next curve previous curve 2D curves 3D curves surfaces fractals polyhedra

CENTRED TROCHOID


Curve studied by Dürer (1525), Desargues (1640), Huygens (1679), Leibniz, Newton (1686), de L'Hospital (1690), Jacob Bernoulli (1690), la Hire(1694), Johann Bernoulli (1695), Daniel Bernoulli (1725), Euler (1745, 1781).

 
 
Complex parametrization:  for an epitrochoid and  for a hypotrochoid ().

The term centred trochoid allows to regroup the epi- and hypotrochoids. The centred trochoids are therefore the trajectories of the motions composed of two uniform circular movements.
They include the centred cycloids (case , equal speeds) and the roses (case , equal radii).
When  (equal centripetal accelerations), we get the trochoids with a meplat.
 
 
Epitrochoids with a meplat (starting with the limaçon)
Hypotrochoids with a meplat

 
The expression can be seen as a vectorial sum...
... or as the middle of two points, when written ; the two points describe concentric uniform circular motions (opposite a case  (rose), the two circles are identical).
Writing , we can separate the circles described by each of the points.
Writing , inversely, we obtain all the centred trochoids as the loci of the barycentres with given weights of two uniform circular motions on the same circle.
Besides, all the barycentres with fixed coefficients of two points describing uniform circular motions describe centred trochoids.
Image drawn with geogebra by Andre Chauviere.

The centred trochoids are also the projections on the plane xOy of the satellite curves.

This notion can be generalised to a finite number n of uniform circular motions, in any directions, under the name polytrochoid.
 

The Ceva trisectrix, Freeth's nephroid and the torpedo are examples of tritrochoids, as well as this elegant dissymmetrical quintifolium:
.

An example of a 2n +1-trochoid is the 2n +1-sectrix of Ceva with complex parametrization .
 
Another example of tritrochoid









It is a trajectory of this type that describe the fans of magical cauldron at the Asterix park:

The cauldrons describe epitrochoids; if the cauldron turns around itself, the curve described by its occupants is a tritrochoid.

The 3D generalisation is the notion of spherical trochoid.
 
next curve previous curve 2D curves 3D curves surfaces fractals polyhedra

© Robert FERRÉOL  2017