# Surface (geometry)

In common language, a **surface** is the exterior face of an object in space (a body),
and is usually considered as part of that object.

Some examples of surfaces are:

- The surface of a ball — called a sphere — is completely uniform.
- The surface of a cube can be seen as six squares that are glued together along their edges.
- The surface of a ring — called a torus — is markedly different from both the sphere and the cube because it has a "hole".
- A piece of rock or mineral may be irregular, crumpled, distorted, but it still has a surface.

Starting from this intuitive idea, over the centuries, the mathematical notion – or rather: several related mathematical notions – of a surface has emerged.

The essential feature of a surface (as an abstract geometrical object) is its two-dimensionality: It has length and breadth, but no depth — and this is also the common property of the mathematical definitions.

Surfaces that are the face of a body are *two-sided*: They have an interior and an exterior side.
Such surfaces are called *orientable*.

But not all abstract surfaces defined in mathematics can be interpreted as the outside hull of some body.
Such surfaces are *one-sided* and are called *non-orientable*.
A well-known example of a non-orientable surface is the Moebius strip.

**Remark:**

In higher-dimensional spaces (dimension ≥ 4)
the term *surface* or **hypersurface** is used for higher-dimensional analogues of common surfaces.

## Mathematical definitions

In analytic geometry and in differential geometry a surface can be described

*explicitly*by a real function of two variables

and the surface is the graph of the function, i.e., the set of points

- ,

or it can be defined

*implicitly*by the zeroes of a function of three variables

i.e., the surface consists of the points

- .

In topology a surface is defined as a topological space such that

- every point has a neighbourhood that is homeomorphic to the (open) unit disk in .

i.e., the space locally "looks" like a plane.

## Mathematical example: the sphere

A sphere (with radius *r*) is implicitly given by

using the function

- .

An explicit form using the function

is

which, however, only describes the upper half of the sphere.