Normal vector in curvilinear coordinates 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.