 0, 1);
}
EX transmatrix eupush(hyperpoint h, ld co IS(1)) {
if(nonisotropic) return nisot::translate(h, co);
if(hyperbolic) { return co ? parabolic13_at(deparabolic13(h)) : inverse(parabolic13_at(deparabolic13(h))); }
transmatrix T = Id;
for(int i=0; i<GDIM; i++) T[i][LDIM] = h[i] * co;
return T;
}
EX transmatrix eupush3(ld x, ld y, ld z) {
if(sl2) return slr::translate(slr::xyz_point(x, y, z));
return eupush(point3(x, y, z));
}
EX transmatrix euscalezoom(hyperpoint h) {
transmatrix T = Id;
T[0][0] = h[0];
T[0][1] = -h[1];
T[1][0] = h[1];
T[1][1] = h[0];
return T;
}
EX transmatrix euaffine(hyperpoint h) {
transmatrix T = Id;
T[0][1] = h[0];
T[1][1] = exp(h[1]);
return T;
}
EX transmatrix cpush(int cid, ld alpha) {
if(gproduct && cid == 2)
return scale_matrix(Id, exp(alpha));
transmatrix T = Id;
if(nonisotropic)
return eupush3(cid == 0 ? alpha : 0, cid == 1 ? alpha : 0, cid == 2 ? alpha : 0);
T[LDIM][LDIM] = T[cid][cid] = cos_auto(alpha);
T[cid][LDIM] = sin_auto(alpha);
T[LDIM][cid] = -curvature() * sin_auto(alpha);
return T;
}
EX transmatrix cmirror(int cid) {
transmatrix T = Id;
T[cid][cid] = -1;
return T;
}
// push alpha units to the right
EX transmatrix xpush(ld alpha) { return cpush(0, alpha); }
EX bool eqmatrix(transmatrix A, transmatrix B, ld eps IS(.01)) {
for(int i=0; i<MXDIM; i++)
for(int j=0; j<MXDIM; j++)
if(std::abs(A[i][j] - B[i][j]) > eps)
return false;
return true;
}
// push alpha units vertically
EX transmatrix ypush(ld alpha) { return cpush(1, alpha); }
EX transmatrix zpush(ld z) { return cpush(2, z); }
EX transmatrix matrix3(ld a, ld b, ld c, ld d, ld e, ld f, ld g, ld h, ld i) {
#if MAXMDIM==3
return transmatrix {{{a,b,c},{d,e,f},{g,h,i}}};
#else
if(GDIM == 2 || MDIM == 3)
return transmatrix {{{a,b,c,0},{d,e,f,0},{g,h,i,0},{0,0,0,1}}};
else
return transmatrix {{{a,b,0,c},{d,e,0,f},{0,0,1,0},{g,h,0,i}}};
#endif
}
EX transmatrix matrix4(ld a, ld b, ld c, ld d, ld e, ld f, ld g, ld h, ld i, ld j, ld k, ld l, ld m, ld n, ld o, ld p) {
#if MAXMDIM==3
return transmatrix {{{a,b,d},{e,f,h},{m,n,p}}};
#else
return transmatrix {{{a,b,c,d},{e,f,g,h},{i,j,k,l},{m,n,o,p}}};
#endif
}
EX transmatrix parabolic1(ld u) {
if(euclid)
return ypush(u);
else if(cgi.emb->is_hyp_in_solnih() && !geom3::flipped) {
return ypush(u);
}
else {
ld diag = u*u/2;
return matrix3(
-diag+1, u, diag,
-u, 1, u,
-diag, u, diag+1
);
}
}
EX transmatrix parabolic13(ld u, ld v) {
if(euclid)
return eupush3(0, u, v);
else if(cgi.emb->is_euc_in_hyp()) {
ld diag = (u*u+v*v)/2;
return matrix4(
1, 0, -u, u,
0, 1, -v, v,
u, v, -diag+1, diag,
u, v, -diag, diag+1
);
}
else {
ld diag = (u*u+v*v)/2;
return matrix4(
-diag+1, u, v, diag,
-u, 1, 0, u,
-v, 0, 1, v,
-diag, u, v, diag+1
);
}
}
EX hyperpoint kleinize(hyperpoint h) {
#if MAXMDIM == 3
return point3(h[0]/h[2], h[1]/h[2], 1);
#else
if(GDIM == 2) return point3(h[0]/h[2], h[1]/h[2], 1);
else return point31(h[0]/h[3], h[1]/h[3], h[2]/h[3]);
#endif
}
EX hyperpoint deparabolic13(hyperpoint h) {
if(euclid) return h;
if(cgi.emb->is_euc_in_hyp()) {
h /= (1 + h[LDIM]);
h[2] -= 1;
h /= sqhypot_d(LDIM, h);
h[2] += .5;
return point3(h[0] * 2, h[1] * 2, log(2) + log(-h[2]));
}
h /= (1 + h[LDIM]);
h[0] -= 1;
h /= sqhypot_d(LDIM, h);
h[0] += .5;
return point3(log(2) + log(-h[0]), h[1] * 2, LDIM==3 ? h[2] * 2 : 0);
}
EX hyperpoint parabolic13(hyperpoint h) {
if(euclid) return h;
else if(cgi.emb->is_euc_in_hyp()) {
return parabolic13(h[0], h[1]) * cpush0(2, h[2]);
}
else if(LDIM == 3)
return parabolic13(h[1], h[2]) * xpush0(h[0]);
else
return parabolic1(h[1]) * xpush0(h[0]);
}
EX transmatrix parabolic13_at(hyperpoint h) {
if(euclid) return rgpushxto0(h);
else if(cgi.emb->is_euc_in_hyp()) {
return parabolic13(h[0], h[1]) * cpush(2, h[2]);
}
else if(LDIM == 3)
return parabolic13(h[1], h[2]) * xpush(h[0]);
else
return parabolic1(h[1]) * xpush(h[0]);
}
EX transmatrix spintoc(const hyperpoint& H, int t, int f) {
transmatrix T = Id;
ld R = hypot(H[f], H[t]);
if(R >= 1e-15) {
T[t][t] = +H[t]/R; T[t][f] = +H[f]/R;
T[f][t] = -H[f]/R; T[f][f] = +H[t]/R;
}
return T;
}
/** an Euclidean rotation in the axes (t,f) which rotates
* the point H to the positive 't' axis
*/
EX transmatrix rspintoc(const hyperpoint& H, int t, int f) {
transmatrix T = Id;
ld R = hypot(H[f], H[t]);
if(R >= 1e-15) {
T[t][t] = +H[t]/R; T[t][f] = -H[f]/R;
T[f][t] = +H[f]/R; T[f][f] = +H[t]/R;
}
return T;
}
/** an isometry which takes the point H to the positive X axis
* \see rspintox
*/
EX transmatrix spintox(const hyperpoint& H) {
if(GDIM == 2 || gproduct) return spintoc(H, 0, 1);
transmatrix T1 = spintoc(H, 0, 1);
return spintoc(T1*H, 0, 2) * T1;
}
/** inverse of hr::spintox
*/
EX transmatrix rspintox(const hyperpoint& H) {
if(GDIM == 2 || gproduct) return rspintoc(H, 0, 1);
transmatrix T1 = spintoc(H, 0, 1);
return rspintoc(H, 0, 1) * rspintoc(T1*H, 0, 2);
}
/** for H on the X axis, this matrix pushes H to C0
* \see gpushxto0
*/
EX transmatrix pushxto0(const hyperpoint& H) {
transmatrix T = Id;
T[0][0] = +H[LDIM]; T[0][LDIM] = -H[0];
T[LDIM][0] = curvature() * H[0]; T[LDIM][LDIM] = +H[LDIM];
return T;
}
/** set the i-th column of T to H */
EX void set_column(transmatrix& T, int i, const hyperpoint& H) {
for(int j=0; j<MXDIM; j++)
T[j][i] = H[j];
}
EX hyperpoint get_column(const transmatrix& T, int i) {
hyperpoint h;