erdos.C

/*
 * erdos.C
 *
 * FUNCTION:
 * Display lambert series involving divisors
 * More generally, 
 * show Weierstrass elliptic function g_2 and g_3 invariants
 * Also show the modular discriminant.
 *
 * HISTORY:
 * quick hack -- Linas Vepstas October 1989
 * modernize -- Linas Vepstas March 1996
 * more stuff -- January 2000
 * more stuff -- October 2004
 */

#include <math.h>
#include <stdio.h>
#include <stdlib.h>

#include "brat.h"
#include "modular.h"
#include "coord-xforms.h"

long double erdos_series (long double re_q, long double im_q)
{
	long double rep, imp;
	erdos_series_c (re_q, im_q, 0, &rep, &imp);
	// return sqrt (rep*rep+imp*imp);
	// return imp;
	long double phase = atan2 (imp, rep);
	phase += M_PI;
	phase /= 2.0*M_PI;
	return phase;
}


long double discriminant (long double re_q, long double im_q)
{
	long double rep, imp;
	disc_c (re_q, im_q, &rep, &imp);

	// return sqrt (rep*rep+imp*imp);
	return rep;
	// return imp;
	// long double phase = atan2 (imp, rep);
	// phase += M_PI;
	// phase /= 2.0*M_PI;
	// return phase;
}

/* Weierstrass elliptic invarient g_2, where q is the nome */
long double gee_2 (long double re_q, long double im_q)
{
	long double rep, imp;
	gee_2_c (re_q, im_q, &rep, &imp);

	// return sqrt (rep*rep+imp*imp);
	return rep;
	// return imp;
	// long double phase = atan2 (imp, rep);
	// phase += M_PI;
	// phase /= 2.0*M_PI;
	// return phase;
}

long double gee_3 (long double re_q, long double im_q)
{
	long double rep, imp;
	gee_3_c (re_q, im_q, &rep, &imp);

	// return sqrt (rep*rep+imp*imp);
	return rep;
	// return imp;
	// long double phase = atan2 (imp, rep);
	// phase += M_PI;
	// phase /= 2.0*M_PI;
	// return phase;
}

double klein_j (double re_q, double im_q)
{
	double rep, imp;
	klein_j_invariant_c (re_q, im_q, &rep, &imp);

// #define POWER_OF
#ifdef POWER_OF
	double xponent = 2.0;
	double mag = sqrt (rep*rep + imp*imp);
	double arg = atan2 (imp, rep);
	mag = pow (mag, xponent);
	double rez = mag * cos (xponent*arg);
	double imz = mag * sin (xponent*arg);

	rep = rez - 2.0*rep;
	imp = imz - 2.0*imp;
#endif
	return sqrt (rep*rep+imp*imp);
	// return rep;
	// return imp;
	// long double phase = atan2 (imp, rep);
	// phase += M_PI;
	// phase /= 2.0*M_PI;
	// return phase;
}

double domain_bounce (double re_t, double im_t, 
                      double (*f)(double,double), int depth)
{
#define DET 5
	depth --;

	double m = re_t*re_t+im_t*im_t;
	double sre = -re_t / m;
	double sim = im_t / m;

	double acc = 0.0;
	int n;
	for (n=-DET; n<=DET; n++)
	{
		double en = n;
		acc += (f) (sre+n, sim);
		if (depth)
			acc += domain_bounce (sre+n, sim, f, depth);
	}

	return acc;
}

double bound (double re_tau, double im_tau)
{
#define BND 0.01
	double m = sqrt (re_tau*re_tau+im_tau*im_tau);
	
	if (fabs(1.0-m) < BND) return 1.0;
	if (m>1.0 && fabs(re_tau-0.5) < BND) return 1.0;
	if (m>1.0 && fabs(re_tau+0.5) < BND) return 1.0;
	return 0.0;

#if 0
	double dist = 0.0;
	dist += BND*fabs (1.0-m);
	dist += BND*fabs (re_tau-0.5);
	dist += BND*fabs (re_tau+0.5);
	return dist;
#endif

}

double domain (double re_q, double im_q)
{
	/* get tau */
	double phase = atan2 (im_q, re_q);
	double r = sqrt (re_q*re_q + im_q*im_q);
	if (1.0<=r) return 0.5;

	r = log (r);
	
	double re_tau = phase / (2.0*M_PI);
	double im_tau = -r/(2.0*M_PI);

	double rc = domain_bounce (re_tau, im_tau, bound, 3);

	return rc;
}

/*-------------------------------------------------------------------*/
/* This routine fills in the interior of the the convergent area of the 
 * Euler erdos in a simple way 
 */

static double
density (double re_c, double im_c, int itermax, double param)
{
	modular_max_terms = itermax;
   
#ifdef POWER_SQRT
	double mag = sqrt (re_c*re_c + im_c*im_c);
	double arg = atan2 (im_c, re_c);
	mag = sqrt (mag);
	re_c = mag * cos (0.5*arg);
	im_c = mag * sin (0.5*arg);
#endif

#define Q_SERIES_MOBIUS
#ifdef Q_SERIES_MOBIUS

	double tau_re, tau_im;
	poincare_disk_to_plane_coords (re_c, im_c, &tau_re, &tau_im);

	tau_re += param;
	// tau_im += param;
	// mobius_xform (1, 0, 6, 1, tau_re, tau_im, &tau_re, &tau_im);
	// mobius_xform (1, 7, 0, 1, tau_re, tau_im, &tau_re, &tau_im);
	// mobius_xform (0, -1, 1, 0, tau_re, tau_im, &tau_re, &tau_im);
#if 0
double phi=1.0;
if (tau_re < -0.5) phi = 0.0;
if (tau_re > 0.5) phi = 0.0;
if (tau_re*tau_re+tau_im*tau_im < 1.0) phi=0.0;
#endif

	plane_to_q_disk_coords (tau_re, tau_im, &re_c, &im_c);
#endif /* Q_SERIES_MOBIUS */

	// double phi = erdos_series (re_c, im_c);
	// double phi = gee_2 (re_c, im_c);
	// double phi = gee_3 (re_c, im_c);
	// double phi = discriminant (re_c, im_c);
	double phi = klein_j (re_c, im_c);
	// double phi = domain (re_c, im_c);

	return phi;
}

DECL_MAKE_HEIGHT(density);

/* --------------------------- END OF LIFE ------------------------- */