I was trying to find a non-trivial example of a unit speed plane curve. The reason is I want something to work with but if I start with a non-unit speed curve and then do the arc length parameterisation I end up with something impossible.
The trivial example is of course the unit circle $(\cos t, \sin t)$ but this is indeed trivial as the curvature is $1$ and also, the circle is too obvious (can determine the curvature just by looking at it).
Does anyone know a unit speed curve that is not the circle?
Best Answer
Consider the spiral $\gamma: \Bbb R \to \Bbb R^2$ given by $$\gamma(t) = (e^t\cos t, e^t \sin t).$$ Then $$\gamma'(t) = (e^t(\cos t - \sin t),e^t (\cos t+\sin t)),$$ and so: $$\|\gamma'(t)\| = e^t\sqrt{(\cos t -\sin t)^2+(\cos t+\sin t)^2} = e^t\sqrt{2}.$$ With this: $$s(t) = \int_0^t e^\xi \sqrt{2}\,{\rm d}\xi = \sqrt{2}(e^t-1) \implies t = \ln\left(\frac{s}{\sqrt{2}}+1\right).$$ So we get a nice and non-trivial unit speed curve: $$\gamma(t(s)) \equiv \gamma(s) = \left(\left(\frac{s}{\sqrt{2}}+1\right)\cos\left(\ln\left(\frac{s}{\sqrt{2}}+1\right)\right),\left(\frac{s}{\sqrt{2}}+1\right)\sin\left(\ln\left(\frac{s}{\sqrt{2}}+1\right)\right)\right).$$