I have a problem here, I need it by monday, I'll appreciate any help.

I'll define first a BERTRAND CURVE, it is a curve $\displaystyle \alpha$in $\displaystyle R^{3}$ such that there exists another curve $\displaystyle \beta$, distinct from it and a bijection f between $\displaystyle \alpha$ and $\displaystyle \beta$ such that for each corresponding points in f, $\displaystyle \alpha$ and $\displaystyle \beta$ have the same principal normal.

I need to show this things.

1. Every plane curve is a Bertrand curve.

I already know that If $\displaystyle \alpha(t)$ is the plane curve, I can use any of its involute of the form $\displaystyle \alpha(t)+ k*N(t)$. $\displaystyle N(t)$here is the principal normal of $\displaystyle \alpha$. The bijection is also obvious. But my problem is, we haven't discussed involutes in class, so I have to show that their principal normals are the same. I'm using the formulas but I can't prove it.

2. A curve with a nonzero curvature $\displaystyle \kappa$ and a nonzero torsion $\displaystyle \tau$ is a Bertrand curve if and only if $\displaystyle a \kappa + b \tau = $ where a,b are some constants.