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CYCLOID
Curve studied by Charles Bouvelles in
1501, Mersenne and Galileo in 1599, Roberval in 1634, Torricelli in
1644 etc...! This curve was not known by the Greeks.
From the Greek kuklos: circle, wheel; name given by Galileo. 
Cartesian parametrization:
(see here for the calculation).
Other parametrization: (see brachistochrone of given length). Complex parametrization: . Cartesian equation: . Differential equation: or , itself a first integral of the equation , that represents the fact that R_{C} = –2N (where N is the "normal"). It also satisfies and . Curvilinear abscissa: 1) (i.e. ) or 2) . Cartesian tangential angle: 1) or 2) . Curvature radius: 1) or 2) . Intrinsic equation 1 (1st form)): . Intrinsic equation 2 (1st form)): . Length of an arch: 8R; area: 3pR^{2}. 
The cycloid is the curve described by a point on a circle with radius R rolling without slipping on a line (D) (here the axis Ox); it is therefore a special case of roulette.
The cycloid can also be defined as the trajectory of a movement composed of a uniform linear motion and a uniform circular motion of equal speed (with complex parametrization ). In other words, if you move forward evenly along a blackboard while holding a chalk stick in your hand and moving it according to an even circular motion with the same speed as your proper motion (in a direction or the other), you trace a cycloid (without the condition that the speeds be equal, we would get a trochoid).
The cycloid is a special case of cycloidal curve, together with the epicycloids and the hypocycloids; it is also a Ribaucour curve.
The cycloid can also be defined by the fact that given two parallel lines, the two points of intersection I and N of the tangent and the normal with these two lines are such that the line (IN) remains orthogonal to these two lines. 
The evolute of the cycloid is a translated cycloid:
The involute of a cycloid passing by one of its vertices is therefore also a translated cycloid.
Therefore, the cycloid is, in a way, a fixed point for the function involute, and the associated "convergence" theorem, opposite, holds:  If we start from a compact parametric curve D_{0} of class C^{2} without inflection points and for which the tangents at the extremities A = A_{0} and B = B_{0} are orthogonal, its involute D_{1} that passes by A will have the same properties: let A_{1}=A and B_{1} be its extremities. We consider the involute D_{2} of D_{1} that passes by B_{1}, and we repeat the process. Every curve D_{n} obtained is brought back into one rectangle thanks to appropriate translations of direction the tangent at B to D, in such a way that the origin of D_{2n+1} is A and the origin of D_{2n} is B. Then D_{2n+1} and D_{2n} converge uniformly towards two semiarcs of a right cycloid. 
The involute passing by the cuspidal point is the curve (parallel to a cycloid) with parametrization: .
It so happens that it also is, up to homothety, the roulette of the cuspidal point of a cardioid rolling externally on a cycloid of equal length! (If the cardioid rolled internally, the roulette would be linear) 
The diameter of the tracing circle envelopes another cycloid, which is the image of the initial cycloid by a homothety with ratio 1/2. It so happens that the caustic at infinity for rays perpendicular to the rolling axis is the very same cycloid.

The radial curve of the cycloid is a circle with radius 2R:
Its Mannheim curve is also a circle, with radius 4R. It is the underlying property thanks to which this strange curvature of a disk into a scallop shell with cycloidal edges is possible. The animation is due to Gérard Lavau and based on an idea of Samuel Boureau.
The roulette of the tip of a cardioid rolling on a cycloid of equal length is linear:
See other remarkable properties of the cycloid at brachistochrone, tautochronous, isochronous, caustic and radial.
See also the minimal surface of Catalan, the only minimal surface for which the cycloid is a geodesic.
Proof due to Roberval (1636) of the fact that the area of an arch of a cycloid is equal to that of 3 generating disks:




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© Robert FERRÉOL, Jacques MANDONNET 2017