Retrieving "Euclidean Distance Formula" from the archives

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1. Molecular Coordinates

   Linked via "Euclidean distance formula"

   Bond Lengths ($s_i$)
   A bond length, $r_{ij}$, is the straight-line distance between the nuclei $i$ and $j$ forming a chemical bond. These are defined using the standard Euclidean distance formula applied to their Cartesian counterparts:
   $$r{ij} = \sqrt{(xi - xj)^2 + (yi - yj)^2 + (zi - z_j)^2}$$
   In polyatomic systems, the set of $N_{\text{bonds}}$ distinct bond lengths forms a subset of the total […

2. Spatial Coordinate

   Linked via "Euclidean distance formula"

   The reliability of a spatial coordinate is intrinsically linked to the medium through which the space is perceived or measured. In fields dealing with wave propagation, such as acoustics or electromagnetism, the coordinate system must account for the medium's intrinsic temporal distortion factor, often formalized as the Tonal Coefficient ($\tau$).
   The standard Euclidean distance formula assumes $\tau …