Euler spiral (Clothoid)

Clothoid (Euler spiral) is a curve whose curvature k changes linearly with its curve length (denote s or L).

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Clothoids are widely used as transition curves in railroad engineering for connecting and transiting the geometry between a tangent and a circular curve. Clothoid has the desirable property that the curvature k is linearly related to the arc length L. Although its defining formulas for coordinates are transcendental functions (Fresnel integrals), the important characteristics can be derived easily from equation k = L/A² where A² is constant. Some applications avoid using transcendental functions by proposing polynomial approximations to the clothoid, i.e.

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Task 1

Determine the length L of Euler spiral

for transition between straight road and circular arc of radius r = 9 m. Solution: Curvature of a bend must be the same as curvature of a clothoid, i.e.

and L = 16 m.

Clothoid k = L/A

Task 2

Determine an angle length α between the tangent of Euler spiral

at L = 16 m and x-axis. Solution: Formula for direction part of a clothoid.

and α = 50,92°.

Clothoid between two circumferences

Circumferences must be tangent to clothoids, and radius of curvature must be the same at each tangent point.

Task 3: Cubic Approximation

Replace the Euler spiral

with a cubic parabola. Solution: From the Taylor expansion of degree 3, we get an approximation of the form

. Substituting A=15, we get the third degree algebraic transient expressed explicitly as

.

Proof:

Slide 60, prezentation 07_Curves