# Surface

One of the basic concepts in geometry. The definitions of a surface in various fields of geometry differ substantially.

In elementary geometry, one considers planes, multi-faced surfaces, as well as certain curved surfaces (for example, spheres). Each curved surface is defined in a special way, very often as a set of points or lines. The general concept of surface is only explained, not defined, in elementary geometry: One says that a surface is the boundary of a body, or the trace of a moving line, etc.

In analytic and algebraic geometry, a surface is considered as a set of points the coordinates of which satisfy equations of a particular form (see, for example, Surface of the second order; Algebraic surface).

In three-dimensional Euclidean space $E ^ {3}$, a surface is defined by means of the concept of a surface patch — a homeomorphic image of a square in $E ^ {3}$. A surface is understood to be a connected set which is the union of surface patches (for example, a sphere is the union of two hemispheres, which are surface patches).

Usually, a surface is specified in $E ^ {3}$ by a vector function

$$\mathbf r = \mathbf r ( x( u , v), y( u , v), z( u , v)),$$

where $0 \leq u , v \leq 1$, while

$$x = x( u, v),\ \ y = y( u, v),\ \ z = z( u, v)$$

are functions of parameters $u$ and $v$ that satisfy certain regularity conditions, for example, the condition

$$\mathop{\rm rank} \left \| \begin{array}{lll} x _ {u} ^ \prime &y _ {u} ^ \prime &z _ {u} ^ \prime \\ x _ {v} ^ \prime &y _ {v} ^ \prime &z _ {v} ^ \prime \\ \end{array} \right \| = 2$$

From the point of view of topology, a surface is a two-dimensional manifold.