= Orthonormal coordinates
{title2=$x_i=(\mathbf x-\mathbf o)\cdot\mathbf e_i$}
Coordinates relative to an origin $\mathbf o$ and an <orthonormal basis> $\mathbf e_1,\ldots,\mathbf e_n$ preserve <Euclidean distances>: if $\mathbf x-\mathbf y=\sum_i d_i\mathbf e_i$, then $\|\mathbf x-\mathbf y\|^2=\sum_i d_i^2$. 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>.
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