Cubic Surfaces
A cubic surface is the set of solutions of a cubic equation in four variables, like
We consider this object in projective space.
Sometimes we consider an affine version of the surface, given by setting one variable to one.
In this case, we use lower case letters for the remaining variables.
The drawing above is the Clebsch surface (named after Alfred Clebsch,
a German mathematician from the 19th century,
depicted on the main page of this app).
The equation is
−3+9 (x+ y+ z)
+3 (x2+ y2+ z2)
−42( x y+ x z+ y z)
−9 (x3+ y3+ z3)
+21 (x2 y+x2 z+x y2+x z2+ y2 z+ y z2)
−6 x y z = 0
The coordinate axes (x, y, and z) are shown in red, green and blue.
The brown axis is an axis of symmetry and represents (1,1,1).
File translated from
TEX
by
TTH,
version 4.08.
On 4 Jun 2017, 11:23.