HOLDITCH CURVE

 Curve studied by Holditch in 1858, H. Benitez in 1980, et Monterde & Rochera in 2017. Animations below made by Alain Esculier.

 To obtain the Holditch curve of , of equation f(x, y) = 0 : eliminate between the equations , with .

 The Holditch curves associated with a given curve are the locations of the fixed points M of a straight line whose two fixed points A, B, belong to . Holditch considered these curves because when is closed, and under certain conditions, he prooved that the difference between the area enclosed by and that enclosed by the Holditch curve is , where (see the Holditch theorem). These curves are cases of glissettes, the two points A and B "sliding" on the curve (see the special case number 2 in the page on the glissettes). Obtaining the equation of the Holditch curve is usually difficult, but here are two simple cases:

 For an ellipse: , with a chord of length 2c, the plotter point being at the center, the Holditch curve has a Cartesian equation: (quartic of genius 2). Polar equation : . For an hyperbola, change the sign in front of y². Loss of convexity for c > p = b²/a. For a parabola: , the Holditch curve, for a chord of length 2c, the tracer point being located in the center, has a Cartesian equation: (quartic). Loss of convexity for c > p . Here is the evolution of the complete Holdich curve of the ellipse, for a plotter point located between A and B, out of the center, rope length 2c. For , it is formed of two continuous curves symmetrical with respect to the axis of the ellipse.
Each curve is ...

 convex for , closed without a double point, traversed in a movement without retrograde phase, where the rope returns to its starting point after having made a complete turn, for , closed without a double point, traversed by a movement with retrograde phase, where the rope returns to its starting point after having made a complete turn, for . The number d is the minimum half-length of a chord whose end is perpendicular to the ellipse (instant ending the retrograde phase, see the animation opposite). eight shaped, traversed by a movement with a retrograde phase where the rope returns to its point of departure without rotation, for b < c < a.  Animation of the deformation of the complete curve for 0 < c < a. 