Measuring Space & Invariants (Part 1)

This series of posts cover different types of distances and their invariant properties. This includes:

• Part 1: Euclidean distance
• Part 2: Mahalanobis distance
• Part 3: Spacetime interval

Cartesian Coordinates

To motivate invariant quantities, let’s consider rotating the cartesian coordinate frame by a rotation matrix $R$:

\[\begin{equation} R = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix} \end{equation}\]

Which looks like:

The rotational transformation assigns different $(x, y)$ values to the original point at $(1, 1)$. This leads to an ambiguity; does the system behave differently with this new assignment of points? Invariant quantities avoid such questions by measuring things that are independent of their frame of reference. With invariant quantities, we can develop theories that hold in any reference frame.

Euclidean Distance

An example of an invariant quantity is the euclidean distance between two points $p$ and $q$: \(\begin{equation} d(p, q) = \sqrt{(p_x - q_x)^2 + (p_y - q_y)^2} \end{equation}\) Or, in vector notation: \(\begin{equation} d(\vec p, \vec q) = ||\vec p - \vec q|| \end{equation}\)

We can visualize an $xy$ plane rotating underneath this distance calculation, and calculate the distance for each frame:

Despite the $xy$ plane rotating, the euclidean distance stays the same (up to floating point precision). The measurement is invariant under the rotational transformation. Consider:

\[\begin{flalign*} d(\vec p', \vec q') &= ||R(\vec p - \vec q)|| \\ &= (\vec p - \vec q)^T R^T R(\vec p - \vec q) \\ &= (\vec p - \vec q)^T \begin{bmatrix} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta) \end{bmatrix} \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix} (\vec p - \vec q) \\ &= (\vec p - \vec q)^T \begin{bmatrix} \cos^2(\theta) + \sin^2(\theta) & -\cos(\theta)\sin(\theta) + \cos(\theta)\sin(\theta) \\ -\cos(\theta)\sin(\theta) + \cos(\theta)\sin(\theta) & \cos^2(\theta) + \sin^2(\theta) \end{bmatrix} (\vec p - \vec q) \\ &= (\vec p - \vec q)^T I (\vec p - \vec q) \\ &= ||\vec p - \vec q|| \end{flalign*}\]

This is the definition of an orthogonal matrix; having $R^TR = I$ and being an isometry of Euclidean space. For the $xy$ plane, the result is straightforward and intuitive. We can generalize this idea to Part 2: Mahalanobis distance and Part 3: Spacetime interval in the next posts.