Math 2.0

Differential geometry Curves in the plane and in space Theorem

Frenet–Serret formulas

Builds on Curvature of a curveTorsion of a curveArc length parametrisationCross productOrthonormal basis

Statement

Let I⊆ℝI \subseteq \R be an interval and let γ ⁣:I→ℝ3\gamma \colon I \to \R^3 be a curve of class C3C^3 parametrised by arc length ss. Assume that its curvature κ(s)\kappa(s) is positive for all s∈Is \in I. Let T=γ′T = \gamma', N=T′/κN = T' / \kappa and B=T×NB = T \times N, and let τ\tau be the torsion of γ\gamma. Then (T(s),N(s),B(s))(T(s), N(s), B(s)) is a positively oriented orthonormal basis of ℝ3\R^3 for each ss, and T′=κN,N′=−κT+τB,B′=−τN.T' = \kappa N, \qquad N' = -\kappa T + \tau B, \qquad B' = -\tau N.

Remarks

The first formula is the definition of NN, and the third is the definition of τ\tau. For the second, differentiate the identities ⟨N,N⟩=1\ip{N}{N} = 1, ⟨N,T⟩=0\ip{N}{T} = 0 and ⟨N,B⟩=0\ip{N}{B} = 0. They give ⟨N′,N⟩=0\ip{N'}{N} = 0, ⟨N′,T⟩=−⟨N,T′⟩=−κ\ip{N'}{T} = -\ip{N}{T'} = -\kappa and ⟨N′,B⟩=−⟨N,B′⟩=τ\ip{N'}{B} = -\ip{N}{B'} = \tau. In matrix form the formulas say that (T,N,B)′=(T,N,B)A(T, N, B)' = (T, N, B) A with a skew-symmetric matrix AA built from κ\kappa and τ\tau. For the helix (acos⁡t,asin⁡t,bt)(a \cos t, a \sin t, bt) with a>0a > 0, both κ=a/(a2+b2)\kappa = a / (a^2 + b^2) and τ=b/(a2+b2)\tau = b / (a^2 + b^2) are constant. The hypothesis κ>0\kappa > 0 is needed, since the normal NN is not defined where κ=0\kappa = 0. For example, two plane arcs in different planes, smoothly joined through a straight segment, form a curve with τ=0\tau = 0 wherever κ>0\kappa > 0 that lies in no plane.

Going deeper

For a student

The formulas form a linear system of ordinary differential equations for the frame (T,N,B)(T, N, B) with coefficients κ\kappa and τ\tau. This is why curvature and torsion determine the curve up to a rigid motion, by the fundamental theorem of space curves. They give quick proofs of basic facts. A curve with κ>0\kappa > 0 and τ=0\tau = 0 lies in a plane. A curve with constant κ>0\kappa > 0 and τ=0\tau = 0 is an arc of a circle. In ℝn\R^n there is an analogous frame with n−1n - 1 generalised curvatures, under a nondegeneracy hypothesis on the derivatives.

Reading path

To understand this page from the ground up, read these 34 pages in order. The foundations come first.

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Curvature of a curve, concept, no evidence levelCurvature of a curveconceptEuclidean space, object, no evidence levelEuclidean spaceobjectEuclidean vector, concept, no evidence levelEuclidean vectorconceptFunction, concept, no evidence levelFunctionconceptTorsion of a curve, concept, no evidence levelTorsion of a curveconceptVector space, definition, no evidence levelVector spacedefinitionFréchet derivative, definition, no evidence levelFréchet derivativedefinitionCross product, definition, no evidence levelCross productdefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptInner product, definition, no evidence levelInner productdefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptOrthogonal vectors, definition, no evidence levelOrthogonal vectorsdefinitionLinear independence, definition, no evidence levelLinear independencedefinitionLinear span, definition, no evidence levelLinear spandefinitionContinuity in R^n, definition, no evidence levelContinuity in R^ndefinitionDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionOrthonormal family, definition, no evidence levelOrthonormal familydefinitionPartial derivative, definition, no evidence levelPartial derivativedefinitionSpanning set, definition, no evidence levelSpanning setdefinitionDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionBasis, definition, no evidence levelBasisdefinitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionIterated derivative, definition, no evidence levelIterated derivativedefinitionOrthonormal basis, definition, no evidence levelOrthonormal basisdefinitionC^k function characterised by continuous derivatives, statement, no evidence levelC^k function characte…statementParametrised curve, definition, no evidence levelParametrised curvedefinitionRectifiable curve, definition, no evidence levelRectifiable curvedefinitionTangent vector of a curve, definition, no evidence levelTangent vector of a c…definitionRegular curve, definition, no evidence levelRegular curvedefinitionArc length parametrisation, definition, no evidence levelArc length parametris…definitionFrenet–Serret formulas, statement, no evidence levelFrenet–Serret formulasstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Curvature of a curve concept
  2. Euclidean space object
  3. Euclidean vector concept
  4. Function concept
  5. Torsion of a curve concept
  6. Vector space definition
  7. Fréchet derivative definition
  8. Cross product definition
  9. Function of several variables concept
  10. Vector-valued function concept
  11. Inner product definition
  12. Linear combination definition
  13. Linear subspace definition
  14. Limit of a function of several variables concept
  15. Orthogonal vectors definition
  16. Linear independence definition
  17. Linear span definition
  18. Continuity in R^n definition
  19. Differentiable function of several variables definition
  20. Orthonormal family definition
  21. Partial derivative definition
  22. Spanning set definition
  23. Differentiable on an open set definition
  24. Basis definition
  25. Higher partial derivatives definition
  26. Iterated derivative definition
  27. Orthonormal basis definition
  28. C^k function characterised by continuous derivatives statement
  29. Parametrised curve definition
  30. Rectifiable curve definition
  31. Tangent vector of a curve definition
  32. Regular curve definition
  33. Arc length parametrisation definition
  34. Frenet–Serret formulas statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Frenet%E2%80%93Serret_formulas
About this page
TagZM6W89YNS7
URIm2://ZM6W89YNS7
Kindstatement, theorem
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MSC53A04
StatusThis copy of the site has no build data. Run m2 status ZM6W89YNS7.
AboutCurvature of a curve, Torsion of a curve
UsesCurvature of a curve, Torsion of a curve, Arc length parametrisation, Cross product, Orthonormal basis
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Changed2 November 2026

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