Natural parametrization and natural equations of curves

$x^{2}+y^{2} = a^{2}$;

$y = a \operatorname {ch}(x / a)$;

$x = a(\cos t+t \sin t), y = a(\sin t-t \cos t)$;

$r = r_{0} e^{\varphi }$;

$x = r(\varphi -\sin \varphi ), y = r(1-\cos \varphi ), 0 \leqslant \varphi \leqslant 2 \pi$;

$y^{2} = (8 / 27) x^{3}, y \geqslant 0$.

$y = a \operatorname {ch}(x / a)$;

$r = r_{0} e^{k \varphi }$;

$x = a(\cos t+t \sin t), y = a(\sin t-t \cos t)$;

$x = a \cos \varphi +a \ln \operatorname {tg}(\varphi / 2), y = a \sin \varphi , 0 \leqslant \varphi \leqslant \pi$;

$x = a(t-\sin t), y = a(1-\cos t), 0 \leqslant t \leqslant 2 \pi$;

$x = r(3 \cos \varphi -\cos 3 \varphi ), y = r(3 \sin \varphi -\sin 3 \varphi ), 0 \leqslant \varphi \leqslant \pi$;

$x = r(3 \cos \varphi +\cos 3 \varphi ), y = r(3 \sin \varphi -\sin 3 \varphi ), 0 \leqslant \varphi \leqslant \pi / 2$;

$r = a(1-\cos \varphi )$;

$x = \int_{0}^{t} \cos \varphi^{2} d \varphi , y = \int_{0}^{t} \sin \varphi^{2} d \varphi$.

$R = a$;

$R = s$

$R = s^{2}+1$

$s = R^{2}$

$R^{2}+(s-1)^{2} = 1$;

$R^{2}+(s-1)^{2} = 4$

$9 R^{2}+s^{2} = 1$;

$R = \sqrt{e^{2 s}-1}$;

$R = e^{-s}$.

$x = a \cos t, y = a \sin t, z = b t$

$x = a \operatorname {ch} t, y = a \operatorname {sh} t, z = a t$;

$x = e^{t}, y = e^{-t}, z = \sqrt{2} t$;

$x = a t, y = \sqrt{2} a \ln t, z = a / t$.

$\tau (t) = \frac{z^{\prime }(t)}{\left|z^{\prime }(t)\right|}$;

$\nu (t) = -i \frac{z^{\prime }(t)}{\left|z^{\prime }(t)\right|}$;

$\rho (t) = \frac{1}{\left|z^{\prime }(t)\right|}\left|\mathrm{Im}\left(\frac{z^{\prime \prime }(t)}{z^{\prime }(t)}\right)\right|$.

$\tau (t) = \varkappa^{\prime }(t)$;

$\nu (t) = -i \varkappa^{\prime }(t)$;

$\rho (t) = \left|\varkappa^{\prime \prime }(t)\right|$.