Just like a curve is the basic building block for
figures in a plane, a surface is the
basic building block for figures in space. A
surface is essentially a curve with depth. Curves and surfaces are analogous in
many ways. If you think of a curve as being the trace of the motion of a
point in a plane, a surface is like the trace
of the motion of a curve in space. Surfaces are
continuous, meaning that given two points on a
surface, you can start from one and reach the other without leaving that
surface. Just like a curve is still one-dimensional, a surface, although it exists in
three dimensions, is still two-dimensional. For example, when you build a curve
by tracing the motion of a point, that curve, although it spans both length and
width, has no width of its own. The curve doesn't have area, it only has
length, one dimension. Similarly, a surface can span more than one plane, but
it still does not have depth of its own. It only has two dimensions, length and
width. We will work mostly with the simplest surface, a plane. Below various
surfaces are pictured.

Surfaces can be classified as being closed or simple closed surfaces. The
surfaces that form the boundaries of geometric solids are simple closed
surfaces, so we'll focus on them. A simple closed surface is one that divides
space into three distinct regions:

The set of all points inside the surface (the interior of the surface).

The set of all points outside the surface (the exterior of the surface).

The set of all points on the surface.

A point is only interior if it can be joined to any other interior point by a
segment of finite length. This is not true
for exterior points--a segment joining exterior points could have infinite
length, since the endpoints could be anywhere in space, and space is infinite.

A simple closed surface can also be either convex or concave. The rules
are very similar to those we saw in Polygons. A convex
surface is one in which any two points on that surface can be joined by a segment that lies either on the
surface or in the interior of the surface. A concave surface has a segment
between points on the surface that lies in the exterior of the surface.

One more note on surfaces: a surface, even if it is a simple closed surface,
does not include the space in its interior. When a simple closed surface
is united with its interior points, it is no longer a surface, it is a geometric
solid.

Lines and Planes

So far we've only discussed parallelism and
perpendicularity with respect to
lines, but planes can be parallel and
perpendicular, too. To understand relationships between planes, one must
understand relationships between lines and planes.

A line and a plane are parallel if and only if they do not intersect. A line
l and a plane are perpendicular if and only if the line l is
perpendicular to every line in the plane that contains the intersection point of
line l and the plane. These cases are drawn below.