Definition:Surface (Geometry)

Definition
As Euclid defined it:

"A surface is that which has length and breadth only."

Plane Surface
Usually referred to as a plane.

As Euclid defined it:

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

Care needs to be taken with this definition. It is possible to create non-plane surfaces which can be generated solely straight lines. Twist a deck of cards and the edges of the deck will no longer be plane surfaces, although the edges of the individual cards are as straight as before. The point to this definition is that any straight line lying partly in a plane surface lies wholly in that surface (unless intersecting it in just one place).

The Plane
The plane is the term used for the general infinite plane surface. This is isomorphic to the Real Vector Space $$\mathbb{R}^2$$.