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
,
a surface is defined by means of the concept of a
surface patch
— a
homeomorphic image of a square in
.
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
by a vector function
where
,
while
are functions of parameters
and
that satisfy certain regularity conditions, for example, the condition
(see also
Differential geometry;
Theory of surfaces;
Riemannian geometry).
From the point of view of topology, a surface is a
two-dimensional manifold.