Math 2.0

Differential geometry Curves in the plane and in space Theorem

Fundamental theorem of space curves

Builds on Curvature of a curveTorsion of a curveArc length parametrisationEuclidean groupFrenet–Serret formulas

Statement

Let I⊆ℝI \subseteq \R be an open interval, and let κ ⁣:I→ℝ\kappa \colon I \to \R and τ ⁣:I→ℝ\tau \colon I \to \R be smooth functions with κ(s)>0\kappa(s) > 0 for all s∈Is \in I. Then there is a smooth curve γ ⁣:I→ℝ3\gamma \colon I \to \R^3 parametrised by arc length whose curvature is κ\kappa and whose torsion is τ\tau. If γ~ ⁣:I→ℝ3\tilde\gamma \colon I \to \R^3 is another such curve, then γ~=ρ∘γ\tilde\gamma = \rho \circ \gamma for a unique orientation-preserving rigid motion ρ\rho of ℝ3\R^3. Here a rigid motion is a map x↦Rx+bx \mapsto Rx + b with RR a rotation matrix and b∈ℝ3b \in \R^3.

Remarks

For existence, solve the Frenet–Serret formulas as a linear system of ordinary differential equations for (T,N,B)(T, N, B), with an orthonormal positive initial frame at some s0∈Is_0 \in I. The solution stays orthonormal because the coefficient matrix is skew-symmetric; then set γ(s)=∫s0sT(u) du\gamma(s) = \int_{s_0}^s T(u) \, du. For uniqueness, move γ~\tilde\gamma by a rigid motion so that both curves have the same point and frame at s0s_0. Then the uniqueness of solutions of the linear system gives equal frames and equal curves. Example: constant κ=a/(a2+b2)\kappa = a/(a^2+b^2) and τ=b/(a2+b2)\tau = b/(a^2+b^2) with a>0a > 0 give exactly the helices (acos⁡t,asin⁡t,bt)(a \cos t, a \sin t, bt), t=s/a2+b2t = s / \sqrt{a^2 + b^2}, up to rigid motion. The hypothesis κ>0\kappa > 0 matters: if κ\kappa vanishes on an interval, the torsion is not defined there and the curve is not determined by κ\kappa alone. An orientation-reversing motion, such as a reflection, changes the sign of τ\tau, so it does not keep the torsion.

Going deeper

For a student

The theorem says that (κ,τ)(\kappa, \tau) is a complete system of invariants of unit-speed space curves with positive curvature. The symmetry group here is the group of orientation-preserving rigid motions, a subgroup of the Euclidean group. For plane curves the analogue is simpler: the signed curvature determines a unit-speed curve up to orientation-preserving rigid motion, through θ′=k\theta' = k and γ′=(cos⁡θ,sin⁡θ)\gamma' = (\cos \theta, \sin \theta). The same idea, a frame solving a linear system, reappears in the theory of surfaces, where the Gauss–Codazzi equations are the integrability conditions.

Reading path

To understand this page from the ground up, read these 38 pages in order. The foundations come first.

Show the path as a graph
Curvature of a curve, concept, no evidence levelCurvature of a curveconceptEuclidean isometry, concept, no evidence levelEuclidean isometryconceptEuclidean 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 curveconceptTranslation, concept, no evidence levelTranslationconceptVector 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 functionconceptEuclidean group, object, no evidence levelEuclidean groupobjectInner 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 formulasstatementFundamental theorem of space curves, statement, no evidence levelFundamental theorem o…statement
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 isometry concept
  3. Euclidean space object
  4. Euclidean vector concept
  5. Function concept
  6. Torsion of a curve concept
  7. Translation concept
  8. Vector space definition
  9. Fréchet derivative definition
  10. Cross product definition
  11. Function of several variables concept
  12. Vector-valued function concept
  13. Euclidean group object
  14. Inner product definition
  15. Linear combination definition
  16. Linear subspace definition
  17. Limit of a function of several variables concept
  18. Orthogonal vectors definition
  19. Linear independence definition
  20. Linear span definition
  21. Continuity in R^n definition
  22. Differentiable function of several variables definition
  23. Orthonormal family definition
  24. Partial derivative definition
  25. Spanning set definition
  26. Differentiable on an open set definition
  27. Basis definition
  28. Higher partial derivatives definition
  29. Iterated derivative definition
  30. Orthonormal basis definition
  31. C^k function characterised by continuous derivatives statement
  32. Parametrised curve definition
  33. Rectifiable curve definition
  34. Tangent vector of a curve definition
  35. Regular curve definition
  36. Arc length parametrisation definition
  37. Frenet–Serret formulas statement
  38. Fundamental theorem of space curves statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Fundamental_theorem_of_curves
About this page
TagTNYQ5R0NVM
URIm2://TNYQ5R0NVM
Kindstatement, theorem
WritingW0 stub
MSC53A04
StatusThis copy of the site has no build data. Run m2 status TNYQ5R0NVM.
AboutCurvature of a curve, Torsion of a curve
UsesCurvature of a curve, Torsion of a curve, Arc length parametrisation, Euclidean group, Frenet–Serret formulas
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Changed2 November 2026

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