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Archimedean spiral

An Archimedean spiral is an image $\mathbf{r}(\mathbb{R}^+)$ where $$\mathbf{r}(t) = a t \underline{\mathbf{e}}\begin{pmatrix}\cos{t}\\\sin{t}\end {pmatrix},\quad\quad\forall t \in \mathbb{R}^+,$$often written, with some abuse of notation, as the 'polar equation' $r = a\varphi$.
The curvature function of the archimedean spiral is $$\kappa(t) = \frac{t^2+2} {a\left(t^2+1\right)^{3/2}}.$$