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GOURSAT CURVE

Homemade name, given as a tribute to Goursat, who studied the surfaces with the symmetries of the regular polyhedra.
Other names: curve with rotation symmetry, curve with radial symmetry.

The Goursat curves of order n are the curves having the symmetries of a regular polygon with n sides, i.e. for which the group of isometries that leave it invariant is that of this polygon, namely the dihedral group of order 2n.

A curve is therefore a Goursat curve of order n iff it is invariant by a rotation of an n-th of a turn (and non-invariant by a rotation by a smaller angle) and it has an axis of symmetry, or iff it has exactly n axes of symmetry.
 
 
General polar equation of a Goursat curve of order a multiple of n: Cartesian equation:
with f even (or odd) and with minimal period with respect to , with f even with respect to y and 
hence the general form: 
(Brocard transformation of the curve ).
general form:
with ;
for even values of n,
for odd values of n

Examples: the roses and their inverses the epispirals, the sinusoidal spirals.

Special cases:

1) n multiple of 2:
General polar equation:  (or ) General Cartesian equation: 
in other words:  (axes Ox and Oy), or also  (axes ).

Examples with exactly two symmetry axes:
degree 2: ellipses, hyperbolas.
degree: the lemniscate of Bernoulli, the lemniscate of Gerono,  the bullet nose curve, the Kampyle Eudoxus, the Kappa curve, the double U, the Kulp quartic, the Delanges trisectrix, the Alain curves, the devil's curves.
ETC....
 

2) n multiple of 3:
General polar equation: General Cartesian equation : 
Remark: 

Examples with exactly three symmetry axes:
degree 3: the equilateral trefoil: and the Humbert cubic: (that are the only cubical Goursat 3-curves up to similarity) .

degree 4: the hypotrochoids with parameter q = 3: (including the deltoid, k = 1 and the regular trifolium, k = 2) , the conchoids of a regular trifolium, the Klein quartic, the Loriga quartic.

3) n multiple of 4:
General polar equation:   or , or also  with f even with respect to the second variable. General Cartesian equation: , in other words  , or also  with f symmetrical.

Examples with exactly four symmetry axes:
degree 4: the rectangular crosscurve, the Salmon quartics .

degree 6:  the hypotrochoids with parameter q = 4: (including the astroid and the quadrifolium), the conchoids of roses with parameter 4 with b non-zero , the windmill,
the Loriga sextic :.

4) n multiple of 5:
General polar equation: General Cartesian equation: .

Examples with exactly five symmetry axes:
degree 5: general Cartesian equation: (one of the 3 numbers k, k', k" being arbitrarily chosen).
(k,k',k")=(1,0,0) gives the epispiral of order 5  gives the curve with 5 double points: 
(k,k',k")=(0,-1,0.2) gives 
 

5) n multiple of 6:
General polar equation:  General Cartesian equation: .

Examples with exactly six symmetry axes:
degree 6:  and 
 

Examples of infinite families:

    - the conchoids of roses are Goursat curves of order n (except for even values of n and b = 0: order 2n)

    - all the families of curves defined symmetrically from n points , vertices of a regular polygon (cf. the Curie principle); in particular, the curves (: isophonic curves: Cayley equipotential lines: ?, : circles), the Loriga curves, and the Cassinian curves: .
 
 
All the curves with polar equation  where f is -periodic (m and n are coprime) and even (or odd) are Goursat curves of order a multiple of n.
Example opposite: with n = 5, m = 3.
More generally, given a complex function f, with the same properties, the curve with complex parametrization  is a Goursat curve of order a multiple of n (the previous case being the case where f is real).
This case includes the epi- and hypotrochoids ().
The more general case gives the polytrochoids.
Opposite, the tritrochoid obtained for .
Given f with the same properties, the curve with complex parametrization  is, also, a Goursat curve of order a multiple of n.

This case includes the generalised curves with sinusoidal radius.

The curves defined by an intrinsic equation:  with f even, L-periodic and such that  is a rational number m/n that is not an integer (m and n are coprime) are Goursat curves of order a multiple of n. Furthermore, the integer m is the rotation index of the curve.
Equivalent form:  with f even,-periodic with zero mean.
Example opposite: n = 5, m = 3, .
We get this way all the Goursat curves of order n and nonzero rotation index.

Classification of the spherical generic curves, Goursat curves of order n, having exactly n double points.

There are exactly 3 kinds, one of which is composed of curves with odd order n:
 
2 double points 3 double points 4 double points 5 double points
First kind
Hypotrochoid with parameter q = n, except for n = 2.
Rotation index: n -1

Lissajous x = cos t , y = sin 3t.

Second kind
Epitrochoid with parameter q = n.
Rotation index: n +1
Third kind
Conchoid of a rose with parameter n/2
Rotation index: 2
 
 

See also the Goursat surfaces, and the surfaces with rotation symmetry.
 
 
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© Robert FERRÉOL 2017