Math 2.0

Differential geometry Definition

Torsion of a curve

Builds on Curvature of a curveArc length parametrisationCross product

Definition

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, with curvature κ(s)>0\kappa(s) > 0 for all s∈Is \in I. Its unit tangent is T=γ′T = \gamma', its unit normal is N=T′/κN = T' / \kappa, and its binormal is B=T×NB = T \times N, where ×\times is the cross product. The vector B′(s)B'(s) is a multiple of N(s)N(s). The torsion of γ\gamma at ss is the number τ(s)\tau(s) with B′(s)=−τ(s) N(s).B'(s) = -\tau(s) \, N(s).

Remarks

For a regular C3C^3 curve with any parameter tt and γ′×γ′′≠0\gamma' \times \gamma'' \ne 0, the torsion is τ=det⁡(γ′,γ′′,γ′′′)∥γ′×γ′′∥2.\tau = \frac{\det(\gamma', \gamma'', \gamma''')}{\norm{\gamma' \times \gamma''}^2}. Example: take the helix γ(t)=(acos⁡t,asin⁡t,bt)\gamma(t) = (a \cos t, a \sin t, bt) with a>0a > 0 and b∈ℝb \in \R. Then γ′×γ′′=(absin⁡t,−abcos⁡t,a2)\gamma' \times \gamma'' = (ab \sin t, -ab \cos t, a^2) and det⁡(γ′,γ′′,γ′′′)=a2b\det(\gamma', \gamma'', \gamma''') = a^2 b, so τ=b/(a2+b2)\tau = b / (a^2 + b^2). Non-example: on the curve t↦(t,t3,0)t \mapsto (t, t^3, 0) the curvature vanishes at t=0t = 0, so NN and BB are not defined there and the torsion is not defined at that point. Some books use the opposite sign convention B′=τNB' = \tau N. A curve with κ>0\kappa > 0 lies in a plane if and only if τ=0\tau = 0 everywhere.

Reading path

To understand this page from the ground up, read these 22 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 levelFunctionconceptFréchet 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 functionconceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptContinuity in R^n, definition, no evidence levelContinuity in R^ndefinitionDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionPartial derivative, definition, no evidence levelPartial derivativedefinitionDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionIterated derivative, definition, no evidence levelIterated derivativedefinitionC^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…definitionTorsion of a curve, definition, no evidence levelTorsion of a curvedefinition
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. Fréchet derivative definition
  6. Cross product definition
  7. Function of several variables concept
  8. Vector-valued function concept
  9. Limit of a function of several variables concept
  10. Continuity in R^n definition
  11. Differentiable function of several variables definition
  12. Partial derivative definition
  13. Differentiable on an open set definition
  14. Higher partial derivatives definition
  15. Iterated derivative definition
  16. C^k function characterised by continuous derivatives statement
  17. Parametrised curve definition
  18. Rectifiable curve definition
  19. Tangent vector of a curve definition
  20. Regular curve definition
  21. Arc length parametrisation definition
  22. Torsion of a curve definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Torsion_of_a_curve
About this page
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Kinddefinition
WritingW0 stub
MSC53A04
DefinesTorsion of a curve
UsesCurvature of a curve, Arc length parametrisation, Cross product
Changed2 November 2026

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