Plücker's Conoid

Plücker's conoid is a ruled surface sometimes also called the cylindroid, conical wedge, or conocuneus of Wallis. von Seggern (1993, p. 288) gives the general functional form as

(1)

whereas Fischer (1986) and Gray (1997) give

(2)

A polar parameterization therefore gives

			(3)
			(4)
			(5)

The cylindroid is the inversion of the cross-cap (Pinkall 1986).

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

			(6)
			(7)
			(8)

(Gray 1997), which is a slight variation of the form called the "conical wedge" by von Seggern (1993, p. 302).

The coefficients of the first fundamental form of the generalized Plücker's conoid are

			(9)
			(10)
			(11)

and of the second fundamental form are

			(12)
			(13)
			(14)

The Gaussian and mean curvatures are given by

			(15)
			(16)