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Logarithmic spiral

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