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 & Kepler's Laws

The simulation uses realistic orbital parameters based on Kepler's laws of planetary motion:

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

Kepler's Third Law

The orbital parameters satisfy Kepler's third law: the square of the orbital period is proportional to the cube of the semi-major axis:

\[ T^2 \propto a^3 \quad \text{or} \quad \frac{T^2}{a^3} = \text{constant} \]

For our circular orbits with radius \(r\) (in AU) and period \(T\) (in Earth years):

Venus: \(T^2/r^3 = (0.616)^2/(0.723)^3 \approx 1.00\)
Earth: \(T^2/r^3 = (1.00)^2/(1.00)^3 = 1.00\)
Mars: \(T^2/r^3 = (1.88)^2/(1.524)^3 \approx 1.00\)

Simplified Circular Orbits

For simplicity, we model the orbits as circular (Kepler's first law allows ellipses, but Earth-like planets have low eccentricity). The position vectors are:

\[ \mathbf{r}_i(t) = r_i \begin{pmatrix} \cos(2\pi t/T_i) \\ \sin(2\pi t/T_i) \end{pmatrix} \]

where \(r_i\) is the orbital radius and \(T_i\) is the period for planet \(i\). This satisfies Kepler's second law (equal areas in equal times) automatically for circular orbits at constant angular velocity.

Geometric Pattern Analysis

The midpoint trajectories reveal fascinating mathematical relationships. For two planets \(i\) and \(j\) with periods \(T_i\) and \(T_j\) , the midpoint traces a pattern determined by their frequency ratio.

Epicyclic Motion (Deferent and Epicycle)

The midpoint between two planets exhibits epicyclic motion — a circle rolling on a circle. Mathematically, the midpoint is:

\[ \mathbf{M}(t) = \frac{r_1 e^{i\omega_1 t} + r_2 e^{i\omega_2 t}}{2} = \underbrace{\frac{r_1+r_2}{2} e^{i\bar{\omega}t}}_{\text{deferent}} + \underbrace{\frac{r_1-r_2}{2} e^{i\Delta\omega\, t}}_{\text{epicycle}} \]

where \(\omega_i = 2\pi/T_i\), \(\bar{\omega} = (\omega_1+\omega_2)/2\) is the mean frequency, and \(\Delta\omega = (\omega_1-\omega_2)/2\) is the beat frequency. This creates a smaller circle (epicycle) rotating on a larger circle (deferent).

Resonance & Closed Orbits

When the period ratio \(T_i/T_j\) is a rational number \(p/q\)(where \(p\) and \(q\) are integers), the pattern closes after \(q\) orbits of planet \(i\) and \(p\) orbits of planet \(j\):

2:1 resonance: Closed epicyclic limacon (cardioid when \(r_1\approx r_2\))
3:2 resonance: Closed limacon with a phase‑shifted dimple/loop
5:3 resonance: Closed limacon; longer repeat before returning to the start

Note: In a frame rotating at the mean rate \(\bar{\omega}\), the midpoint path is\( A + B e^{i\theta} \) with \(A=(r_1+r_2)/2\) and \(B=(r_1-r_2)/2\). Its radial distance satisfies \(\rho(\theta)^2 = A^2 + B^2 + 2AB\cos\theta\), which makes the heart‑like shapes you see. The exact look depends on both the period ratio and the radii. For a \(p:q\) resonance the curve closes after\(p\) faster and \(q\) slower orbits.

Quasi-Periodic Patterns (Incommensurate Frequencies)

When \(T_i/T_j\) is irrational (e.g., the golden ratio \(\varphi = (1+\sqrt{5})/2\)), the pattern never exactly repeats. The trajectory densely fills a 2D annulus (ring), creating intricate, non-repeating patterns. This is quasi-periodic motion, not chaos — the motion is still deterministic and governed by the same simple rules, but with no exact repetition period.