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LIMAÇON (OR SNAIL) OF PASCAL
cercle, limašon Ó boucle, cardio´de, limašon haricot, et 2 limašons convexes

Curve studied by Dürer in 1525 (under the name Arachne), Etienne Pascal in 1630, and Roberval (who named the curve) in 1650.
Étienne Pascal (1588 -1651): magistrate and amateur mathematician, father of Blaise.

 
Polar equation:  (a, e > 0).
Complex parametrization:  ().
Cartesian equation: .
Rational bicircular quartic (singular point at O).
Area for e1: .

 
The limaçons of Pascal are the conchoids of a circle with respect to one of its points O (here, the circle with diameter [OA] with A(ea, 0)).

Hence the mechanical construction suggested opposite, thanks to a stick of length 2a sliding through a point of a circle with diameter ea and the middle of which is constrained to describe the circle.

The curves assume different shapes depending on the cases:
 
e < 1: elliptic limaçon (inverse of an ellipse with respect to its focus), or with an isolated point, or acnodal;
     .e < 1/2: convex curve that "tends" to the circle C(O,a) when e goes to 0.
     .e = 1/2: limaçon of Pascal with a meplat. 
      .1/2 < e < 1: curve with the shape of a bean;
e = 1: cuspidal limaçon more commonly called cardioid.

e > 1: hyperbolic limaçon (inverse of a hyperbola with respect to its focus) or with a loop, or crunodal; it is a closed curve with a loop; for e = 2, we get the trisectrix limaçon, which is also a rose.

cercle, limašon Ó boucle, cardio´de, limašon haricot, et 2 limašons convexes

The complex parametrization above shows that the limaçons of Pascal are the epitrochoids generated by a point linked to a circle rolling without slipping around a circle with the same radius (here the fixed circle is the circle with diameter [OA] and the tracing point is located at distance a from the centre of the moving circle).

They are also, like all rational bicircular quartics:
    - the pedals of circles (here pedal of the circle with centre A and radius a with respect to O).
therefore, they are the envelopes of circles one of the ends of the diameters of which is fixed (here, O) while the other is a point of a circle (here, of the circle with centre A and radius a).
     - the inverses of conics with respect to one of their foci (here, of the conic with eccentricity e with respect to O).

     - the cissoids of two circles, one of which passes through the centre of the other, with respect to this centre (here, cissoid with respect to O of the circle with centre (-ea/2, 0) passing by O and the circle with centre O and radius a - this definition is in fact equivalent to the conchoidal definition).

The limaçons of Pascal are the rational Cartesian curves and are special cases of complete Cartesian ovals; when e1 (i.e. when it is not a cardioid), the bipolar equation with respect to O and O' (, 0) is: .

More precisely:
 - the hyperbolic limaçon is the reunion of two Cartesian ovals:  (big loop) and  (small loop).
 - the elliptic limaçon is the Cartesian oval:  (plus the isolated point O).

We also find the limaçons of Pascal in the Bélidor drawbridge system.

The evolutes of the limaçons are the caustics by reflection of circles (light source at finite distance).
 

The Arachnes...

The curves with equation  (cissoids of a circle passing by O and a rose) provide an elegant filing of the limaçon; here, k = 55/37 (idea of Thierry LEGAY) .
 

See here a gear shaped like a limaçon of Pascal, and on this site a technical application to the "24 hour clock".

See also the general conchoids of circles.


Image made by par Daniel Alexis


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© Robert FERRÉOL 2017