Consider a curve $\beta:I\subseteq R \rightarrow \mathbb{E}^3:s\mapsto\beta(s)$ parametrized by its arc length defined on a sphere with radius r. We define the curve $\alpha$ as: $$\alpha (t)=\int_a^t \beta (s)\times\beta'(s)ds$$ Prove that $\alpha$ has velocity $r$ and torsion $-r^2$.
I don't know how to begin? Normally, I can find the velocity very easily by calculating $\parallel \alpha' \parallel$. But I don't know how to proceed when dealing with an integral and a vector product in the parametrization of the curve?
Hint: let $\beta$ be a Frenet curve with non-vanishing torsion $\tau$ and let $r$ be a positive number. Then show:
If $\beta$ is contained in the sphere with center $p_0$ with radius $r$ (i.e., $\langle \beta-p_0,\beta-p_0\rangle=r^2$), then $$\beta=p_0-\frac{1}{\kappa}N-\left(\frac{1}{\kappa}\right)'\cdot\frac{1}{\tau}\cdot B,$$ where $N$ and $B$ denote the normal and binormal vector, resp., and hence $$ \left(\frac{1}{\kappa}\right)^2+ \left(\left(\frac{1}{\kappa}\right)'\right)^2\cdot\left(\frac{1}{\tau}\right)^2=r^2.$$