How to show the normal spherical image x is never constant?
I assume your curve is parametrized by arc-lenght?
If so: Suppose the unit normal vector N was constant. This normal vector is always perpendicular to the tangent vector T, so the vector T must be in the plane perpendicular to N. But if T is containing entirely in a plane, then the normal unit vector will also be in this plane (being the derivative of T), a contradiction.