Math 2.0

Differential geometry Definition

Total curvature

Builds on Curvature of a curveRegular curveClosed curve

Definition

Let γ ⁣:[a,b]→ℝn\gamma \colon [a, b] \to \R^n be a regular curve of class C2C^2 with curvature κ\kappa. The total curvature of γ\gamma is ∫abκ(t) ∥γ′(t)∥ dt,\int_a^b \kappa(t) \, \norm{\gamma'(t)} \, dt, that is, the integral of κ\kappa with respect to arc length. For a plane curve with signed curvature kk, the total signed curvature is ∫abk(t) ∥γ′(t)∥ dt\int_a^b k(t) \, \norm{\gamma'(t)} \, dt.

Remarks

Example: the circle γ(t)=(rcos⁡t,rsin⁡t)\gamma(t) = (r \cos t, r \sin t), t∈[0,2π]t \in [0, 2\pi], has κ=1/r\kappa = 1/r and ∥γ′∥=r\norm{\gamma'} = r, so its total curvature is 2π2\pi. Non-example: the boundary of a square is a closed curve that is not of class C2C^2, so the definition does not apply. For convex polygons one uses instead the sum of the exterior angles, which is again 2π2\pi. The total curvature does not change under a regular reparametrisation or a rigid motion. For a closed regular plane curve, the total signed curvature equals 2π2\pi times the turning number of the curve.

Reading path

To understand this page from the ground up, read these 19 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 spaceobjectFunction, concept, no evidence levelFunctionconceptFréchet derivative, definition, no evidence levelFréchet derivativedefinitionFunction 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 curvedefinitionClosed curve, definition, no evidence levelClosed curvedefinitionTangent vector of a curve, definition, no evidence levelTangent vector of a c…definitionRegular curve, definition, no evidence levelRegular curvedefinitionTotal curvature, definition, no evidence levelTotal curvaturedefinition
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. Function concept
  4. Fréchet derivative definition
  5. Function of several variables concept
  6. Vector-valued function concept
  7. Limit of a function of several variables concept
  8. Continuity in R^n definition
  9. Differentiable function of several variables definition
  10. Partial derivative definition
  11. Differentiable on an open set definition
  12. Higher partial derivatives definition
  13. Iterated derivative definition
  14. C^k function characterised by continuous derivatives statement
  15. Parametrised curve definition
  16. Closed curve definition
  17. Tangent vector of a curve definition
  18. Regular curve definition
  19. Total curvature definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Total_curvature
About this page
TagEMVDSQJM8S
URIm2://EMVDSQJM8S
Kinddefinition
WritingW0 stub
MSC53A04
DefinesTotal curvature
UsesCurvature of a curve, Regular curve, Closed curve
Changed2 November 2026

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