nLab Frenet-Serret formulas

Definition

Given a curve in $n$-dimensional Euclidean space $E^n$, with a $C^{n+1}$-differentiable parametrization by the arc length from a given point on the curve

$$\array{ F &\colon& I & \longrightarrow & E^n \\ && s &\mapsto& F(s) }$$

(hence a naturally parametrized curve of smoothness class $C^{n+1}$), Frenet-Serret formulas express the derivatives with respect to the arc length of the Frenet moving frame vectors as a linear combination of the Frenet moving frame vectors, where the coefficients form an antisymmetric matrix whose nonzero coefficients are (up to sign) the curvature of the curve, its torsion and their higher analogues.

This works under the assumption that the derivatives

$$F^{(k)}(s) \;=\; d^k F/d s^k \;\in\; T^k_{F(s)} E^n \;\cong\; \mathbf{R}^n, \,\,\,\,\, k = 1,\ldots,n$$

are linearly independent at each point (using the canonical identification with $\mathbf{R}^n$ with any of its tangent spaces).

The Gram-Schmidt orthogonalization procedure then transforms the n-tuple $(F^{(1)}(s),\ldots,F^{(n)}(s))$ into an orthogonal $n$-frame $(V_1(s),\ldots,V_n(s))$ called the Frenet $n$-frame.

Writing down the $n$-tuple of the derivatives with respect to $s$ of the Frenet frame in a $C^0(\mathbf{R})$-basis formed by the Frenet frame reveals an antisymmetric matrix of coefficients over $C^0(\mathbf{R})$. Its nonzero coefficients are, up to a sign, the curvature of the curve, the torsion and their higher analogues.

In dimension $n=3$, the Frenet frame vectors are called the tangent, normal and binormal unit vectors of the curve. In dimension 4 the 4th vector is called the trinormal unit vector. The plane spanned by the tangent and normal unit vectors is called the osculating plane at $F(s)$.

References

• Herman Gluck, Higher curvatures of curves in Euclidean space, The American Mathematical Monthly, 73:7 (1966) 699-704, doi

See also:

• en.wikipedia/Frenet-Serret_formulas