Definition:Surface (Geometry)
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Definition
In the words of Euclid:
(The Elements: Book $\text{I}$: Definition $5$)
and:
(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)