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POLYGASTEROID
Name given by Loria in 1930.
Other names: curve with n antinodes, or curve with n overhangs ( Laboulaye, 1849), generalized conic (Mamikon, 2012). 
Polar equation:
with positive
n.
Algebraic curve iff n is rational. 
The polygasteroids are the inverses of the
conchoids
of roses.
They also are the conical projections of cylindric
sine waves on a plane perpendicular to the axis of the cylinder, when
the centre is on the axis.

Conchoid of a rose in blue, polygasteroid in red 

Remark: the "monogasteroids" (n = 1) are none other than the conics. Therefore, the polygasteroids are the Brocard transforms of conics with respect to one of their poles.
The curve is composed of a base pattern symmetrical about
Ox,
obtained for :
base pattern for e < 1 
base pattern for e = 1 (parabolic branch) 
base pattern for e > 1 (branch with asymptotes) 
transformed by all the rotations by ,
when k is an integer.
When n is rational and its numerator is p,
p rotations generate the whole curve.
Case e < 1: we get shapes similar to inverse
conchoids
of roses.
For n = p / q, the polygasteroid with parameter
n
is one of the possible projection of the
Turk's
head knot of type (p,q); its p external vertices and
its p internal vertices form a regular polygon, and it has
p(q
– 1) double points.
n = 1: ellipse 
n = 2: Booth oval 
n = 3 
n = 4 
n = 5 
n = 1/2 
n = 3/2 projection of the trefoil knot 
n = 5/2 projection of the 5.1 knot 
n = 7/2 
n = 9/2 
n = 1/3 
n = 2/3 projection of the eight knot 
n = 4/3 projection of the 8.18 knot 
n = 5/3 projection of the 10.123 knot 
n = 7/3 
n = 1/4 
n = 3/4 projection of the 9.40 knot 
n = 5/4 
n = 7/4 
n = 9/4 
n = 1/5 
n = 2/5 
n = 3/5 
n = 4/5 
n = 6/5 
Case e = 1 : equation becomes (compare to the epispirals).
n = 1: parabola 
n = 2: Kampyle of Eudoxus 
n = 3 
n = 4 
n = 5 
n = 1/2 
n = 3/2 
n = 5/2 
n = 7/2 
n = 9/2 
n = 1/3 
n = 2/3 
n = 4/3 
n = 5/3 
n = 7/3 
n = 1/4 
n = 3/4 
n = 5/4 
n = 7/4 
n = 9/4 
n = 1/5 
n = 2/5 
n = 3/5 
n = 4/5 
n = 6/5 
Case e > 1:
n = 1: hyperbola 
n = 2 
n = 1/2 
n = 3/4 
The polygasteroids are the planar expansions of the planar section of the cone of revolution.
If the plane of the conic is winded into a cone with vertex O, halfangle at the vertex , and axis Oz, then the projection on xOy of this winded conic is the polygasteroid , which provides a construction of the latter in the case n < 1.
These curves can also be obtained as the mating profile of a line.
The evolutes of the tractories
of circles as well as the projections on the symmetry plane of the
rhumb
lines of the open torus are polygasteroids.
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© Robert FERRÉOL 2024