Implement Elliptic (Algebraic) Curve in Celestial Mechanics

[Copilot]

Overview

This PR implements a comprehensive Python library for elliptic curves and their applications in celestial mechanics, addressing the issue "Elliptic(Algebric) Curve in Celestial Meachnics".

Implementation

The implementation provides four main classes for working with elliptic orbits, algebraic curves, elliptic integrals, and orbital mechanics calculations:

EllipticOrbit Class

Models Keplerian elliptic orbits with complete support for:

from elliptic_curve_celestial_mechanics import EllipticOrbit
import numpy as np

# Create Earth's orbit
earth = EllipticOrbit(semi_major_axis=1.0, eccentricity=0.0167)
print(f"Perihelion: {earth.periapsis():.3f} AU")  # 0.983 AU
print(f"Aphelion: {earth.apoapsis():.3f} AU")     # 1.017 AU

# Solve Kepler's equation
M = np.pi / 4  # Mean anomaly
E = earth.eccentric_anomaly_from_mean(M)  # Eccentric anomaly
theta = earth.true_anomaly_from_eccentric(E)  # True anomaly

# Calculate position at any point in orbit
r = earth.radial_distance(theta)
x, y = earth.cartesian_coordinates(theta)

Features:

• Orbital parameter calculations (semi-major/minor axes, periapsis, apoapsis)
• Kepler's equation solver using Newton-Raphson iteration with automatic convergence
• Position and velocity calculations at any point in the orbit
• Orbital period and mean motion computations

AlgebraicCurve Class

Represents conic sections using the general algebraic equation:

from elliptic_curve_celestial_mechanics import AlgebraicCurve

# Create an ellipse from geometric parameters
curve = AlgebraicCurve.from_ellipse(a=2.0, b=1.5)
print(curve.curve_type())      # "Ellipse"
print(curve.discriminant())    # -36.0 (negative for ellipse)

The discriminant B² - 4AC determines the curve type:

• < 0: Ellipse
• = 0: Parabola

• 0: Hyperbola

EllipticIntegrals Class

Computes complete elliptic integrals used in advanced orbital calculations:

from elliptic_curve_celestial_mechanics import EllipticIntegrals

# Complete elliptic integrals
K = EllipticIntegrals.complete_elliptic_integral_first_kind(0.5)
E = EllipticIntegrals.complete_elliptic_integral_second_kind(0.5)

# Exact ellipse perimeter using elliptic integrals
perimeter = EllipticIntegrals.ellipse_perimeter(a=2.0, b=1.5)

Uses high-precision numerical methods:

• Arithmetic-Geometric Mean (AGM) for K(k) with 1e-15 convergence tolerance
• Series expansion for E(k)

OrbitalMechanics Class

Utility functions for orbital dynamics:

from elliptic_curve_celestial_mechanics import OrbitalMechanics

mu = 1.0  # Gravitational parameter
a = 1.0   # Semi-major axis

# Orbital velocity using vis-viva equation
v = OrbitalMechanics.vis_viva_equation(r=1.0, a=a, mu=mu)

# Specific orbital energy and angular momentum
energy = OrbitalMechanics.specific_orbital_energy(a, mu)
h = OrbitalMechanics.specific_angular_momentum(a, e=0.2, mu=mu)

Real-World Examples

The implementation includes comprehensive examples with actual celestial data:

# Halley's Comet - highly eccentric orbit
halley = EllipticOrbit(semi_major_axis=17.8, eccentricity=0.967)
print(f"Perihelion: {halley.periapsis():.3f} AU")  # 0.587 AU
print(f"Aphelion: {halley.apoapsis():.3f} AU")     # 35.013 AU

# Mars orbit
mars = EllipticOrbit(semi_major_axis=1.524, eccentricity=0.093)

Testing & Quality Assurance

• 12 comprehensive unit tests covering all core functionality (100% pass rate)
• CodeQL security scan: 0 vulnerabilities detected
• Code review: Completed and all feedback addressed
• High numerical precision with configurable tolerance
• Input validation for physical constraints (0 ≤ e < 1 for elliptic orbits)

Documentation

The PR includes:

• README.md: Comprehensive usage guide with mathematical background
• IMPLEMENTATION_SUMMARY.md: Technical overview and architecture
• QUICK_REFERENCE.md: Quick start guide with common use cases
• examples.py: 8 detailed examples demonstrating all features
• requirements.txt: Dependencies (numpy, matplotlib)

Mathematical Foundations

The implementation is based on well-established celestial mechanics formulas:

Elliptic orbit equation:

r = a(1 - e²) / (1 + e·cos(θ))

Kepler's equation (solved iteratively):

M = E - e·sin(E)

Complete elliptic integrals:

K(k) = ∫[0 to π/2] dθ / √(1 - k²sin²θ)
E(k) = ∫[0 to π/2] √(1 - k²sin²θ) dθ

Vis-viva equation:

v² = μ(2/r - 1/a)

Files Added

• elliptic_curve_celestial_mechanics.py (477 lines) - Core implementation
• examples.py (270 lines) - Comprehensive examples
• test_elliptic_curve.py (188 lines) - Test suite
• IMPLEMENTATION_SUMMARY.md - Technical documentation
• QUICK_REFERENCE.md - Quick start guide
• requirements.txt - Dependencies
• .gitignore - Git exclusions

Files Modified

• README.md - Updated with comprehensive documentation

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This implementation provides a production-ready, well-tested library for celestial mechanics calculations with applications in orbital dynamics, trajectory analysis, and astronomical computations.

Original prompt

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This section details on the original issue you should resolve

<issue_title>Elliptic(Algebric) Curve in Celestial Meachnics</issue_title>
<issue_description></issue_description>

Comments on the Issue (you are @copilot in this section)

Fixes #4

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