In
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:
:
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
:
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
:
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'
()
External links
*
Surfaces
Geometric shapes
Eponyms in geometry
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