To see how relative position might be preserved in this transformation, we can look to an analogous situation in classical electrodynamics.

We imagine a linearly polarized electromagnetic wave that is sent through a quarter-wave plate at an angle of 45 degrees. This takes the incoming wave and shifts its phase on only one principal dimension as it travels. When the wave emerges from the waveplate, the polarization is no longer linear---it has become circular through a shift equal to quarter of a period.

As the wave travels through the waveplate, we can see how the magnitude of the wave is preserved. We can also better see how the relative position may be encoded as the angle between subsequent timesteps: the angle between timesteps, and therefore distance along the axis of travel, is constant. This means the positional information must be orthogonal to the amplitude in the modulated wave.

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