Plücker's Conoid

A Ruled Surface sometimes also called the Cylindroid. von Seggern (1993) gives the general functional form as

$$ax^2+by^2-zx^2-zy^2=0,$$ (1)

whereas Fischer (1986) and Gray (1993) give

$$z={2xy\over(x^2+y^2)}.$$ (2)

A polar parameterization therefore gives

$$x(r,\theta) = r\cos\theta$$ (3)

$$y(r,\theta) = r\sin\theta$$ (4)

$$z(r,\theta) = 2\cos\theta\sin\theta.$$ (5)

A generalization of Plücker's conoid to $n$ folds is given by

$$x(r,\theta) = r\cos\theta$$ (6)

$$y(r,\theta) = r\sin\theta$$ (7)

$$z(r,\theta) = \sin(n\theta)$$ (8)

(Gray 1993). The cylindroid is the inversion of the Cross-Cap (Pinkall 1986).

See also Cross-Cap, Right Conoid, Ruled Surface

References

Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, pp. 4-5, 1986.

Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 337-339, 1993.

Pinkall, U. Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, p. 64, 1986.

von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 288, 1993.