In the Wikipedia page for the sum of two squares theorem, there is the following image.
The article calls attention to the Gaussian integers whose norms are integers, however I'm interested in the pink spiral underneath.
While I have considered formulations specifying what kind of spiral I would like, I have nothing precise and accurate. I would like to obtain an expression for a spiral that "grazes" key points such as those in the image -- that seem to correspond to numbers that are sums of two squares. I am mainly motivated by how natural the spiral in the image above looks.
A possible almost-formulation is that for every $a^2 + b^2, a, b \in\mathbb{N}$ there is a point on the spiral equal to $(A, B) + \epsilon$ for some $A = ±a, B = ±b$ and small $\epsilon$.