Sinusoidal cone

SINUSOIDAL CONE

n = 1

n = 2

n = 3
n = 4

n =1/2 n =1/2, but the figure only shows the half-cone; the directrix is a Viviani curve, and this cone could be called Viviani cone.

Cylindrical equation: (surface with rotation symmetry).
Cartesian parametrization: ().
Algebraic surface when n is rational.

A sinusoidal cone is a cone a directrix of which is a sine wave cylinder (here ) and vertex the center of this curve.

The section by a cylinder with axis Oz is the reunion of two sine wave cylinders (symmetric images of one another about O).

The section by a sphere centered on O is the reunion of two cylindric sine waves (symmetric images of one another about O).
The section by a horizontal plane z = b is an epispiral: .
Compare to Plücker's conoid.

We can also consider the secantoidal cone, with cylindrical equation , i.e. , for which the section by a horizontal plane is a rose: .

Pencil shaving, by Levi Capareda

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n = 1	n = 2
n = 3	n = 4
n =1/2	n =1/2, but the figure only shows the half-cone; the directrix is a Viviani curve, and this cone could be called Viviani cone.