Centred trochoid

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).
See alsao: Eric Guiot, Trajectory under harmonic potential and magnetic force.

Complex parametrization: , , and for an epitrochoid, for a hypotrochoid.
Differential equation: , so .

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

Interprétation électromagnétique (voir lien Éric Guiot).
En écrivant l'équation différentielle sous la forme , on voit que ces courbes sont les trajectoires d'une particule chargée soumise à une force centrale d'intensité poportionnelle à la distance, et à un champ magnétique d'intensité constante perpendiculaire au plan de la courbe.
Avec , on a donc , et . On obtient alors une hypotrochoïde pour k < 0, et une épitrochoïde pour .
Notons que si , alors sont complexes conjugués, et on obtient alors une spirale de la tige en rotation.
Electromagnetic interpretation (see Éric Guiot link).
By writing the differential equation in the form , we see that these curves are the trajectories of a
charged particle subjected to a central force of intensity proportional to the distance, and to a magnetic field of constant intensity perpendicular to the plane of the curve.
With , we therefore have , and . We then obtain a hypotrochoid for k < 0, and an epitrochoid for .
Note that if , then are complex conjugate, and then we get a spiral of the rotating rod.

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.


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.