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 $0\le u,v\le 1$, 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

$$\mathrm{rank}\Vert \begin{array}{lll}{x}_u^{\prime }& y_u^{\prime }& z_u^{\prime }\\ x_v^{\prime }& y_v^{\prime }& z_v^{\prime }\end{array}\Vert =2$$

(see also Differential geometry; Theory of surfaces; Riemannian geometry).

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

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