curvature determines the curve

The curvaturePlanetmathPlanetmath ( of plane curve determines uniquely the form and of the curve, i.e. one has the

Theorem.  If  sk(s)  is a continuous real function, then there exists always plane curves satisfying the equation

κ=k(s) (1)

between their curvature κ and the arc length s.  All these curves are congruentMathworldPlanetmathPlanetmath (

Proof.  Suppose that a curve C satisfies the condition (1).  Let the value  s=0  correspond to the point P0 of this curve.  We choose O as the origin of the plane.  The tangentMathworldPlanetmath and the normal of C in O are chosen as the x-axis and the y-axis, with positive directions the directions of the tangent and normal vectors of C, respectively.  According to (1) and the definition of curvature, the equation


for the direction angle θ of the tangent of C is valid in this coordinate systemMathworldPlanetmath; the initial conditionMathworldPlanetmath is

θ= 0whens= 0.

Thus we get

θ=0sk(t)𝑑t:=ϑ(s), (2)

which implies

dxds=cosϑ(s),dyds=sinϑ(s). (3)

Since  x=y=0  when  s=0, we obtain

x=0scosϑ(t)𝑑t,y=0ssinϑ(t)𝑑t. (4)

Thus the function  sk(s)  determines uniquely these functions x and y of the parameter s, and (4) represents a curve with definite form and .

The above reasoning shows that every curve which satisfies (1) is congruent ( with the curve (4).

We have still to show that the curve (4) satisfies the condition (1).  By differentiating ( the equations (4) we get the equations (3), which imply  (dxds)2+(dyds)2=1,  or  ds2=dx2+dy2  which means that the parameter s represents the arc length of the curve (4), counted from the origin.  Differentiating (3) we get, because  ϑ(s)=k(s)  by (2),

d2xds2=-k(s)sinϑ(s),d2yds2=k(s)cosϑ(s). (5)

The equations (3) and (5) then yield


i.e. the curvature of (4), according the parent entry (, satisfies


Thus the proof is settled.


  • 1 Ernst Lindelöf: Differentiali- ja integralilasku ja sen sovellutukset I.  WSOY. Helsinki (1950).
Title curvature determines the curve
Canonical name CurvatureDeterminesTheCurve
Date of creation 2016-02-22 16:14:25
Last modified on 2016-02-22 16:14:25
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Theorem
Classification msc 53A04
Synonym fundamental theorem of plane curvesMathworldPlanetmath
Related topic FundamentalTheoremOfSpaceCurves
Related topic ErnstLindelof