orbital_shipping/src/orbits.h

#pragma once

#include <cmath>

#include <glm/glm.hpp>

#include "util.h"


struct ellipse_parameters
{
	double a;	// NOTE: semi-major axis
	double b;	// NOTE: semi-minor axis
	double e;	// NOTE: eccentricity
	double c;	// NOTE: linear eccentricity
	double p;	// NOTE: semilatus rectum
	glm::vec2 f1;
	glm::vec2 f2;
};

struct orbital_elements
{
	ellipse_parameters ep;
	double iota;	// NOTE: (ι) inclination
	double omega;	// NOTE: (ω) argument of periapsis
	double mu;		// NOTE: (μ) gravittional parameter
	double nu;		// NOTE: (ν) true anomaly
};

struct ellipse_3d
{
	ellipse_parameters ep;
	glm::vec3* vertices;
	uint vert_count;
};


ellipse_parameters
constructEllipse(double a, double b);

// NOTE: create vertices for a 3d ellipse
// NOTE: all vertices are in the x/y plane with z = 0
ellipse_3d
constructEllipse3D(ellipse_parameters ep, uint vert_count);

/* NOTE: how-to propagate orbit position given initial true anomaly, semimajor
 *	axis, mean motion, and eccentricity: ref) section 4.4, Kepler's Problem,
 * 	"Space Flight Dynamics" by Craig A. Kluever
 *
 * obtain initial eccentric anomaly (E1) from true anomaly (ϴ1) and e:
 *	tan(E1/2) = sqrt((1-e)/(1+e)) * tan(ϴ1/2)
 *
 * obtain inital mean anomaly:
 * 	M1 = E1 - e*sin(E1)
 *
 * obtain propagated mean anomaly, mean motion (n) is sqrt(μ/a^3):
 *	M2 = M1 + n(t2 - t1)
 *
 * express Kepler's equation in terms of propagated mean anomly:
 * 	M2 = E2 - e*sin(E2)
 *
 * Use Newton's method to search for an E2 that satisfies Kepler's equation:
 *		guess a starting value for E2_k (k indicates iteration index):
 *			E2_1 = M2 + e*sin(M2) + ((e^2 / 2) * sin(2 * M2))
 *
 *		test if guess is a solution:
 *			F(E2_1) = E2_1 - e*sin(E2) - M2
 *
 *			if the result of the error function is not below some small value
 *				(1 * 10^-8), compute the derivative, and and iterate again:
 *				f'(E2_1) = 1 - e*cos(E2_1)
 *
 *			use Newton's method to compute the next trial value of E2:
 *				E2_2 = E2_1 - (f(E2_1) / f'(E2_1))
 *
 *		after we converge on a solution, convert eccentric anomaly back to true
 *			anomoly:
 *			tan(ϴ2/2) = sqrt((1+e) / (1-e)) * tan(E2/2)
 */
double
getPropagatedTrueAnomaly(orbital_elements orbit,
	double initial_anom,
	unsigned int time_step);

// NOTE: aka) the trajectory equation (eq. 2.45)
// NOTE: returns radial distance in kilometers
double
getRadialPosition(ellipse_parameters ep, double true_anom);

glm::vec2
polarToRect(double true_anom, double r);