next curve  previous curve  2D curves  3D curves  surfaces  fractals  polyhedra 
ARTICULED ANTIPARALLELOGRAM CURVE
Home named. 
Pour des données , si on donne un mouvement
circulaire uniforme à
P : ,
le point Q a pour mouvement : . Le point M est défini par . Quartique bicirculaire rationnelle ayant un axe de symétrie (et on les obtient toutes ainsi). For data A(0,
a),
B(0, –a),
AP = BQ = 2b, AB = PQ =2a, if we give
a uniform circular motion to P: ,


Une courbe de l'antiparallélogramme
(ou du contreparallélogramme) articulé est le lieu
d'un point
M de la bielle [PQ] d'un quadrilatère articulé
(APQB) dont les côtés opposés sont de même
longueur,
A et B étant fixes  c'est donc une courbe
du troisbarres à manivelles de même longueur dont la
bielle a une longueur égale à la distance entre les pivots.
C'est le seul cas où les bielles du mécanisme du troisbarres
effectuent des rotations complètes (voir [Pécaud]).
On ne regarde que le cas ou (APQB) est croisé, le cas du parallélogramme donnant de simples cercles. Lorsque le point M est au milieu de la bielle,
on obtient les courbes de Booth.


Case where the connecting rod is shorter than the cranks
(a < b).
The two cranks intersect at a point I; the triangles IAB and IPQ being congruent, we have 2IA + 2IB = IA + IQ + IB + IP =AP + BQ = 4b: the point I describes an ellipse with foci A and B, semimajor axis b and semiminor axis . The point M is the mirror image of the point with respect to the tangent in I to the ellipse which is the axis of symmetry of the antiparallelogram. The curve described by M is therefore the orthotomic curve of the ellipse with respect to the point N. The symmetrical ellipse, with foci P and Q, rolls without slipping on the fixed ellipse. The curves considered here are therefore the pedal
curves of an ellipse with respect to a point located on the focal axis.



Case where the connecting rod is longer than the cranks (a > b). The two extended cranks intersect at a point I of the axis of symmetry of the antiparallelogram and IP = IB by symmetry. So IA – IB = IP + PA – IB = AP = 2b, and point I describes a hyperbola with foci A and B, with focal halfaxis b and transverse halfaxis . The point M is the mirror image of the point with respect to the tangent in I to the hyperbola which is the axis of symmetry of the antiparallelogram. The curve described by M is therefore the orthotomic curve of the hyperbola with respect to the point N. The symmetrical hyperbola, with foci P and Q, rolls without slipping on the fixed hyperbola. The curves considered here are therefore pedal curves of hyperbola with respect to a point located on the focal axis. The point N is a singular point of the curve: it is isolated if it is inside the ellipse, a cusp point if it is located on the ellipse (cuspidal curve), crossing point if it is outside the ellipse (crunodal curve). The isolated case therefore occurs if . Right, example of a cuspidal case: . 

Like any curve of the threebar, these curves have a triple generation by articulated system. Animation opposite due to Keishiro Ueki. 

We can generalize to a wheel of the plane associated with the connecting rod. We thus obtain all the rational bicircular quartics. 

next curve  previous curve  2D curves  3D curves  surfaces  fractals  polyhedra 
© Robert FERRÉOL
2023