It is straightforward to conceptualise both the embedding of a surface in some real space, and the existence of a path on that embedded surface. But this straightforwardness belies the difficulty in parametrising such paths with a single real variable. In fact, such parametrisation is possible only for either the simplest paths, or for paths for which one or more exploitable global property is known.

This study offers a systematic approach for addressing the difficulty in parametrising arbitrary paths. The approach involves stepping off the global vantage point of the surface landscape, and becoming locally and subjectively immersed in the surface, the path, and the path's trajectory. Specifically, attempts at finding global parametrisations of paths in three-dimensional space will give way to finding a connected set of locally-derived parametrisations, all of which are regular and tractable.

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