In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, especially in
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, 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
:
where is a polynomial of degree 4, such as . This is a surface in
affine space .
On the other hand, a projective quartic surface is a surface in
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
of the same form, but now is a
''homogeneous'' polynomial of 4 variables of degree 4, so for example .
If the base field is or the surface is said to be ''
real'' or ''
complex'' respectively. One must be careful to distinguish between algebraic
Riemann surfaces, which are in fact
quartic curve
In algebraic geometry, a quartic plane curve is a plane algebraic curve of the fourth degree. It can be defined by a bivariate quartic equation:
:Ax^4+By^4+Cx^3y+Dx^2y^2+Exy^3+Fx^3+Gy^3+Hx^2y+Ixy^2+Jx^2+Ky^2+Lxy+Mx+Ny+P=0,
with at least one o ...
s over , and quartic surfaces over . For instance, the
Klein quartic is a ''real'' surface given as a quartic curve over . If on the other hand the base field is finite, then it is said to be an ''arithmetic quartic surface''.
Special quartic surfaces
*
Dupin cyclides
* The
Fermat quartic, given by ''x''
4 + ''y''
4 + ''z''
4 + ''w''
4 =0 (an example of a K3 surface).
* More generally, certain
K3 surfaces are examples of quartic surfaces.
*
Kummer surface
*
Plücker surface
*
Weddle surface
See also
*
Quadric surface
In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is de ...
(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)
References
*
*{{Citation , last1=Jessop , first1=C. M. , title=Quartic surfaces with singular points , url=http://digital.library.cornell.edu/cgi/t/text/text-idx?c=math;idno=04290002 , publisher=Cornell University Library , isbn=978-1-4297-0393-2 , year=1916
Complex surfaces
Algebraic surfaces