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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 nth of a turn (and noninvariant
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 3curves 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 nonzero ,
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, Lperiodic 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