hyperbolic_tilings/hyperbolic.c
2023-01-17 11:20:25 -05:00

401 lines
12 KiB
C

#include <complex.h>
#include <stdio.h>
#include <math.h>
#include "triangle.h"
#include "linalg.h"
#define POINCARE 1
#define LOOP(i) for(int i = 0; i < 3; i++)
#define CUTOFF(x) ((x)>1000.0?1000.0:(x)<-1000.0?-1000.0:(x))
#define CONV(x) (CUTOFF(x)*490+500)
#define DCONV(x) (CUTOFF(x)*490)
void cartan_matrix(gsl_matrix *cartan, double a1, double a2, double a3, double s)
{
gsl_matrix_set(cartan, 0, 0, -2);
gsl_matrix_set(cartan, 0, 1, 2*s*cos(a3));
gsl_matrix_set(cartan, 0, 2, 2/s*cos(a2));
gsl_matrix_set(cartan, 1, 0, 2/s*cos(a3));
gsl_matrix_set(cartan, 1, 1, -2);
gsl_matrix_set(cartan, 1, 2, 2*s*cos(a1));
gsl_matrix_set(cartan, 2, 0, 2*s*cos(a2));
gsl_matrix_set(cartan, 2, 1, 2/s*cos(a1));
gsl_matrix_set(cartan, 2, 2, -2);
}
void initialize_triangle_generators(gsl_matrix **gen, gsl_matrix *cartan)
{
LOOP(i) {
gsl_matrix_set_identity(gen[i]);
LOOP(j) *gsl_matrix_ptr(gen[i], i, j) += gsl_matrix_get(cartan, i, j);
}
}
typedef struct { double x[3]; } point;
point apply(gsl_matrix *g, point x)
{
point out;
LOOP(i) out.x[i] = 0.0;
LOOP(i) LOOP(j) out.x[i] += gsl_matrix_get(g, i, j) * x.x[j];
return out;
}
point incidence(point a, point b)
{
point c;
LOOP(i) c.x[i] = a.x[(i+1)%3]*b.x[(i+2)%3] - a.x[(i+2)%3]*b.x[(i+1)%3];
return c;
}
double applyForm(point x, point y, gsl_matrix *form)
{
double out = 0.0;
LOOP(i) LOOP(j) out += x.x[i] * gsl_matrix_get(form, i, j) * y.x[j];
return out;
}
void boundary(point x, point y, point *a, point *b, gsl_matrix *form)
{
double formX = applyForm(x, x, form);
double formY = applyForm(y, y, form);
double formXY = applyForm(x, y, form);
double t1 = (- formXY + sqrt(formXY * formXY - formX * formY)) / formX;
double t2 = (- formXY - sqrt(formXY * formXY - formX * formY)) / formX;
LOOP(i) a->x[i] = t1 * x.x[i] + y.x[i];
LOOP(i) b->x[i] = t2 * x.x[i] + y.x[i];
}
point column(gsl_matrix *m, int j)
{
point out;
LOOP(i) out.x[i] = gsl_matrix_get(m, i, j);
return out;
}
point row(gsl_matrix *m, int i)
{
point out;
LOOP(j) out.x[j] = gsl_matrix_get(m, i, j);
return out;
}
double coord(point p, int k, gsl_matrix *frame)
{
double tmp[3] = {0, 0, 0};
double factor = 1.0;
LOOP(i) LOOP(j) tmp[i] += gsl_matrix_get(frame, i, j)*p.x[j];
#ifdef POINCARE
double norm2 = (tmp[0]*tmp[0] + tmp[1]*tmp[1]) / (tmp[2]*tmp[2]);
if(fabs(norm2) < 0.5) // just to avoid dividing by 0
factor = 1.0 / (1.0 + sqrt(1.0 - norm2));
else if(fabs(norm2) < 1.0)
factor = (1.0 - sqrt(1.0 - norm2)) / norm2;
else
factor = 1.0;
#endif
return factor*tmp[k]/tmp[2];
}
void print_tex_header() {
char *header =
"\\documentclass{standalone}\n"
"\\usepackage[utf8]{inputenc}\n"
"\\usepackage[dvipsnames]{xcolor}\n"
"\\usepackage{tikz}\n"
"\\begin{document}\n"
"\\begin{tikzpicture}[scale=5]\n";
printf("%s", header);
}
void print_tex_footer() {
char *footer =
"\\draw[thick] (0,0) circle (1.0);\n"
"\\end{tikzpicture}\n"
"\\end{document}\n";
printf("%s", footer);
}
void print_svg_header() {
char *header =
"<?xml version=\"1.0\" encoding=\"UTF-8\"?>\n"
"<svg width=\"1000\" height=\"1000\" xmlns=\"http://www.w3.org/2000/svg\">\n";
printf("%s", header);
}
void print_svg_footer() {
printf("<circle cx=\"%f\" cy=\"%f\" r=\"%f\" style=\"fill:none;stroke-width:3;stroke:black;\" />\n",
CONV(0.0), CONV(0.0), DCONV(1.0));
printf("</svg>\n");
}
char *poincare_arc(double ax, double ay, double bx, double by, char *buffer)
{
// find m = (a+b)/2 + tc with c orthogonal to a-b and |m|^2 = |m-a|^2 + 1
// m^2 = m^2 - 2<m,a> + a^2 + 1
// 0 = -2<m,a> + a^2 + 1 = -<a+b,a> - 2t<c,a> + a^2 + 1 = -<b,a> - 2t<c,a> + 1
// t = (1 - <a,b>)/<a,c>/2
double cx = ay - by;
double cy = bx - ax;
double t = (1 - ax*bx - ay*by)/(ax*cx + ay*cy)/2;
double mx = ax/2 + bx/2 + t*cx; // center of circle
double my = ay/2 + by/2 + t*cy;
double r = sqrt((ax-mx)*(ax-mx) + (ay-my)*(ay-my));
//double phix = atan2(x2-m2,x1-m1);
//double phiy = atan2(y2-m2,y1-m1);
// if(phix - phiy > M_PI)
// phiy += 2*M_PI;
// else if(phiy - phix > M_PI)
// phix += 2*M_PI;
// sprintf(buffer, "%f:%f:%f", phix/M_PI*180, phiy/M_PI*180, r);
sprintf(buffer, "%f %f 0 0 %d %f %f", DCONV(r), DCONV(r), t>0, CONV(bx), CONV(by));
return buffer;
}
void draw_triangle(point *p, gsl_matrix *frame, const char *arguments)
{
#ifdef POINCARE
char buffer1[100], buffer2[100], buffer3[100];
double x1 = coord(p[0], 0, frame);
double y1 = coord(p[0], 1, frame);
double x2 = coord(p[1], 0, frame);
double y2 = coord(p[1], 1, frame);
double x3 = coord(p[2], 0, frame);
double y3 = coord(p[2], 1, frame);
// TikZ version
/* printf("\\draw[%s] (%f, %f) arc (%s) arc (%s) arc (%s);\n", arguments, x1, y1,
poincare_arc(x1, y1, x2, y2, buffer1),
poincare_arc(x2, y2, x3, y3, buffer2),
poincare_arc(x3, y3, x1, y1, buffer3)); */
// SVG version
printf("<path d=\"M %f %f A %s A %s A %s Z\" style=\"%s\" />\n", CONV(x1), CONV(y1),
poincare_arc(x1, y1, x2, y2, buffer1),
poincare_arc(x2, y2, x3, y3, buffer2),
poincare_arc(x3, y3, x1, y1, buffer3),
arguments);
#else
printf("\\draw[%s] (%f, %f) -- (%f, %f) -- (%f, %f) -- cycle;\n", arguments,
coord(p[0], 0, frame), coord(p[0], 1, frame),
coord(p[1], 0, frame), coord(p[1], 1, frame),
coord(p[2], 0, frame), coord(p[2], 1, frame));
#endif
}
void draw_dot(point p, gsl_matrix *frame, const char *arguments)
{
double x = coord(p, 0, frame);
double y = coord(p, 1, frame);
#ifdef POINCARE
printf("<circle cx=\"%f\" cy=\"%f\" r=\"6\" style=\"%s\"/>\n", CONV(x), CONV(y), arguments);
#endif
}
void draw_line(point p1, point p2, gsl_matrix *frame, const char *arguments)
{
char buffer[100];
double x1 = coord(p1, 0, frame);
double y1 = coord(p1, 1, frame);
double x2 = coord(p2, 0, frame);
double y2 = coord(p2, 1, frame);
#ifdef POINCARE
// printf("\\draw[%s] (%f, %f) arc (%s);\n", arguments, x1, y1,
// poincare_arc(x1, y1, x2, y2, buffer));
printf("<path d=\"M %f %f A %s\" style=\"%s\"/>\n", CONV(x1), CONV(y1),
poincare_arc(x1, y1, x2, y2, buffer), arguments);
#else
printf("\\draw[%s] (%f, %f) -- (%f, %f);\n", arguments, x1, y1, x2, y2);
#endif
}
void compute_word(workspace_t *ws, gsl_matrix *result, gsl_matrix **gen, const char *word, int modifier, int inverse)
{
gsl_matrix_set_identity(result);
if(inverse) {
for(int i = 0; word[i] != 0; i++)
multiply_left(gen[(word[i]-'a'+modifier)%3], result, ws);
} else {
for(int i = 0; word[i] != 0; i++)
multiply_right(result, gen[(word[i]-'a'+modifier)%3], ws);
}
}
int main(int argc, const char *argv[])
{
groupelement_t *group;
gsl_matrix **matrices;
gsl_matrix *cartan;
gsl_matrix *gen[3];
gsl_matrix **special;
gsl_matrix **special_eigenvectors;
point *special_attracting, *special_repelling, *special_rotation;
gsl_matrix *frame;
workspace_t *ws;
int p,q,r;
int elements, nspecial, nspecial_hyp, nspecial_rot;
if(argc < 5) {
fprintf(stderr, "Usage: %s <p> <q> <r> <n_elements> <word1> <word2> ...\n", argv[0]);
exit(1);
}
nspecial = argc - 5;
elements = atoi(argv[4]);
p = atoi(argv[1]), q = atoi(argv[2]), r = atoi(argv[3]);
group = malloc(elements*sizeof(groupelement_t));
matrices = malloc(elements*sizeof(gsl_matrix*));
for(int i = 0; i < elements; i++)
matrices[i] = gsl_matrix_alloc(3, 3);
cartan = gsl_matrix_alloc(3, 3);
frame = gsl_matrix_alloc(3, 3);
LOOP(i) gen[i] = gsl_matrix_alloc(3, 3);
ws = workspace_alloc(3);
special = malloc(3*nspecial*sizeof(gsl_matrix*));
special_eigenvectors = malloc(3*nspecial*sizeof(gsl_matrix*));
special_attracting = malloc(3*nspecial*sizeof(point));
special_repelling = malloc(3*nspecial*sizeof(point));
special_rotation = malloc(3*nspecial*sizeof(point));
for(int i = 0; i < 3*nspecial; i++) {
special[i] = gsl_matrix_alloc(3, 3);
special_eigenvectors[i] = gsl_matrix_alloc(3, 3);
}
generate_triangle_group(group, elements, p, q, r);
cartan_matrix(cartan, M_PI/p, M_PI/q, M_PI/r, 1.0);
initialize_triangle_generators(gen, cartan);
diagonalize_symmetric_form(cartan, frame, ws); // choose frame of reference which diagonalizes the form
gsl_matrix_set_identity(matrices[0]);
for(int i = 1; i < elements; i++)
multiply(matrices[group[i].parent->id], gen[group[i].letter], matrices[i]);
nspecial_hyp = nspecial_rot = 0;
for(int i = 0; i < nspecial; i++) {
int j = 0;
int nreal;
compute_word(ws, special[3*i+j], gen, argv[i+5], j, 0);
nreal = eigenvectors(special[3*i+j], special_eigenvectors[3*i+j], ws);
if(nreal == 3) {
special_attracting[nspecial_hyp] = column(special_eigenvectors[3*i+j], 0);
// repelling = attracting of inverse
compute_word(ws, special[3*i+j], gen, argv[i+5], j, 1);
eigenvectors(special[3*i+j], special_eigenvectors[3*i+j], ws);
special_repelling[nspecial_hyp] = column(special_eigenvectors[3*i+j], 0);
nspecial_hyp++;
} else {
special_rotation[nspecial_rot] = column(special_eigenvectors[3*i+j], 0);
nspecial_rot++;
}
}
fprintf(stderr, "%d special elements, %d rotations, %d hyperbolic\n", nspecial, nspecial_rot, nspecial_hyp);
/*
// let's correct the frame of reference by using hyperbolic transformations
center = apply(frame, triangle_points[2]);
double angle = atan2(center.x[1], center.x[0]);
double boost = atanh(-sqrt(center.x[0]*center.x[0]+center.x[1]*center.x[1])/center.x[2]);
gsl_matrix *frame_correction = gsl_matrix_alloc(3, 3);
gsl_matrix_set_identity(frame_correction);
gsl_matrix_set(frame_correction, 0, 0, cos(angle-M_PI/2));
gsl_matrix_set(frame_correction, 0, 1, sin(angle-M_PI/2));
gsl_matrix_set(frame_correction, 1, 0, -sin(angle-M_PI/2));
gsl_matrix_set(frame_correction, 1, 1, cos(angle-M_PI/2));
gsl_matrix_set(frame_correction, 2, 2, 1);
// multiply_left(frame_correction, frame, ws);
gsl_matrix_set_identity(frame_correction);
gsl_matrix_set(frame_correction, 0, 0, cosh(-boost));
gsl_matrix_set(frame_correction, 0, 2, sinh(-boost));
gsl_matrix_set(frame_correction, 1, 1, 1);
gsl_matrix_set(frame_correction, 2, 0, sinh(-boost));
gsl_matrix_set(frame_correction, 2, 2, cosh(-boost));
multiply_left(frame_correction, frame, ws);
gsl_matrix_free(frame_correction);
*/
// the actual drawing
point transformed[3];
point reflection_lines[3];
point triangle_points[3];
print_svg_header();
for(int k = 0; k < elements; k++) {
if(group[k].length % 2)
continue;
LOOP(i) reflection_lines[i] = row(cartan, i);
LOOP(i) triangle_points[i] = incidence(reflection_lines[(i+1)%3], reflection_lines[(i+2)%3]);
LOOP(i) transformed[i] = apply(matrices[k], triangle_points[i]);
draw_triangle(transformed, frame, "fill:#cfcfcf;");
}
char stylestring[100];
char colors[3][20] = {"red", "blue", "green"};
// draw special elements
for(int k = 0; k < elements; k++) {
// if(group[k].length % 2)
// continue;
for(int i = 0; i < nspecial_hyp; i+=3) {
// draw_dot(apply(matrices[k], special_repelling[i]), frame, "fill:red;stroke-width:1;");
// draw_dot(apply(matrices[k], special_attracting[i]), frame, "fill:blue;stroke-width:1;");
snprintf(stylestring, sizeof(stylestring), "fill:none;stroke:%s;stroke-width:1;", colors[i%3]);
draw_line(apply(matrices[k], special_repelling[i]),
apply(matrices[k], special_attracting[i]),
frame, stylestring);
}
for(int i = 0; i < nspecial_rot; i+=3) {
snprintf(stylestring, sizeof(stylestring), "fill:%s;stroke:%s;stroke-width:1;", colors[i%3], colors[i%3]);
// fprintf(stderr, "%f %f\n", special_rotation[i].x[0], special_rotation[i].x[1]);
draw_dot(apply(matrices[k], special_rotation[i]), frame, stylestring);
}
}
print_svg_footer();
// clean up
free(group);
for(int i = 0; i < elements; i++)
gsl_matrix_free(matrices[i]);
free(matrices);
gsl_matrix_free(cartan);
gsl_matrix_free(frame);
LOOP(i) gsl_matrix_free(gen[i]);
workspace_free(ws);
for(int i = 0; i < 3*nspecial; i++) {
gsl_matrix_free(special[i]);
gsl_matrix_free(special_eigenvectors[i]);
}
free(special);
free(special_eigenvectors);
free(special_attracting);
free(special_repelling);
free(special_rotation);
}