In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, especially in
algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, a quartic surface is a
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
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
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relat ...
.
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
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
'' respectively. One must be careful to distinguish between algebraic
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
s, which are in fact
quartic curves 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 cyclide
In mathematics, a Dupin cyclide or cyclide of Dupin is any Inversive geometry, geometric inversion of a standard torus, Cylinder (geometry), cylinder or cone, double cone. In particular, these latter are themselves examples of Dupin cyclides. They ...
s
* 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 is a generalization of conic sections (ellipses, parabolas, and hyperbolas). In three-dimensional space, quadrics include ellipsoids, paraboloids, and hyperboloids.
More generally, a quadric hyper ...
(The union of two quadric surfaces is a special case of a quartic surface)
*
Cubic surface
In mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic surfaces are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than ...
(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