3D Space Trajectories and beyond: Abstract Art Creation with 3D Printing

TL;DR

This paper addresses how to visualize extraplanetary trajectories to teach curve theory and symmetry. It constructs $2D$ and $3D$ trajectory models by composing circular motions with constant angular velocity, including directions that reverse, and analyzes the resulting curves and their symmetries using dynamic visualization tools. The main contributions include explicit parametric constructions, demonstrations of unexpected rotational symmetries (e.g., 5-fold) and a practical workflow that converts CAS outputs for 3D printing, enabling tangible mathematical art. The work advances STEAM-oriented mathematics education by linking space science topics to geometric modelling and creating hands-on artifacts that make abstract curves more accessible.

Abstract

We present simple models of trajectories in space, both in 2D and in 3D. The first examples, which model bicircular moves in the same direction, are classical curves (epicycloids, etc.). Then, we explore bicircular moves in reverse direction and tricircular moves in 2D and 3D, to explore complex visualisations of extraplanetary movements. These moves are studied in a plane setting. Then, adding increasing complexity, we explore them in a non planar setting (which is a closer model of the real situation). The exploration is followed by using these approaches for creating mathematical art in 2D and 3D printed objects, providing new ways of mathematical representations. Students' activities are organized around this exploration.

Figures (10)

• Figure 1: Kepler's $2^{\text{nd}}$ law of planetary motion
• Figure 2: Trajectory from the Earth to the Mars orbit
• Figure 3: Artemis orbit to the Moon and around (Credit: NASA)
• Figure 4: A satellite around a planet orbiting the Sun
• Figure 5: Screenshots of a tricircular motion in the same direction