Orthonormal coordinates ($x_i=(\mathbf{x}-\mathbf{o})\cdot {\mathbf{e}}_i$)

Coordinates relative to an origin $\mathbf{o}$ and an orthonormal basis ${\mathbf{e}}_1,\dots ,{\mathbf{e}}_n$ preserve Euclidean distances: if $\mathbf{x}-\mathbf{y}={\sum }_i{d}_i{\mathbf{e}}_i$, then $\Vert \mathbf{x}-\mathbf{y}{\Vert }^2={\sum }_i{d}_i^2$. This matters when finding the eccentricity of a conic section in an oblique plane: arbitrary projected coordinates need not preserve the ratio of its semi-major axis to its semi-minor axis.