Math 2.0

Differential geometry Definition

Involute

Builds on Arc length parametrisationTangent vector of a curveRegular curveEvolute

Definition

Let γ ⁣:I→ℝn\gamma \colon I \to \R^n be a regular curve of class C2C^2, parametrised by arc length ss on an interval II, and let c∈ℝc \in \R. The involute of γ\gamma with parameter cc is the curve J(s)=γ(s)+(c−s) γ′(s),s∈I.J(s) = \gamma(s) + (c - s) \, \gamma'(s), \qquad s \in I. Here γ′(s)\gamma'(s) is the unit tangent vector, so J(s)J(s) lies on the tangent line at γ(s)\gamma(s), at distance ∣c−s∣\abs{c - s} from γ(s)\gamma(s).

Remarks

Example: for the unit circle γ(s)=(cos⁡s,sin⁡s)\gamma(s) = (\cos s, \sin s) and c=0c = 0, one gets J(s)=(cos⁡s+ssin⁡s,sin⁡s−scos⁡s)J(s) = (\cos s + s \sin s, \sin s - s \cos s). Check: J′(s)=(scos⁡s,ssin⁡s)J'(s) = (s \cos s, s \sin s), which is orthogonal to γ′(s)=(−sin⁡s,cos⁡s)\gamma'(s) = (-\sin s, \cos s). In general J′(s)=(c−s)γ′′(s)J'(s) = (c - s) \gamma''(s), so JJ meets each tangent line of γ\gamma at a right angle, and JJ is singular at s=cs = c. Non-example: the circle of radius 22 about the origin is not an involute of the unit circle. Indeed, it meets each tangent line of the unit circle at an angle that is not a right angle. For a plane curve with nonzero curvature, the evolute of each involute is the original curve, on the part where s≠cs \ne c.

Reading path

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Euclidean space, object, no evidence levelEuclidean spaceobjectEvolute, concept, no evidence levelEvoluteconceptFunction, 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…definitionInvolute, definition, no evidence levelInvolutedefinition
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. Euclidean space object
  2. Evolute concept
  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. Rectifiable curve definition
  17. Tangent vector of a curve definition
  18. Regular curve definition
  19. Arc length parametrisation definition
  20. Involute definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Involute
About this page
TagJ3NHX0R6YD
URIm2://J3NHX0R6YD
Kinddefinition
WritingW0 stub
MSC53A04
DefinesInvolute
UsesArc length parametrisation, Tangent vector of a curve, Regular curve, Evolute
Changed2 November 2026

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