In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, Plücker's conoid is a
ruled surface
In geometry, a surface is ruled (also called a scroll) if through every point of there is a straight line that lies on . Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, the ...
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 disc ...
. 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:
:
This function has an
essential singularity
In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits odd behavior.
The category ''essential singularity'' is a "left-over" or default group of isolated singularities that a ...
at the
origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
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.
By using
cylindrical coordinates
A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference di ...
in space, we can write the above function into parametric equations
:
Thus Plücker's conoid is a
right conoid
In geometry, a right conoid is a ruled surface generated by a family of straight lines that all intersect perpendicularly to a fixed straight line, called the ''axis'' of the right conoid.
Using a Cartesian coordinate system in three-dimensional ...
, 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
:
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 surface is ruled (also called a scroll) if through every point of there is a straight line that lies on . Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, the ...
*
Right conoid
In geometry, a right conoid is a ruled surface generated by a family of straight lines that all intersect perpendicularly to a fixed straight line, called the ''axis'' of the right conoid.
Using a Cartesian coordinate system in three-dimensional ...
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'
()
External links
*
Surfaces
Geometric shapes
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