Definition:Surface (Geometry)

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In the words of Euclid:

A surface is that which has length and breadth only.

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


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