The curve has constant torsion.

Question

Question:

Show that when the curve $c_1=c_1(t)$ has constant torsion $\tau$, the curve $$c_2=c_2(t)=-\frac{1}{\tau}N+\int_{t_0}^{t}B(u)du$$ has constant curvature $-\tau$ or $+\tau$.

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What I know:

$$\dot T=\kappa N$$ $$\dot{N}=-\kappa T +\tau B$$ $$\dot B=-\tau N$$

In fact, I posted what i did. But, I am not sure. There may be some mistakes.

Answer 1

You are on the right track. Remember unit tangent $T=c_2^\prime$, so when you get $$c_2^\prime=\frac{\kappa}{\tau}T$$ simply do the substitution and get $$T=\frac{\kappa}{\tau}T$$ then square both sides: $$1=\left(\frac{\kappa}{\tau}\right)^2$$