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Problem posed by Jean Bernoulli in 1716, solved by Taylor in 1717; curve then studied by Ossian Bonnet in 1844 and Ribaucour in 1880.
Albert Ribaucour (1845-1893): French mathematician and engineer.
See also this link.

Second order differential equation: .
First order equation: .
Cartesian equation: .
Cartesian parametrization: , with .
Algebraic curve for k odd integer.
Curvilinear abscissa given by .
Radius of curvature: .

recognize the circle and the cycloid
For k a positive integer, we can take any value of t; the curve is closed when k is odd, periodic when k is even.
When k is positive and not an integer, take .

Case k = -1/4, -1/2, -1 (catenary), -2 (parabola).
When k negative, take ; the curve has a vertical asymptote when k> -1, and a parabolic branch otherwise.

A Ribaucour curve is a curve for which any point M on the curve satisfies  where I is the centre of curvature at M, N the intersection point between (MI) and a given line (D) (here: Ox) and k is a given constant.

In other words, with Ox as the line (D), they are the curves such that the radius of curvature is proportional to the normal: Rc = k N.

The Ribaucour curve of index k is also the locus of the pole of the sinusoidal spiral rolling without slipping on (D), with .

Finally, it is the solution to the calculus of variations problem that consists in determining the curve for which the integral  is extremal; this is why it is also the trajectory of a light ray in an inhomogeneous medium: see this page devoted to the Fermat principle.

Special case:
Index of the Ribaucour curve Nature of this curve Index
of the rolling sinusoidal spiral
Nature of this spiral figure Integral for which the Ribaucour curve is an extremum Interpretation
k = 2 parabola with directrix (D) n = 1/3 Tschirnhausen cubic, with tracing point at the focus
k = 1 catenary with base (D) n = 1/2 parabola, with tracing point at the focus
curve joining A to B for which the area of the surface spanned by its rotation around (D) is minimal, such a surface is a catenoid.
k = 1/2 see the last column n = 2/3 central negative pedal of the rectangular hyperbola
homogeneous massive wire joining A to B with minimal moment of inertia
k = 0 point n = 1 line .... ....
k = 1/2 rectangular Sturm roulette
or right lintearia
n = 2 rectangular hyperbola
k = 1 circle centred on (D) n infinite ....   geodesic in the Poincaré half-plane
k = 3/2   n = 2 lemniscate of Bernoulli see the animation above ?
k = 2 cycloid with cuspidal points on (D) n = 1 circle, with tracing point on the circle brachistochrone curve
k = 3 sextic 
n = 1/2 cardioid, with tracing point at the tip ?
k infinite line n = 0 point   the line is the shortest path from a point to another...

Nota: the equation  shows that the Ribaucour curves, for k different from 0, can be defined by ; we can then consider that the curve of order 0 is the curve defined by  which is none other than the catenary of equal strength.

Derivation of the equations from the 3 definitions above:
Proof starting with the definition by the property of the normals. Proof starting with the rolling sinusoidal spiral  Proof starting with the calculus of variations problem 
The equation Rc = - k N can be written , hence  and  which give the equation with separated variables  which leads to the solution given in the boxed paragraph above. The equation of the roulette of the pole of the curve with polar equation  is (1); but for the sinusoidal spiral and  so (1) becomes , QED. The Euler-Lagrange equation applied to the integral  leads to the differential equation , which gives, with , the differential equation . Writing , we get the above parametrization.

The Mannheim curve of a Ribaucour curve is a Ribaucour curve with parameter k 1.
The rotation of a Ribaucour curve around the Ox axis gives a rotation surface whose principal curvatures are
proportional at each point.
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© Robert FERRÉOL 2017