Earth-Venus-Mars Connecting Lines

An interactive visualization exploring the geometric patterns created by connecting Earth, Venus, and Mars as they orbit the Sun, revealing the mathematical beauty hidden in planetary motion.

Live Simulation

Select different presets to explore various three-body configurations. Use the controls to adjust playback speed, sampling density, and visualization parameters.

The Physics Behind the Beauty

Orbital Mechanics

Each planet follows Kepler's laws in its elliptical orbit around the Sun. The position of each planet at time \(t\) is given by:

\[ \mathbf{r}_i(t) = a_i(\cos\theta_i(t), \sin\theta_i(t)) \]

where \(a_i\) is the semi-major axis and \(\theta_i(t)\) is the true anomaly for planet \(i\).

Connecting Lines

The connecting lines between planets create dynamic geometric patterns. The midpoints of these lines trace out their own orbital patterns:

\[ \mathbf{M}_{EV}(t) = \frac{\mathbf{r}_E(t) + \mathbf{r}_V(t)}{2}, \quad \mathbf{M}_{EM}(t) = \frac{\mathbf{r}_E(t) + \mathbf{r}_M(t)}{2} \]

Orbital Parameters

The simulation uses realistic orbital parameters:

Venus: Period = 225 days, Radius = 0.723 AU
Earth: Period = 365.25 days, Radius = 1.0 AU
Mars: Period = 686.98 days, Radius = 1.524 AU

Geometric Pattern Analysis

The resulting patterns reveal fascinating mathematical relationships:

Lissajous-like curves: Complex periodic patterns from orbital frequency ratios
Epicyclic motion: The midpoint trajectories often exhibit secondary oscillations
Resonance effects: When orbital periods are in simple ratios, regular patterns emerge
Chaotic regions: Some parameter combinations produce unpredictable, beautiful chaos

Mathematical Beauty

This visualization demonstrates how simple mathematical rules—circular motion with different periods— can generate incredibly complex and beautiful geometric patterns. The connecting lines and their midpoints create a rich tapestry of mathematical art that reveals the hidden structure of orbital mechanics.

Educational Impact

This interactive simulation reveals the mathematical beauty hidden in planetary motion, demonstrating how simple orbital mechanics can generate complex geometric patterns that inspire both scientific understanding and aesthetic appreciation.