Goursat curve

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

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

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.

	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