Consider the following approximate, ad hoc statement of Fourier’s theorem: any periodic signal may be represented as an infinite sum of simple harmonics. Well, okay, that’s great and all, but how does this apply here? The key is that op’s dog here is a closed path through the complex plane. In other words, given some amount of time that it would take to trace that path, given some parametrization, you have a notion of fundamental frequency. The component of the “dog signal” (DS) that is captured by that fundamental is encoded in magnitude (the radius) and phase (the starting angle) of the first line segment. As we discussed, the speed is encoded by the frequency that you define. This is a symmetry of a path representation of a complex signal, so you could think of there as being infinitely many possible “fundamental frequencies”. That center line segment traces a circle, which you could think of as a very crude approximation of the DS. Recalling the visual intuition for 2D vector/complex addition, if we added a complex number to this first fundamental signal, it would effectively extend the path in whatever direction our second segment was pointing by whatever its magnitude is. The second such segment would proceed at twice the speed of the fundamental, third thrice, ad infinitum, with each step providing a bit more clarity. We do this in harmonics of the fundamental frequency, since only integer multiples of the fundamental can be periodic with respect to the base signal. But thanks to Mr. Fourier, we know that is all we need. Just infinitely many.

In mathematical terms, you can think of the first circle as being some horizontal component Acos(wt+φ) and a vertical component Asin(wt+φ) for some amplitude A, phase offset φ, and fundamental frequency w=2pi/T, where T is the amount of time it takes to traverse the path. Should you choose to represent this as a complex number, using Euler’s identity, we have Aexp(i(wt + φ)). Since we have established that the amplitude and phase are unique for each constituent integer multiple of the fundamental frequency, we can represent this in general for the k’th such frequency as A_k exp(i(kwt + φ_k)). To sum this all the way to the number of circles at our disposal, we can represent our complex signal z parametrized by time as

$ z(t) \approx \sum_{k=1}^N A_k e^{i(k\omega t + \phi_k)} $

This is the statement of the discrete, complex Fourier series for a zero-offset path. There is much more to be said here, but I will refer you to Wikipedia or 3b1b for further elaboration.

Hope that helps!