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JERABEK CURVE
Curve studied by Jerabek and Neuberg in 1885.
Loria p. 239. 
Polar equation: .
For 0 < k = tha <1: . For k = cotha > 1: . Cartesian equation: . Polar equation in the frame centred on (a,0):

The Jerabek curve (G_{k}) is the locus of the end M of a right angle PAM, when the point P describes a circle (C) with centre O and radius a and the point M is on the line (OP) (here, A(a, 0) and OP = ka).
Therefore, it is the base of the movement of a plane over a fixed plane a fixed point of which is P and a fixed line of which is (PA), this movement is called circular conchoidal movement. 
When A is inside the circle (k > 1), the curve is closed, included in the circle, and has a crunodal double point and a tacnodal double point.
When A is outside the circle, there is a tacnodal double point and two asymptotes.
Besides, these two cases are inverses of one another: more precisely, the curves (G_{k}) and (G_{1/k}) are inverses of one another with respect to the circle (C).
Writing the polar equation where e = 1/k, proves that the Jerabek curve is the conchoid of a centred conic with pole a focus of this conic, and modulus the semimajor axis (see conchoid of a conic).
It is also the pedal of the evolute of a centred conic with respect to one of its foci.
When A is inside the circle, if I is the middle of [MP] , IM = IP = IA, so OI + IA = OP = constant: the point I describes an ellipse and the Jerabek curve can be obtained by decreasing the radius vector of the green ellipse starting at the focus O by a length equal to the radius vector that reaches the other focus A.
Let J be the image of I by the homothety with centre O and ratio 2; J describes the blue ellipse and JM = OP = constant: the Jerabek curve is the conchoid of the blue ellipse with respect to the focus O, the radius vector being decreased by the semimajor axis of this ellipse. M is the symmetrical image of A with respect to the normal to the green ellipse passing by I: the Jerabek curve is therefore the orthotomic with pole A of the involute of the evolute of the green ellipse. See strophoidal curves, for a strophoidal construction of the Jerabek curve using this property.

The Jerabek curve is the base of the circular conchoidal movement.
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© Robert FERRÉOL 2017