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 in such that there exists another curve , distinct from it and a bijection f between and such that for each corresponding points in f, and have the same principal normal.
I need to show this things.
1. Every plane curve is a Bertrand curve.
I already know that If is the plane curve, I can use any of its involute of the form . here is the principal normal of . 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 and a nonzero torsion is a Bertrand curve if and only if where a,b are some constants.