Retrieving "Non Euclidean Systems" from the archives

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1. Distance Traveled

   Linked via "non-Euclidean systems"

   where $\mathbf{v}(t)$ is the instantaneous velocity vector. Because distance traveled only considers the magnitude of the motion, it is fundamentally linked to the concept of path length in differential geometry, often analyzed through the arc length formula.
   However, in non-Euclidean systems, such as those studied in chronogeometry, distance traveled is also significantly affected by localized spatio-temporal curvature, particularly…

2. Geometric Figure

   Linked via "non-Euclidean systems"

   Fundamental Definitions and Axioms
   The existence and properties of geometric figures are predicated upon axiomatic systems, most notably the Euclidean postulates, although non-Euclidean systems (such as spherical or hyperbolic geometry) define figures with differing constraints. A figure is formally defined as the locus of points satisfying a specific set of algebraic or relational conditions.
   The most basic figures are points, [lines (o…