Quartic surface

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In mathematics, especially in algebraic geometry, a quartic surface is a surface defined by an equation of degree 4.

More specifically there are two closely related types of quartic surface: affine and projective. An affine quartic surface is the solution set of an equation of the form

$f(x,y,z)=0\$

where f is a polynomial of degree 4, such as f(x,y,z) = x⁴ + y⁴ + xyz + z² − 1. This is a surface in affine space.

On the other hand, a projective quartic surface is a surface in projective space P³ of the same form, but now f is a homogeneous polynomial of 4 variables of degree 4, so for example f(x,y,z,w) = x⁴ + y⁴ + xyzw + z²w² − w⁴.

If the base field in R or C the surface is said to be real or complex. If on the other hand the base field is finite, then it is said to be an arithmetic quartic surface.

[edit] Special quartic surfaces

Dupin cyclides
The Fermat quartic, given by x⁴ + y⁴ + z⁴ + w⁴ =0 (an example of a K3 surface).
K3 surfaces
Kummer surface
Plücker surface
Weddle surface

[edit] See also

Quadric surface (The union of two quadric surfaces is a special case of a quartic surface)
Cubic surface (The union of a cubic surface and a plane is another particular type of quartic surface)