Introduction

Our theory starts from the following simple geometric

Postulate A single point determines a unique zero-dimensional ( vector) space.

Usually, a little black point or spot is used as an intuitively geometric model of zero-dimensional space. Notice that "point" is an undefined term without length, width and height.

In the physical world, it is reasonable to imagine that there exits two different points. Hence, one has the

Postulate Any two different points determine one and only one straight line.

A straightened loop, extended beyond any finite limit in both directions, is a lively geometric model of a straight line. Mathematically, pick up two different points O and A on a flat paper, imagining extended beyond any limit in any direction, and then, connect O and A by a ruler. Now, we have a geometric model of an unlimited straight line L (see Fig. 1.1).

Figure 1.1

As far as the basic concepts of straight lines are concerned, one should know the following facts (1) (6).

1. There are uncountably infinite points on L.

2. The straight line determined by any two different points on L coincides with L.

3. Any two points P and Q on L decide a segment, denoted by PQ:

   1. If P = Q (i.e. P and Q coincide, and represent the same point), then the segment PQ degenerates into a single P (or Q);

   2. If P ? Q