Math 2.0

Differential geometry Definition

Arc length parametrisation

Builds on Regular curveTangent vector of a curveRectifiable curve

Definition

Let γ ⁣:[a,b]→ℝn\gamma \colon [a, b] \to \mathbb{R}^n be a regular C1C^1 curve. Its arc length function is s(t)=∫at∥γ′(u)∥ du,t∈[a,b],s(t) = \int_a^t \|\gamma'(u)\| \, du, \qquad t \in [a, b], and its length is ℓ=s(b)\ell = s(b). Since s′(t)=∥γ′(t)∥>0s'(t) = \|\gamma'(t)\| > 0, the function s ⁣:[a,b]→[0,ℓ]s \colon [a, b] \to [0, \ell] is a strictly increasing C1C^1 bijection with a C1C^1 inverse t(s)t(s). The arc length parametrisation of γ\gamma is the curve γ~ ⁣:[0,ℓ]→ℝn\tilde\gamma \colon [0, \ell] \to \mathbb{R}^n, γ~(s)=γ(t(s))\tilde\gamma(s) = \gamma(t(s)).

Remarks

The curve γ~\tilde\gamma has the same image and direction as γ\gamma, and ∥γ~′(s)∥=1\|\tilde\gamma'(s)\| = 1 for each ss. So γ~′(s)\tilde\gamma'(s) is the unit tangent vector. The length of γ~\tilde\gamma from γ~(s0)\tilde\gamma(s_0) to γ~(s1)\tilde\gamma(s_1) is s1−s0s_1 - s_0. For the helix γ(t)=(cos⁡t,sin⁡t,t)\gamma(t) = (\cos t, \sin t, t), ∥γ′(t)∥=2\|\gamma'(t)\| = \sqrt{2}, so s=2 ts = \sqrt{2}\, t and γ~(s)=(cos⁡(s/2),sin⁡(s/2),s/2)\tilde\gamma(s) = (\cos(s / \sqrt{2}), \sin(s / \sqrt{2}), s / \sqrt{2}). The arc length parametrisation is unique up to a shift of the parameter. It simplifies the formulas for curvature.

Reading path

To understand this page from the ground up, read these 18 pages in order. The foundations come first.

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Euclidean 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 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…definition
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  1. Euclidean space object
  2. Function concept
  3. Fréchet derivative definition
  4. Function of several variables concept
  5. Vector-valued function concept
  6. Limit of a function of several variables concept
  7. Continuity in R^n definition
  8. Differentiable function of several variables definition
  9. Partial derivative definition
  10. Differentiable on an open set definition
  11. Higher partial derivatives definition
  12. Iterated derivative definition
  13. C^k function characterised by continuous derivatives statement
  14. Parametrised curve definition
  15. Rectifiable curve definition
  16. Tangent vector of a curve definition
  17. Regular curve definition
  18. Arc length parametrisation definition

Where this leads

18 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Differentiable_curve, Length and natural parametrization
  2. Web https://en.wikipedia.org/wiki/Arc_length, Finding arc lengths by integration
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Kinddefinition
WritingW0 stub
MSC53A04
DefinesArc length parametrisation
UsesRegular curve, Tangent vector of a curve, Rectifiable curve

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