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POLYGASTEROID


Name given by Loria in 1930.
Other name, given by Laboulaye in 1849: curve with n antinodes, or curve with n overhangs.

 
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
These curves can be obtained as the mating profiles of ellipses.

Case e = 1 (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, half-angle 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 2017