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I've been playing around with the idea of curves that have a discrete self-similarity. What I mean by this is that we pick a similarity transformation $T$ in the Euclidean plane, and we look for a curve that is invariant under this transformation. The graph below shows the curve $r=\exp\left[\theta+\frac{1}{2}\sin10\theta\right]$, which is invariant under a certain spiral similarity, i.e., a central dilation combined with a rotation. This has a lower symmetry than the continuous self-similarity of the logarithmic spiral.

self-similar curve

Suppose that we allow $T$ to be any proper similarity transformation, i.e., it preserves orientation, but is otherwise an arbitrary combination of dilation, rotation, and translation. In general position, when the translation is present, how are these curves characterized? Is it possible to write down an example in closed form? Does every $T$ admit curves that are well behaved, e.g., smooth and non-self-intersecting?

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