# Euler spiral (Clothoid)

Clothoid is a curve whose curvature .

*k*changes linearly with its curve length (denote*s*or*L*). 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*s*. Although its defining formulas for coordinates are transcendental functions (Fresnel integrals), the important characteristics can be derived easily from equation*k = s/A*where*A*is constant. Some applications avoid working with the transcendental functions by proposing polynomial approximations to the clothoid, e.g.## Task 1

Determine the length for transition between straight road and circular arc of radius and

*s*of Euler spiral*r = 9 m*. Solution: Curvature of a bend must be the same as curvature of a clothoid, i.e.*s = 16 m*.## Clothoid k = s/A

## Task 2

Determine an angle length at and

*α*between the tangent of Euler spiral*s = 16 m*and x-axis. Solution: Formula for direction part of a clothoid.*.**α*= 50,92°## Clothoid between two circumferences

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