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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 halflength 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. 

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© Robert FERRÉOL
2019