A point moves through space, tracing folds, twists, crossings and elongated lines to form part of it. From this visualization emerges an informal definition of a curve: "a piece of space that the eye can follow."
Now imagine a large knife with a more or less flexible blade—really more like a whip or a rope—unfurling through space. A surface appears: a slice of space made up of all the points reached at one time or another by our moving rope. One difference from a curve is that locating a point on the surface now requires two times rather than one: the moment when the whip passes through the point in question, then, once the whip is frozen in that position, the moment when our eye, gliding along the motionless whip, reaches the target point.

A surface with a coordinate system.

Many guises for a surface ---------------------------------