PlanetMath: Plücker's conoid

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Plücker's conoid

(Definition)

Plücker's conoid is a ruled surface that results from taking a straight line connected to an axis, rotating it about that axis and moving it straight up and down the axis to give the desired number of folds. Being an example of a right conoid, Plücker's conoid is sometimes called a conical wedge, or a conocuneus of Wallis or even a cylindroid.

The Cartesian equation for a conoid with two folds is $z = \frac{x^2 - y^2}{x^2 + y^2}$ . This can be generalized to any desired number of folds as $x(r, \theta) = r \cos \theta$ , $y(r, \theta) = r \sin \theta$ and $z(r, \theta) = c \sin (n\theta)$ . Plücker's conoid has applications in mechanical drafting.

Bibliography

1: J. Plücker, ``On a new geometry of space'', Philosophical Transactions of the Royal Society of London 155 (1965): 725 - 791
2: S. P. Radzevich, ``A Possibility of Application of Pliicker's Conoid for Mathematical Modeling of Contact of Two Smooth Regular Surfaces in the First Order of Tangency'', Mathematical and Computer Modelling 42 (2005): 999 - 1022

"Plücker's conoid" is owned by Mravinci.

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Other names:

Plucker's conoid, Plücker conoid, Plucker conoid, conical wedge, conocuneus of Wallis, Wallis conocuneus

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Cross-references: equation, cylindroid, even, right, folds, axis, connected, line, straight, ruled surface
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This is version 1 of Plücker's conoid, born on 2007-02-23.
Object id is 8956, canonical name is PluckersConoid.
Accessed 1332 times total.

Classification:

AMS MSC:	51M04 (Geometry :: Real and complex geometry :: Elementary problems in Euclidean geometries)
	51M20 (Geometry :: Real and complex geometry :: Polyhedra and polytopes; regular figures, division of spaces)
	14J25 (Algebraic geometry :: Surfaces and higher-dimensional varieties :: Special surfaces)

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