0.1 The functions and
In optics, both of them express the .
The Fresnel integrals at infinity have the finite value
a curve called clothoid. Since the equations (2) both define odd functions, the clothoid has symmetry about the origin. The curve has the shape of a “” (see this http://www.wakkanet.fi/ pahio/A/A/clothoid.pngdiagram).
The arc length of the clothoid from the origin to the point is simply
Thus the of the whole curve (to the point ) is infinite.
The curvature (http://planetmath.org/CurvaturePlaneCurve) of the clothoid also is extremely ,
i.e. proportional (http://planetmath.org/Variation) to the arc lenth; thus in the origin only the curvature is zero.
Conversely, if the curvature of a plane curve varies proportionally to the arc length, the curve is a clothoid.
This property of the curvature of clothoid is utilised in way and railway construction, since the form of the clothoid is very efficient when a portion of way must be bent to a turn: the zero curvature of the line can be continuously raised to the wished curvature.
|Date of creation||2014-07-11 21:15:59|
|Last modified on||2014-07-11 21:15:59|
|Last modified by||pahio (2872)|