Plücker’s conoid

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 $n$ 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.

References

• 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
 Title Plücker’s conoid Canonical name PluckersConoid Date of creation 2013-03-22 16:44:02 Last modified on 2013-03-22 16:44:02 Owner Mravinci (12996) Last modified by Mravinci (12996) Numerical id 4 Author Mravinci (12996) Entry type Definition Classification msc 51M04 Classification msc 51M20 Classification msc 14J25 Synonym Plucker’s conoid Synonym Plücker conoid Synonym Plucker conoid Synonym conical wedge Synonym conocuneus of Wallis Synonym Wallis conocuneus