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

RATIONAL BICIRCULAR QUARTIC

En bleu une crunodale, en rouge une cuspidale, en vert une acnodale

Polar equation: .
Cartesian equation: .

Case  depending on whether ??????

The rational bicircular quartics are the bicircular quartics with a real singularity - here, O- that is necessarily unique (the cyclic points are the other two singularities); the quartic is called crunodal, cuspidal, or acnodal, depending on whether this singularity is a double point with different tangents, a cuspidal point, or an isolated point.

Like the rational circular cubics, they have the property of having 4 equivalent remarkable geometrical definitions.

1) They are the pedals of centred conics (here pedal with respect to O of the conic ).
They are crunodal, cuspidal, or acnodal depending on whether the point O is outside, on, or inside the conic.
 

crunodal case

cuspidal case

acnodal case

Examples: when the conic is a circle, we get the limaçons of Pascal (including the cardioid) and when O is the centre of the conic, we get the Booth curves (including the lemniscate of Bernoulli).

This definition as a pedal implies a definition as the roulette of a conic rolling on an equal conic, and also as a curve of the three-bar linkage in the case of the antiparallelogram.

tracing of a point on an ellipse rolling on an equal ellipse: we get a pseudo-cardioid.

2) They are the envelopes of the circles with diameters the ends of which are a fixed point (here O) and a point describing a centred conic.
 

crunodal case

cuspidal case

acnodal case

3) They are the inverses of conics with respect to a point that is not on the conic (here, the conic with equation:  where p is the square of the inversion radius).
The quartic is crunodal, cuspidal, or acnodal depending on whether the conic is a hyperbola, a parabola or an ellipse.
 

crunodal case

cuspidal case

acnodal case

4) They are the cissoids of two circles with respect to one of the points of one of these circles, the first one being the circle with centre  passing by O and the second one being the circle with centre  and radius a. The quartic is crunodal, cuspidal, or acnodal depending on whether these circles intersect, are tangent or are disjoint.
 

crunodal case

cuspidal case

acnodal case

See also, as a special case, the Dürer conchoid.
 
next curve previous curve 2D curves 3D curves surfaces fractals polyhedra

© Robert FERRÉOL 2017