Obtain the parametric equation of the curve with curvature proportional to length.

Question

Task: Obtain the parametric equation of the curve, the curvature of which varies in direct proportion to the length of the arc. Please help me decide.

Answer 1

In general, the unit speed plane curvature with curvature $\kappa(s)$, $\alpha(s_0) = (x_0,y_0)$ and $\alpha'(s_0) = (a,b)$ (where $a^2+b^2=1$) is given by $$\alpha(s) = \left(x_0 + \int_{s_0}^s \cos(\theta(\xi)+\theta_0)\,{\rm d}\xi,y_0 + \int_{s_0}^s \sin(\theta(\xi)+\theta_0)\,{\rm d}\xi\right)$$where $$\theta(\xi) = \int_{s_0}^\xi \kappa(\tau)\,{\rm d}\tau$$and $\theta_0$ is such that $\cos\theta_0 = a$ and $\sin\theta_0 = b$. So let's take $s_0 = 0$, $\kappa(s) = cs$ for some $c \in \Bbb R$, $x_0=y_0 = b = 0$ and $a=1$, so that $\theta_0 = 0$ as well. Then $$\theta(\xi) = \int_0^\xi c\tau\,{\rm d}\tau = \frac{c\xi^2}{2}$$and $$\alpha(s) = \left(\int_0^s \cos\left(\frac{c\xi^2}{2}\right)\,{\rm d}\xi,\int_0^s \sin\left(\frac{c\xi^2}{2}\right)\,{\rm d}\xi\right).$$If you change all the initial data I conveniently chose, you will get an affine reparametrization of something congruent to the above.