The curvature of plane curve determines uniquely the form and size of the curve, i.e. one has the
Theorem. If is a continuous real function, then there exists always plane curves satisfying the equation
between their curvature and the arc length . All these curves are congruent.
Proof. Suppose that a curve satisfies the condition (1). Let the value correspond to the point of this curve. We choose as the origin of the plane. The tangent and the normal of in are chosen as the -axis and the -axis, with positive directions the directions of the tangent and normal vectors of , respectively. According to (1) and the definition of curvature, the equation
for the direction angle of the tangent of is valid in this coordinate system; the initial condition is
Thus we get
Since when , we obtain
Thus the function determines uniquely these functions and of the parameter , and (4) represents a curve with definite form and size.
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 , or which means that the parameter represents the arc length of the curve (4), counted from the origin. Differentiating (3) we get, because by (2),
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).