Definition:Surface (Geometry)

Definition

In the words of Euclid:

A surface is that which has length and breadth only.

(The Elements: Book $\text{I}$: Definition $5$)

and:

An extremity of a solid is a surface.

(The Elements: Book $\text{XI}$: Definition $2$)

Plane Surface

In the words of Euclid:

A plane surface is a surface which lies evenly with the straight lines on itself.

(The Elements: Book $\text{I}$: Definition $7$)

Regular Surface

A subset $S \subseteq \R^3$ is a regular surface if and only if for each $p \in S$ there exist:

a neighborhood $V \subseteq \R^3$ of $p$

an open set $U \subseteq \R^2$

a surjective mapping $\mathbf x : U \to V \cap S$, written as:

$\map {\mathbf x} {u, v} := \struct {\map x {u, v}, \map y {u, v}, \map z {u, v} }$

such that:

$(1): \quad \map x {u, v}, \map y {u, v}, \map z {u, v}$ are smooth

$(2): \quad \mathbf x: U \to V \cap S$ is a homeomorphism

$(3): \quad$ For each $q \in U$, the differential $\d_q \mathbf x: \R^2 \to \R^3$ of $\mathbf x$ at $q$ is one-to-one

Examples

Arbitrary Plane Surface

The locus of the equation in Cartesian $3$-space:

$z = x + y$

is a plane surface.

Spherical Surface

The locus of the equation in Cartesian $3$-space:

$x^2 + y^2 + z^2 - 4 = 0$

is the surface of a sphere of radius $2$ whose center is at the origin.

Cylindrical Surface

The locus of the parametric equations in Cartesian $3$-space:

$\begin {cases} x & = & r \cos \theta \\ y & = & r \sin \theta \\ z & = & \lambda \end {cases}$

is a cylindrical surface whose cross-section is a circle with radius $r$.

Also see

• Results about surfaces can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): surface
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): surface
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): surface (of a solid)