Articuled antiparallelogram

ARTICULED ANTIPARALLELOGRAM CURVE

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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: ,
the point Q has for motion: .
The point M is defined by .
Rational bicircular quartic having an axis of symmetry (and we get them all this way).

Une courbe de l'antiparallélogramme (ou du contre-parallélogramme) 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 trois-barres 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 trois-barres 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.
A curve of the articulated antiparallelogram is the locus of a point M of the connecting rod [PQ] of an articulated quadrilateral (APQB) whose opposite sides are of the same length, A and B being fixed - it is therefore a curve of the three-bar with cranks of the same length whose connecting rod has a length equal to the distance between the pivots. This is the only case where the connecting rods of the three-bar mechanism make full rotations (see [Pecaud]).
We only look at the case where (APQB) is crossed, the case of the parallelogram giving simple circles.
When the point M is in the middle of the connecting rod, we obtain the curves of 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, semi-major axis b and semi-minor 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.
The point N is a singular point of the curve: it is isolated if it is inside the ellipse (case of the figure opposite), a cusp point if it is located on the ellipse (cuspidal curve), a crossing point if it is exterior to the ellipse (crunodal curve).
The isolated case, where the curve has an egg shape, therefore occurs if .
Right, example of a cuspidal case: .

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 half-axis b and transverse half-axis .
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 three-bar, 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.

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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, semi-major axis b and semi-minor 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. The point N is a singular point of the curve: it is isolated if it is inside the ellipse (case of the figure opposite), a cusp point if it is located on the ellipse (cuspidal curve), a crossing point if it is exterior to the ellipse (crunodal curve). The isolated case, where the curve has an egg shape, therefore occurs if . Right, example of a cuspidal case: .
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 half-axis b and transverse half-axis . 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 three-bar, 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.