In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, in the fields of differential geometry and algebraic geometry

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

, the Enneper surface is a surface that can be described parametrically by:

It was introduced by Alfred Enneper

Alfred Enneper

Alfred Enneper earned his PhD from the Georg-August-Universität Göttingen in 1856 for his dissertation about functions with complex arguments. After his habilitation 1859 in Göttingen he was from 1870 on Professor at Göttingen. He studied minimal surfaces and parametrized Enneper's minimal...

 in connection with minimal surface

Minimal surface

In mathematics, a minimal surface is a surface with a mean curvature of zero.These include, but are not limited to, surfaces of minimum area subject to various constraints....

 theory.

Figure 1. An Enneper surface

Implicitization methods of algebraic geometry

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

 can be used to find out that the points in the Enneper surface given above satisfy the degree-9 polynomial

Polynomial

In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

 equation

Figure 2. The Enneper surface in Figure 1 has been rotated 30° around the +z axis.

Figure 3. The Enneper surface in Figure 1 has been rotated 60° around the +z axis.

Dually, the tangent plane at the point with given parameters is where

Its coefficients satisfy the implicit degree-6 polynomial equation

Enneper's is a minimal surface

Minimal surface

In mathematics, a minimal surface is a surface with a mean curvature of zero.These include, but are not limited to, surfaces of minimum area subject to various constraints....

. The Jacobian

Jacobian

In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector- or scalar-valued function with respect to another vector. Suppose F : Rn → Rm is a function from Euclidean n-space to Euclidean m-space...

, Gaussian curvature

Gaussian curvature

In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, κ1 and κ2, of the given point. It is an intrinsic measure of curvature, i.e., its value depends only on how distances are measured on the surface, not on the way...

 and mean curvature

Mean curvature

In mathematics, the mean curvature H of a surface S is an extrinsic measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space....

are