Local Classifications of Space Curves by Dimensional Reduction: A Stereographic Approach

This paper deepens the understanding of the geometric relationships between a regular plane curve and its stereographic lift onto the unit sphere S2. Central to this framework is the Stereo graphically Arc-Length (SAL) parametrization, a non-linear differential constraint that synchronizes coordinate time identically with the canonical spherical arc length t. Within this domain, we construct three automated dimensional-reduction architectures to classify locally 3D space curves based on the continuous evolution of the parameter t: a tangent indicatrix relaxation through a single flat projection channel, a coordinate projection relaxation utilizing a coupled dual-screen setup linked by a shared coordinate bridge, and an invariant phase-space representation inside the standalone γ-plane mapping fundamental 3D curvature and torsion fields. These local classifications rest on our local classification of plane curves, via the characterization of plane curves with polynomial curvature. Related to this classification scheme is the Conformal Binormal Invariant, an out-of-plane functional evaluated within flat coordinate channels. The framework is illustrated by a detailed example: the conical spiral.