Frenet–Serret formulas

Special cases

If the curvature is always zero then the curve will be a straight line. Here the vectors N, B and the torsion are not well defined.
If the torsion is always zero then the curve will lie in a plane.
A curve may have nonzero curvature and zero torsion.  For example, the circle of radius R given by r(t)=(R cos t, R sin t, 0) in the z=0 plane has zero torsion and curvature equal to 1/R.  The converse, however, is false.  That is, a regular curve with nonzero torsion must have nonzero curvature.  (This is just the contrapositive of the fact that zero curvature implies zero torsion.)
A helix has constant curvature and constant torsion.

Plane curves

Given a curve contained on the x-y plane, its tangent vector T is also contained on that plane. Its binormal vector B can be naturally postulated to coincide with the normal to the plane (along the z axis). Finally, the curve normal can be found completing the right-handed system, N = B × T. This form is well-defined even when the curvature is zero; for example, the normal to a straight line on a plane will be perpendicular to the tangent, all co-planar.