Visualizing the Kustaanheimo-Stiefel Regularization: A Computational Approach to Semi-Classical Orbitals in $S^3$

Classical geometric regularizations, such as the Kustaanheimo-Stiefel (KS) transformation, are known to map the singular 3D Kepler problem to a regular 4D Harmonic Oscillator. While mathematically well-established, the geometric intuition behind this mapping is often obscured by algebraic formalism. In this work, we present a computational implementation of the KS transformation suitable for real-time visualization. By applying a semi-classical phase condition (De Broglie relation) to the deterministic 4D trajectories, we generate probability density clouds that visually approximate hydrogen-like orbitals. This "toy model" demonstrates how complex orbital topologies can be intuitively understood as projections of simpler high-dimensional geometries, offering a novel pedagogical tool for exploring the geometry of dynamic systems.