A cubic algebraic surface given by the equation

(1)

with the added constraint

(2)

The implicit equation obtained by taking the plane at infinity as is

(3)

(Hunt 1996), illustrated above.

On Clebsch's diagonal surface, all 27 of the complex lines (Solomon's seal lines) present on a general smooth cubic surface are real. In addition, there are 10 points on the surface where 3 of the 27 lines meet. These points are called Eckardt points (Fischer 1986ab, Hunt 1996), and the Clebsch diagonal surface is the unique cubic surface containing 10 such points (Hunt 1996).

If one of the variables describing Clebsch's diagonal surface is dropped, leaving the equations

			(4)
			(5)

the equations degenerate into two intersecting planes given by the equation

(6)

Fischer, G. (Ed.). Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Kommentarband. Braunschweig, Germany: Vieweg, pp. 9-11, 1986a.

Fischer, G. (Ed.). Plates 10-12 in Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig, Germany: Vieweg, pp. 13-15, 1986b.

Hunt, B. The Geometry of Some Special Arithmetic Quotients. New York: Springer-Verlag, pp. 122-128, 1996.

Nordstrand, T. "Clebsch Diagonal Surface." http://jalape.no/math/clebtxt.

CITE THIS AS:

Weisstein, Eric W. "Clebsch Diagonal Cubic." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ClebschDiagonalCubic.html