Is it true that the normal vector, or, $\ddot{\mathbf r}$ always vanishes for:
- a helix in cylindrical coordinates
- a loxodrome in spherical coordinates
- a torus knot in toroidal coordinates
When does $\ddot{\mathbf r}$ vanish for a curve in curvilinear coordinates?
If $\ddot{\vec{r}}$ vanishes in one coordinate system then it will vanish in all coordinate systems. This is a fundamental property of vectors.