Conoid Plücker

Fig. 1. Obtaining Plücker conoid (k = 2)

Conoid of Plücker (in honor of the German mathematician Julius Plücker ), or a cylindroid - a ruled surface of the third order, described in Cartesian coordinates by the equation:

z={\frac {kxy}{x^{2}+y^{2}}}

{\ displaystyle z = {\ frac {kxy} {x ^ {2} + y ^ {2}}}}

{\ displaystyle z = {\ frac {kxy} {x ^ {2} + y ^ {2}}}}

,

or in polar coordinates:

x(r,\theta )=r\cos \theta ,\quad y(r,\theta )=r\sin \theta ,\quad z(r,\theta )=c\sin(k\theta )

{\ displaystyle x (r, \ theta) = r \ cos \ theta, \ quad y (r, \ theta) = r \ sin \ theta, \ quad z (r, \ theta) = c \ sin (k \ theta )}

{\ displaystyle x (r, \ theta) = r \ cos \ theta, \ quad y (r, \ theta) = r \ sin \ theta, \ quad z (r, \ theta) = c \ sin (k \ theta )}

,

where k is a coefficient that determines the number of "folds" of the surface. The Plücker conoid belongs to the so-called direct conoids , and can be obtained in three-dimensional Cartesian coordinates by rotating a segment, simultaneously making oscillating motions with a period of 2π, around the applicate axis (see Fig. 1). It is used in kinematics to construct the helical axis of the composite motion from the given helical axes of two component movements.

The Plücker conoid equation in cylindrical coordinates:

x=v\cos u,\quad y=v\sin u,\quad z=\sin 2u.

{\ displaystyle x = v \ cos u, \ quad y = v \ sin u, \ quad z = \ sin 2u.}

{\ displaystyle x = v \ cos u, \ quad y = v \ sin u, \ quad z = \ sin 2u.}

Literature

Cylindroid // Brockhaus and Efron Encyclopedic Dictionary : in 86 volumes (82 volumes and 4 additional). - SPb. , 1890-1907.

Conoid Plücker

Literature

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