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CORNU SPIRAL OR CLOTHOID
Curve studied by Jacques Bernoulli in 1705, Euler in
1743, Fresnel in 1818, Cornu in 1874, Cesaro in 1886 (who gave it
the name clothoid).
Other names: Euler spiral, Fresnel spiral, volute spiral, arc radioid. In German: Spinnlinie. Alfred Cornu (18411902): French physicist. Clothoid comes from the Greek klothein: spin (wool), the shape of the curve reminds of the shape of the thread wrapping itself around the weaving loom. Clotho was the one of the three Parcae who spun human beings' destiny. There is no link with the English word cloth. Transcendental curve. 
Kinematic definition:
(2 among these three equations are enough).
Cartesian parametrization: Complex parametrization: . Transcendental curve. Cartesian tangential angle: . Curvilinear abscissa: s = a t. Radius of curvature: . Intrinsic equation 1: . Intrinsic equation 2: . The asymptotic points and can be determined thanks to the Fresnel integrals . 
The Cornu spiral, or clothoid, can be defined as the planar
curve for which the curvature is proportional to the curvilinear abscissa.
Since the distance
between two points on the curve is ,
this curve, once it was traced, gave (before the modern computational abilities)
approximate values of this integral, that appeared in calculations of the
resulting vibration during diffraction phenomena; it is reason why this
curve was used by the physicist Cornu.
Kinematic definition: the clothoid is the curve for which,
when it is travelled along at constant speed v, the curvature varies
in proportion to time.
The angle of rotation of the steering wheel of a car
is proportional to the angle of the wheels, which itself is equivalent,
in a neighbourhood of 0, to the curvature multiplied by the distance between
the front and back wheels of the car. Therefore, the clothoid approximates,
in its central part, the curve described by a car driven at constant speed
by a driver turning the wheel at constant speed. This curve is referred
to as curve of constant gyration,
hence its use in the tracing of curves of highways.
Furthermore, as any curve with a point of zero curvature,
segments of lines and of clothoids can be connected in such a way that
the curvature varies continuously along the curve. Therefore, the centrifugal
force, to which is subject an observer travelling continuously along this
curve, varies continuously, which is not the case for a connection circleline,
hence the use of clothoids for railway tracks too. These curves are, more
generally, referred to as radioids
(hence the name arc radioid given to the clothoid).




Plot (corresponding to the previous diagram) of the curvature against the curvilinear abscissa. It can be noted that for the arcs of clothoids, it is continuous (but not differentiable) along the whole trajectory. 
The clothoid also appears in the problem of determining the shortest trajectory joining two points on the plane, with a bounded derivative of the curvature, given tangents and curvatures at the beginning and the end, and continuous tangent and curvature. The optimal trajectory is an arc of a clothoid or a segment of a line.
The Mannheim curve
of the clothoid is a rectangular hyperbola and its radial
curve a lituus.
A generalisation of the clothoid is the curve for which
the curvature is proportional to the nth power of the curvilinear
abscissa (the classic clothoid being the case n = 1). It is studied
under the name pseudospiral
of Pirondini (opposite, the case n = 2); it so happens that
the evolute of the clothoid is a pseudospiral of index n = 3).
Other generalisation: case where the curvature is a polynomial in s. 
See also the anticlothoid, which is none other than the involute of a circle.
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© Robert FERRÉOL 2017