Math 2.0

Differential geometry Definition

Evolute

Builds on Osculating circleCurvature of a curveRegular curve

Definition

Let γ ⁣:I→ℝ2\gamma \colon I \to \R^2 be a regular curve of class C2C^2 on an interval II. Let N(t)N(t) be its unit tangent rotated by a right angle counterclockwise, and let k(t)k(t) be its signed curvature. Assume that k(t)≠0k(t) \ne 0 for all t∈It \in I. The evolute of γ\gamma is the curve E(t)=γ(t)+1k(t)N(t),t∈I.E(t) = \gamma(t) + \frac{1}{k(t)} N(t), \qquad t \in I. So E(t)E(t) is the centre of the osculating circle of γ\gamma at tt.

Remarks

Example: for γ(t)=(t,t2)\gamma(t) = (t, t^2), one has k(t)=2(1+4t2)−3/2k(t) = 2 (1 + 4t^2)^{-3/2} and N(t)=(−2t,1)/1+4t2N(t) = (-2t, 1) / \sqrt{1 + 4t^2}. Then E(t)=(−4t3,1/2+3t2)E(t) = (-4t^3, 1/2 + 3t^2), so the evolute is the semicubical parabola 27x2=16(y−1/2)327 x^2 = 16 (y - 1/2)^3. Non-example: for γ(t)=(t,t3)\gamma(t) = (t, t^3) on ℝ\R, the curvature vanishes at t=0t = 0, so the evolute is not defined on the whole interval; near t=0t = 0 the centres of curvature go off to infinity. The evolute of a circle of radius rr is the constant curve at its centre. If γ\gamma is of class C3C^3, then E′(t)E'(t) is parallel to N(t)N(t), and E′(t)=0E'(t) = 0 exactly where k′(t)=0k'(t) = 0, that is, at the vertices of γ\gamma.

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 levelFunctionconceptOsculating circle, concept, no evidence levelOsculating circleconceptFréchet 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 curvedefinitionTangent vector of a curve, definition, no evidence levelTangent vector of a c…definitionRegular curve, definition, no evidence levelRegular curvedefinitionEvolute, definition, no evidence levelEvolutedefinition
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. Osculating circle concept
  5. Fréchet derivative definition
  6. Function of several variables concept
  7. Vector-valued function concept
  8. Limit of a function of several variables concept
  9. Continuity in R^n definition
  10. Differentiable function of several variables definition
  11. Partial derivative definition
  12. Differentiable on an open set definition
  13. Higher partial derivatives definition
  14. Iterated derivative definition
  15. C^k function characterised by continuous derivatives statement
  16. Parametrised curve definition
  17. Tangent vector of a curve definition
  18. Regular curve definition
  19. Evolute definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Evolute
About this page
TagYC9E5JJDDF
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Kinddefinition
WritingW0 stub
MSC53A04
DefinesEvolute
UsesOsculating circle, Curvature of a curve, Regular curve
Changed2 November 2026

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