Concept

# Plücker's conoid

Summary
In geometry, Plücker's conoid is a ruled surface named after the German mathematician Julius Plücker. It is also called a conical wedge or cylindroid; however, the latter name is ambiguous, as "cylindroid" may also refer to an elliptic cylinder. Plücker's conoid is the surface defined by the function of two variables: : z=\frac{2xy}{x^2+y^2}. This function has an essential singularity at the origin. By using cylindrical coordinates in space, we can write the above function into parametric equations : x=v\cos u,\quad y=v\sin u,\quad z=\sin 2u. Thus Plücker's conoid is a right conoid, which can be obtained by rotating a horizontal line about the z-axis with the oscillatory motion (with period 2π) along the segment [–1, 1] of the axis (Figure 4). A generalization of Plücker's conoid is given by the parametric equations : x=v \cos u,\quad y=v \sin u,\quad z= \sin nu. where n denotes
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