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geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, Plücker's conoid is a
ruled surface In geometry, a Differential geometry of surfaces, surface in 3-dimensional Euclidean space is ruled (also called a scroll) if through every Point (geometry), point of , there is a straight line that lies on . Examples include the plane (mathemat ...
named after the German mathematician
Julius Plücker Julius Plücker (16 June 1801 – 22 May 1868) was a German mathematician and physicist. He made fundamental contributions to the field of analytical geometry and was a pioneer in the investigations of cathode rays that led eventually to the di ...
. 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. This function has an essential singularity at the origin. By using
cylindrical coordinates A cylinder () has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infinite ...
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 with the oscillatory motion (with period 2''π'') along the segment 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 denotes the number of folds in the surface. The difference is that the period of the oscillatory motion along the is . (Figure 5 for ) File:Plucker conoid (n=2).gif, Animation of Plucker's conoid with File:Plucker's conoid (n=2).jpg, Plucker's conoid with File:Plucker's conoid (n=3).jpg, Plucker's conoid with File:Plucker's conoid (n=2).gif, Animation of Plucker's conoid with File:Plucker's conoid (n=3).gif, Animation of Plucker's conoid with File:Plucker's conoid (n=4).jpg, Plucker's conoid with


See also

*
Ruled surface In geometry, a Differential geometry of surfaces, surface in 3-dimensional Euclidean space is ruled (also called a scroll) if through every Point (geometry), point of , there is a straight line that lies on . Examples include the plane (mathemat ...
* Right conoid *
Wallis's conical edge In geometry, Wallis's conical edge is a ruled surface given by the parametric equations : x=v\cos u,\quad y=v\sin u,\quad z=c\sqrt where , and are constants. Wallis's conical edge is also a kind of right conoid. It is named after the English math ...


References

* A. Gray, E. Abbena, S. Salamon, ''Modern differential geometry of curves and surfaces with Mathematica'', 3rd ed. Boca Raton, Florida:CRC Press, 2006.

() * Vladimir Y. Rovenskii, ''Geometry of curves and surfaces with MAPLE'

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External links

* Surfaces Geometric shapes Eponyms in geometry {{algebraic-geometry-stub