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 (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.