Editing Frenet–Serret formulas (section)

{{Short description|Formulas in differential geometry}}
{{redirect|Binormal|the category-theoretic meaning of this word|normal morphism}}
[[Image:frenet.svg|thumb|300px|right|A space curve; the vectors {{math|'''T''', '''N''', '''B'''}}; and the [[osculating plane]] spanned by {{math|'''T'''}} and {{math|'''N'''}}]]

In [[differential geometry]], the '''Frenet–Serret formulas''' describe the [[kinematic]] properties of a particle moving along a differentiable [[curve]] in three-dimensional [[Euclidean space]] <math>\R^3,</math> or the geometric properties of the curve itself irrespective of any motion. More specifically, the formulas describe the [[derivative]]s of the so-called '''tangent, normal, and binormal''' [[unit vector]]s in terms of each other. The formulas are named after the two French mathematicians who independently discovered them: [[Jean Frédéric Frenet]], in his thesis of 1847, and [[Joseph Alfred Serret]], in 1851. [[Vector notation]] and linear algebra currently used to write these formulas were not yet available at the time of their discovery.

The tangent, normal, and binormal unit vectors, often called {{math|'''T'''}}, {{math|'''N'''}}, and {{math|'''B'''}}, or collectively the '''Frenet–Serret basis''' (or '''TNB basis'''), together form an [[orthonormal basis]] that [[Linear span|spans]] <math>\R^3,</math> and are defined as follows:
* {{math|'''T'''}} is the unit vector [[tangent vector|tangent]] to the curve, pointing in the direction of motion.
* {{math|'''N'''}} is the [[normal vector|normal]] unit vector, the derivative of {{math|'''T'''}} with respect to the [[Rectifiable path|arclength parameter]] of the curve, divided by its length.
* {{math|'''B'''}} is the binormal unit vector, the [[cross product]] of {{math|'''T'''}} and {{math|'''N'''}}.
The above basis in conjunction with an [[Origin (mathematics)|origin]] at the point of evaluation on the curve define a [[moving frame]], the '''Frenet–Serret frame''' (or '''TNB frame''').

The Frenet–Serret formulas are:
<math display=block> \begin{align}
  \frac{\mathrm{d} \mathbf{T} }{ \mathrm{d} s } &= \kappa\mathbf{N}, \\[4pt]
  \frac{\mathrm{d} \mathbf{N} }{ \mathrm{d} s } &= -\kappa\mathbf{T}+\tau\mathbf{B}, \\[4pt]
  \frac{\mathrm{d} \mathbf{B} }{ \mathrm{d} s } &= -\tau\mathbf{N},
\end{align}</math>
where <math>\tfrac{d}{ds}</math> is the derivative with respect to arclength, {{mvar|κ}} is the [[curvature]], and {{mvar|τ}} is the [[torsion of curves|torsion]] of the space curve. (Intuitively, curvature measures the failure of a curve to be a straight line, while torsion measures the failure of a curve to be planar.) The {{math|'''TNB'''}} basis combined with the two [[Scalar (mathematics)|scalars]], {{mvar|κ}} and {{mvar|τ}}, is called collectively the '''Frenet–Serret apparatus'''.