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Frenet–Serret formulas for plane curves

The expression of the curvature In terms of arc-length parametrization is essentially the first Frenet–Serret formula

where the primes refer to the derivatives with respect to the arc length s, and N (s) is the normal unit vector in the direction of T ′(s).

As planar curves have zero torsion, the second Frenet–Serret formula provides the relation

For a general parametrization by a parameter t, one needs expressions involving derivatives with respect to t. As these are obtained by multiplying by ds / dt the derivatives with respect to s, one has, for any proper parametrization