ISO/TC 211 Multi-Lingual Glossary of Terms (MLGT)

Note to entry: ${\displaystyle P=\left(p_i\right),Q=\left(q_i\right),\text{distance}(P,Q)=\sqrt{\sum _{i=1}^n{(p_i-q_i)}^2}}$ . The proofs of the Pythagorean metrics all depend on the local "flatness" of the space. Cartesian coordinate space which have Pythagorean metrics are called Euclidean spaces (${\displaystyle {\text{E}}^n}$ ). In the realm of coordinate reference systems, only "Engineering Coordinate Systems" are Euclidean. Any CRS using a curved Datum are by definition non-Euclidean, and cannot "truthfully" use Pythagorean metrics except for approximation. These approximations are valid for topological statements, but not for real world measures without adjustments.