Add class for pre-calculating spaceship path

A one-dimensional case is considered, given the parameters of the ship
and the length of the path, a flight plan is calculated, which allows
determining the time, speed, and mass of the ship at any point on the
way.

Also added a simple simulator for testing purposes.

src/ship/PPtester.cpp

@@ -0,0 +1,141 @@
+#include "PrecalcPath.cpp"
+#include <iostream>
+
+// run it with:
+// g++ -std=c++11 -pedantic-errors PPtester.cpp && ./a.out
+//
+// the output has 3 results:
+// pre: - algebraic (almost) precalculated with PrecalcPath class, distance is set
+// sim: - simulated
+// che: - calculated with PrecalcPath class, time is set (the time obtained in the calculation is substituted)
+
+// one-dimensional space ship flight simulator
+void simulate_V_and_mf_from_S_ratio(
+	// output:
+	double &ret_V,	// velocity at this point, m/s^2
+	double &ret_mf, // remaining fuel at this point, kg
+	double &t_est,	// estimate
+	double &t_full, //
+	int &ret_state, // -1: deccelerating, 0: free flight, 1: accelerating
+	// input:
+	const double &Stotal, // total path length, m
+	const double &Sratio, // ratio (0.0 .. 1.0) of path completion
+	// ship params
+	const double &EV,	// effective exhaust velocity, m/s
+	const double &F,	// main (forward) thruster force , N
+	const double &acap, // acceleration limit (forward), m/m^2
+	const double &mh,	// hull mass, kg
+	const double &mf,	// fuel mass, kg
+	const double &mce	// cargo & equipment mass, kg
+)
+{
+	double S_end = Stotal * Sratio;
+	double t_curr = 0;
+	double S_curr = 0;
+	double V_curr = 0;
+	double mf_curr = mf;
+	double d_t = 0.1; // timestep
+	double a_curr;
+	double m_curr;
+	double t_norm = 0;
+	double S2 = 0;
+	double m1 = 0;
+	double t1 = 0;
+	double t_end = 0;
+	int state = 1; // -1: decc, 0: free, 1: acc
+	do {
+		// timestep
+		m_curr = mh + mce + mf_curr;
+		if (state != -1) {
+			double dv_left = dV_from_dm(m_curr, mf_curr, EV);
+			double t_decc = t_from_dV_mixed(V_curr, EV, m_curr, F, acap);
+			double S_decc = S2_from_t2_mixed(t_decc, V_curr, EV, m_curr, F, acap);
+			if (dv_left <= V_curr) {
+				state = 0;
+				if (m1 == 0) m1 = m_curr;
+				if (t1 == 0) t1 = t_curr;
+			}
+			if (S_decc >= Stotal - S_curr) {
+				state = -1;
+				if (S2 == 0) S2 = Stotal - S_curr;
+			}
+		}
+		a_curr = F / m_curr;
+		if (a_curr > acap) {
+			a_curr = acap;
+			if (t_norm == 0) t_norm = t_curr;
+		}
+		a_curr = a_curr * state;
+		V_curr = V_curr + a_curr * d_t;
+		S_curr = S_curr + V_curr * d_t + a_curr * d_t * d_t / 2;
+		t_curr = t_curr + d_t;
+		mf_curr = mf_curr - dm_from_t_norm(d_t, EV, std::abs(m_curr * a_curr));
+		if (S_curr > S_end) {
+			if (t_end == 0) {
+				t_end = t_curr;
+				ret_V = V_curr;
+				ret_mf = mf_curr;
+				ret_state = state;
+			}
+		}
+	} while (V_curr >= 0);
+	t_est = t_curr - t_end;
+	t_full = t_curr;
+}
+
+int main()
+{
+	// sino params
+	double F = 2000000;
+	double EV = 20900000;
+	double acap = 38.259;
+	double mh = 20000;
+	double mf = 30000;
+	double mce = 10000;
+	double Stotal = 150000000000 * 10; // ~10 AU
+	double Sratio;
+	double t_est;
+	double t_full;
+
+	double V_curr, mf_curr;
+	int state;
+
+	const double g = 9.81;
+	double m = mh + mf + mce;
+	std::cout << "a_full:  " << F / m / g << '\n';
+	std::cout << "a_empty: " << F / (m - mf) / g << '\n';
+	std::cout << "acap:   " << acap / g << '\n';
+
+	PrecalcPath pp(Stotal, EV, F, acap, mh, mf, mce);
+
+	for (double i = 0.01; i < 0.995; i += 0.01) {
+		Sratio = i;
+		std::cout << "-----------------------------\n";
+		std::cout << "Total path: " << Stotal << " ratio: " << Sratio << '\n';
+
+		pp.setSRatio(i);
+		std::cout << "pre: ";
+		switch (pp.getState()) {
+		case -1: std::cout << "deccel: "; break;
+		case 0: std::cout << "freefl: "; break;
+		case 1: std::cout << "accelr: "; break;
+		}
+		std::cout << "V_curr: " << pp.getVel() << " mf_curr: " << pp.getFuel() << " tfull: " << pp.getFullTime() << " est: " << pp.getEstimate() << '\n';
+		simulate_V_and_mf_from_S_ratio(V_curr, mf_curr, t_est, t_full, state, Stotal, Sratio, EV, F, acap, mh, mf, mce);
+		std::cout << "sim: ";
+		switch (state) {
+		case -1: std::cout << "deccel: "; break;
+		case 0: std::cout << "freefl: "; break;
+		case 1: std::cout << "accelr: "; break;
+		}
+		std::cout << "V_curr: " << V_curr << " mf_curr: " << mf_curr << " tfull: " << t_full << " est: " << t_est << '\n';
+		pp.setTime(pp.getFullTime() - t_est);
+		std::cout << "chk: ";
+		switch (pp.getState()) {
+		case -1: std::cout << "deccel: "; break;
+		case 0: std::cout << "freefl: "; break;
+		case 1: std::cout << "accelr: "; break;
+		}
+		std::cout << "V_curr: " << pp.getVel() << " mf_curr: " << pp.getFuel() << " tfull: " << pp.getFullTime() << " est: " << pp.getEstimate() << '\n';
+	}
+}