triangle_group_limit_set/limit_set.c

#include <math.h>
#include <stdio.h>
#include <sys/time.h>
#include <X11/XKBlib.h>

#include "initcairo.h"
#include "triangle.h"
#include "linalg.h"

#define LOOP(i) for(int i = 0; i < 3; i++)

typedef struct {
	double x;
	double y;
} point_t;

int processEvent(GraphicsInfo *info, XEvent *ev, void *data);
void setup();
void destroy();
void cartanMatrix(gsl_matrix *cartan, double a1, double a2, double a3, double s);
void initializeTriangleGenerators(gsl_matrix **gen, gsl_matrix *cartan);
int compareAngle(const void *x, const void *y);
int computeLimitCurve();
void draw(void *data);
point_t intersect(point_t a, point_t b, point_t c, point_t d);

GraphicsInfo G;

double parameter = 2.8;
int n_group_elements;
double *limit_curve; // x, y, angle triples
int limit_curve_valid = 0;
groupelement_t *group;
workspace_t *ws;
gsl_matrix *gen[3];
gsl_matrix **matrices;
gsl_matrix *cartan, *cob, *coxeter, *coxeter_fixedpoints, *fixedpoints;

int processEvent(GraphicsInfo *info, XEvent *ev, void *data)
{
	int state;
	unsigned long key;

	switch(ev->type) {

	case KeyPress:
		state = ev->xkey.state & (ShiftMask | LockMask | ControlMask);
		key = XkbKeycodeToKeysym(ev->xkey.display, ev->xkey.keycode, 0, !!(state & ShiftMask));
		printf("Key pressed: %ld\n", key);

		switch(key) {
		case '<':
			parameter /= exp(0.002);
			limit_curve_valid = computeLimitCurve();
			break;
		case '>':
			parameter *= exp(0.002);
			limit_curve_valid = computeLimitCurve();
			break;
		case ' ':
			parameter = 2.890053638;
			limit_curve_valid = computeLimitCurve();
			break;
		case XK_Return:
			parameter = 2.76375163;
			limit_curve_valid = computeLimitCurve();
			break;
		}

		draw(0);
	}

	return 0;
}

void setup()
{
	n_group_elements = 10000;
	limit_curve = malloc(3*n_group_elements*sizeof(double));
	group = malloc(n_group_elements*sizeof(groupelement_t));
	ws = workspace_alloc(3);

	LOOP(i) gen[i] = gsl_matrix_alloc(3, 3);
	matrices = malloc(n_group_elements*sizeof(gsl_matrix*));
	for(int i = 0; i < n_group_elements; i++)
		matrices[i] = gsl_matrix_alloc(3, 3);
	cartan = gsl_matrix_alloc(3, 3);
	cob = gsl_matrix_alloc(3, 3);
	coxeter = gsl_matrix_alloc(3, 3);
	coxeter_fixedpoints = gsl_matrix_alloc(3, 3);
	fixedpoints = gsl_matrix_alloc(3, 3);
}

void destroy()
{
	free(limit_curve);
	free(group);
	workspace_free(ws);

	LOOP(i) gsl_matrix_free(gen[i]);
	for(int i = 0; i < n_group_elements; i++)
		gsl_matrix_free(matrices[i]);
	free(matrices);
	gsl_matrix_free(cartan);
	gsl_matrix_free(cob);
	gsl_matrix_free(coxeter);
	gsl_matrix_free(coxeter_fixedpoints);
	gsl_matrix_free(fixedpoints);
}

void cartanMatrix(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 initializeTriangleGenerators(gsl_matrix **gen, gsl_matrix *cartan)
{
	LOOP(i) gsl_matrix_set_identity(gen[i]);
	LOOP(i) LOOP(j) *gsl_matrix_ptr(gen[i], i, j) += gsl_matrix_get(cartan, i, j);
}

int compareAngle(const void *x, const void *y)
{
	return ((double*)x)[2] > ((double*)y)[2] ? 1 : -1;
}

// intersect the lines a-b and c-d
point_t intersect(point_t a, point_t b, point_t c, point_t d)
{
	point_t res;
	double t = ((c.y-d.y)*(c.x-a.x) - (c.x-d.x)*(c.y-a.y)) / ((b.x-a.x)*(c.y-d.y) - (b.y-a.y)*(c.x-d.x));
	res.x = (1-t)*a.x + t*b.x;
	res.y = (1-t)*a.y + t*b.y;
	return res;
}

void draw_segment(point_t a, point_t b)
{
	cairo_move_to(G.ctx, a.x, a.y);
	cairo_line_to(G.ctx, b.x, b.y);
	cairo_stroke(G.ctx);
}

void draw_polygon(int sides, point_t a, point_t b, point_t c, point_t d) // only quadrilaterals so far
{
	draw_segment(a, b);
	draw_segment(b, c);
	draw_segment(c, d);
	draw_segment(d, a);
}

void draw_box(const char *word, char base)
{
	// lower stack[i] reserved for linalg.c is used by multiply_*()
	gsl_matrix *tmp1 = ws->stack[10];
	gsl_matrix *conj = ws->stack[12];
	gsl_matrix *conjinv = ws->stack[13];
	gsl_matrix *coxeter[3] = { ws->stack[14], ws->stack[15], ws->stack[16] }; // fixed points of coxeter elements

	gsl_matrix_set_identity(conj);
	gsl_matrix_set_identity(conjinv);
	for(int i = 0; i < strlen(word); i++) {
		if(word[i] == ' ')
			continue;
		multiply_right(conj, gen[word[i]-'a'], ws);
		multiply_left(gen[word[i]-'a'], conjinv, ws);
	}

	for(int i = 0; i < 3; i++) {
		multiply_many(ws, tmp1, 5, conj, gen[(base-'A'+i)%3], gen[(base-'A'+i+1)%3], gen[(base-'A'+i+2)%3], conjinv);
		eigenvectors(tmp1, coxeter[i], ws);
		multiply_left(cob, coxeter[i], ws);
	}

	point_t p_a[3],p_m[3],p_r[3],p_i[4];

	LOOP(i) p_a[i].x = gsl_matrix_get(coxeter[i], 0, 0) / gsl_matrix_get(coxeter[i], 2, 0);
	LOOP(i) p_a[i].y = gsl_matrix_get(coxeter[i], 1, 0) / gsl_matrix_get(coxeter[i], 2, 0);
	LOOP(i) p_m[i].x = gsl_matrix_get(coxeter[i], 0, 1) / gsl_matrix_get(coxeter[i], 2, 1);
	LOOP(i) p_m[i].y = gsl_matrix_get(coxeter[i], 1, 1) / gsl_matrix_get(coxeter[i], 2, 1);
	LOOP(i) p_r[i].x = gsl_matrix_get(coxeter[i], 0, 2) / gsl_matrix_get(coxeter[i], 2, 2);
	LOOP(i) p_r[i].y = gsl_matrix_get(coxeter[i], 1, 2) / gsl_matrix_get(coxeter[i], 2, 2);

	p_i[0] = intersect(p_a[1], p_r[1], p_m[0], p_r[0]);
	p_i[1] = intersect(p_a[0], p_r[0], p_a[2], p_r[2]);
	p_i[2] = intersect(p_a[1], p_m[1], p_a[2], p_r[2]);
	p_i[3] = intersect(p_a[0], p_r[0], p_a[1], p_m[1]);

	draw_polygon(4, p_a[1], p_i[0], p_r[0], p_i[3]);
}

int computeLimitCurve()
{
	int p = 5, q = 5, r = 5;
	int k = 2, l = 2, m = 2;

	generate_triangle_group(group, n_group_elements, p, q, r);

	// do first in the Fuchsian case to get the angles
	cartanMatrix(cartan, M_PI/p, M_PI/q, M_PI/r, 1.0);
	initializeTriangleGenerators(gen, cartan);
	gsl_matrix_set_identity(matrices[0]);
	for(int i = 1; i < n_group_elements; i++)
		multiply(matrices[group[i].parent->id], gen[group[i].letter], matrices[i]);
	diagonalize_symmetric_form(cartan, cob, ws);
	multiply_many(ws, coxeter, 3, gen[0], gen[1], gen[2]);
	if(!eigenvectors(coxeter, coxeter_fixedpoints, ws))
		return 0;

	for(int i = 0; i < n_group_elements; i++) {
		multiply_many(ws, fixedpoints, 3, cob, matrices[i], coxeter_fixedpoints);
		limit_curve[3*i+2] = atan2(
			gsl_matrix_get(fixedpoints, 0, 0)/gsl_matrix_get(fixedpoints, 2, 0),
			gsl_matrix_get(fixedpoints, 1, 0)/gsl_matrix_get(fixedpoints, 2, 0));
	}

	// now to it again to calculate x and y coordinates
	cartanMatrix(cartan, M_PI*k/p, M_PI*l/q, M_PI*m/r, parameter);
	initializeTriangleGenerators(gen, cartan);
	gsl_matrix_set_identity(matrices[0]);
	for(int i = 1; i < n_group_elements; i++)
		multiply(matrices[group[i].parent->id], gen[group[i].letter], matrices[i]);
	gsl_matrix_memcpy(cob, cartan); // is this a good choice of basis
//	gsl_matrix_set_identity(cob);
	multiply_many(ws, coxeter, 3, gen[0], gen[1], gen[2]);
	if(!eigenvectors(coxeter, coxeter_fixedpoints, ws))
		return 0;

	for(int i = 0; i < n_group_elements; i++) {
		multiply_many(ws, fixedpoints, 3, cob, matrices[i], coxeter_fixedpoints);

		limit_curve[3*i] = gsl_matrix_get(fixedpoints, 0, 0)/gsl_matrix_get(fixedpoints, 2, 0);
		limit_curve[3*i+1] = gsl_matrix_get(fixedpoints, 1, 0)/gsl_matrix_get(fixedpoints, 2, 0);
	}

	qsort(limit_curve, n_group_elements, 3*sizeof(double), compareAngle);

	return 1;
}

void draw(void *data)
{
	struct timeval current_time;
	gettimeofday(&current_time, 0);
	double start_time = current_time.tv_sec + current_time.tv_usec*1e-6;

	cairo_push_group(G.ctx);
	cairo_set_source_rgb(G.ctx, 1, 1, 1);
	cairo_paint(G.ctx);

	cairo_set_matrix(G.ctx, &G.matrix);

	if(limit_curve_valid) {
		int last_inside = 0;
		for(int i = 0; i < n_group_elements; i++) {
			double x = limit_curve[3*i];
			double y = limit_curve[3*i+1];

			cairo_user_to_device(G.ctx, &x, &y);

			if(-x < G.width && x < 3*G.width && -y < G.height && y < 3*G.height) {
				if(!last_inside) {
					cairo_move_to(G.ctx, limit_curve[3*i], limit_curve[3*i+1]);
				} else {
					cairo_line_to(G.ctx, limit_curve[3*i], limit_curve[3*i+1]);
				}
				last_inside = 1;
			} else {
				last_inside = 0;
			}
		}

		cairo_set_line_width(G.ctx, 1.0/G.scalefactor);
		cairo_set_line_join(G.ctx, CAIRO_LINE_JOIN_BEVEL);
		cairo_set_source_rgb(G.ctx, 0, 0, 0);
		cairo_stroke(G.ctx);

		cairo_set_source_rgb(G.ctx, 1, 0, 0);
//		draw_box("c", 'C');
		draw_box("", 'B');
		draw_box("a", 'A');
//		draw_box("", 'C');

		cairo_set_source_rgb(G.ctx, 0, 0, 1);
		draw_box("ca", 'A');
		draw_box("cac", 'C');
		draw_box("caca", 'A');
		draw_box("cacac", 'C');
		draw_box("acac", 'A');
		draw_box("aca", 'C');

		cairo_set_source_rgb(G.ctx, 0, 1, 0);
		draw_box("ac", 'C');
		draw_box("aca", 'A');
		draw_box("acac", 'C');
		draw_box("acaca", 'A');
		draw_box("caca", 'C');
		draw_box("cac", 'A');
	}

	cairo_identity_matrix(G.ctx);

	cairo_move_to(G.ctx, 15, 30);
	cairo_set_source_rgb(G.ctx, 0, 0, 0);
	cairo_set_font_size(G.ctx, 16);
	char buf[100];
	sprintf(buf, "t = %.8f", parameter);
	cairo_show_text(G.ctx, buf);

	cairo_pop_group_to_source(G.ctx);
	cairo_paint(G.ctx);
	cairo_surface_flush(G.sfc);

	gettimeofday(&current_time, 0);
	double end_time = current_time.tv_sec + current_time.tv_usec*1e-6;
	printf("draw() finished in %g seconds\n", end_time - start_time);
}

int main()
{
	setup();
	limit_curve_valid = computeLimitCurve();

	if(!initCairo(0, KeyPressMask, 200, 200, "Triangle group", &G))
		return 1;

	cairo_matrix_init_identity(&G.matrix);
	G.matrix.xx = G.matrix.yy = G.scalefactor = 1100.0;
	G.matrix.x0 = 1150.0;
	G.matrix.y0 = 900.0;

	startTimer(&G);

	while(!checkEvents(&G, processEvent, draw, 0)) {
		waitUpdateTimer(&G);
	}

	destroyCairo(&G);

	destroy();

	return 0;
}