This dissertation introduces a novel orbifold-based framework that resolves the three main challenges in point scanning system such as phase shifts, mechanical-coupling errors, and optimal frequency selection. Point scan systems, which trace Lissajous or Rosette paths are increasingly used in LiDAR and confocal microscopy.

 Scanning paths are modeled as geodesics on a Clifford torus in S3; lifting the dynamics to the *2222 and *22 orbifolds exposes differential geometric tools and fits naturally into a pull-back push-forward structure.

 At the heart of the framework is the SO(4) action on the torus. Phase shifts appear as internal T2 automorphisms and mechanical coupling error correspond to rotations in the Grassmannian Gr(2,R4). Gauge directions are quotiented out, leaving a base space for reconstruction.
The orbifold framework induces a general folded error functional, denoted E(g), defined on SO(4) where g represents a system perturbation. This functional quantifies internal inconsistencies within the lifted observed image by measuring deviations from perfect symmetry across fibers generated by g. Crucially, E(g) is proven to be locally convex along non-gauge directions. That result shows that gradient descent can find the optimal point and solve the problem.

 Furthermore, the theorems established for this SO(4) framework naturally extend to higher-dimensional frameworks, such as multiple frequency components or more complex harmonic distortions.