hyperbolic_tilings/hyperbolic.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);
}