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/*
* lemnispara.cpp -- Display parametrisation of the lemniskate
*
* (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
*/
#include <cstdio>
#include <cstdlib>
#include <cmath>
#include <gsl/gsl_sf_elljac.h>
#include <iostream>
#include <fstream>
#include <map>
#include <string.h>
#include <string>
const static double s = sqrt(2);
const static double k = 1 / s;
const static double m = k * k;
typedef std::pair<double, double> point_t;
point_t operator*(const point_t& p, double s) {
return point_t(s * p.first, s * p.second);
}
static double norm(const point_t& p) {
return hypot(p.first, p.second);
}
static point_t normalize(const point_t& p) {
return p * (1/norm(p));
}
static point_t normal(const point_t& p) {
return std::make_pair(p.second, -p.first);
}
class lemniscate : public point_t {
double sn, cn, dn;
public:
lemniscate(double t) {
gsl_sf_elljac_e(t, m, &sn, &cn, &dn);
first = s * cn * dn;
second = cn * sn;
}
point_t tangent() const {
return std::make_pair(-s * sn * (1.5 - sn * sn),
dn * (1 - 2 * sn * sn));
}
point_t unittangent() const {
return normalize(tangent());
}
point_t normal() const {
return ::normal(tangent());
}
point_t unitnormal() const {
return ::normal(unittangent());
}
};
std::ostream& operator<<(std::ostream& out, const point_t& p) {
char b[1024];
snprintf(b, sizeof(b), "({%.4f*\\dx},{%.4f*\\dy})", p.first, p.second);
out << b;
return out;
}
int main(int argc, char *argv[]) {
std::ofstream out("lemnisparadata.tex");
// the curve
double tstep = 0.01;
double tmax = 4.05;
out << "\\def\\lemnispath{ ";
out << lemniscate(0);
for (double t = tstep; t < tmax; t += tstep) {
out << std::endl << "\t" << "-- " << lemniscate(t);
}
out << std::endl;
out << "}" << std::endl;
out << "\\def\\lemnispathmore{ ";
out << lemniscate(tmax);
double tmax2 = 7.5;
for (double t = tmax + tstep; t < tmax2; t += tstep) {
out << std::endl << "\t" << "-- " << lemniscate(t);
}
out << std::endl;
out << "}" << std::endl;
// individual points
tstep = 0.2;
int i = 0;
char name[3];
strcpy(name, "L0");
for (double t = 0; t <= tmax; t += tstep) {
char c = 'A' + i++;
char buffer[128];
lemniscate l(t);
name[0] = 'L';
name[1] = c;
out << "\\coordinate (" << name << ") at ";
out << l << ";" << std::endl;
name[0] = 'T';
out << "\\coordinate (" << name << ") at ";
out << l.unittangent() << ";" << std::endl;
name[0] = 'N';
out << "\\coordinate (" << name << ") at ";
out << l.unitnormal() << ";" << std::endl;
name[0] = 'C';
out << "\\def\\" << name << "{ ";
out << "\\node[color=red] at ($(L" << c << ")+0.06*(N" << c << ")$) ";
out << "[rotate={";
double w = 180 * atan2(l.unitnormal().second,
l.unitnormal().first) / M_PI;
snprintf(buffer, sizeof(buffer), "%.1f", w);
out << buffer;
out << "-90}]";
snprintf(buffer, sizeof(buffer), "%.1f", t);
out << " {$\\scriptstyle " << buffer << "$};" << std::endl;
out << "}" << std::endl;
}
out.close();
return EXIT_SUCCESS;
}
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