Clothoid is a curve whose curvature [color=#0000ff][i]k[/i][/color] changes linearly with its curve length (denote [color=#0000ff][i]s[/i][/color] or [color=#0000ff][i]L[/i][/color]). Clothoids are widely used as transition curves in railroad engineering for connecting and transiting the geometry between a tangent and a circular curve.[br]Clothoid has the desirable property that the curvature [color=#0000ff][i]k[/i][/color] is linearly related to the arc length [color=#0000ff][i]s[/i][/color]. Although its defining formulas for coordinates are transcendental functions (Fresnel integrals), the important characteristics can be derived easily from equation[color=#0000ff] [i]k = s/A [/i][/color] where [i]A[/i] is constant. [br]Some applications avoid working with the transcendental functions by proposing polynomial approximations to the clothoid, e.g. [math]y=\frac{x^3}{6A}[/math].[br][img]https://www.geogebra.org/resource/vvdswmpf/Yu4eL8i7AD6OlKYu/material-vvdswmpf.png[/img]

Determine the length [color=#0000ff][i]s[/i][/color] of Euler spiral [math]k=\frac{s}{12^2}[/math] for transition between straight road and circular arc of radius [color=#0000ff][i]r = 9 m[/i][/color].[br]Solution: Curvature of a bend must be the same as curvature of a clothoid, i.e. [math]\frac{1}{9}=\frac{s}{12^2}[/math] and [color=#0000ff][i]s = 16 m[/i][/color].

Determine an angle length [color=#0000ff][i]α [/i][/color] between the tangent of Euler spiral [math]k=\frac{s}{12^2}[/math] at [color=#0000ff][i]s = 16 m[/i][/color] and x-axis. [br]Solution: Formula for direction part of a clothoid. [math]\alpha=\frac{s^2}{2.A^2}=\frac{16^2}{2.12^2}=\frac{8}{9}[/math] and [color=#0000ff][i][color=#0000ff][i]α[/i][/color] = 50,92°[/i][/color].