Polygasteroid

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

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

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, 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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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

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