Chapter 5 Geometry

Section 5.1 Elements of Plane Geometry

5.1.2 Points and Lines

In geometry, a place or a position in a plane is idealised to the most basic object, namely a point. A single point itself cannot be characterised any further.
For several points, relations between these points can be considered in different ways - and points can be used to define new objects such as line segments and lines (see figure below). Mathematically, these objects are sets of points.

First, we consider a line segment and the distance between points. To do this, we need a comparison tool for measuring distance. In mathematics, this tool is a comparative length called the unit length. For applications, appropriate length units such as metres or centimetres are chosen, depending on the task in hand.
Line Segments and Distances 5.1.1
Given two points A and B, the line segment AB between A and B is the shortest path between the two points A and B.

The length of the line segment AB is denoted by [ AB ]. The line length equals the distance between the two points A and B.

A ray of light emitted by a distant star or by the sun is an appropriate notion of a ray starting at the initial point A and proceeding through a second point B indefinitely. A ray is also called a half-line.

Continuing the path of a line segment AB on both ends indefinitely results in a line.
Line 5.1.2
Let A and B be two points (i.e. point A is different from point B). Then, A and B define exactly one line AB.

Considering, beside A and B, an additional point P, we can ask for the distance d of the point P from the line AB, which is defined as the shortest path between P and one of the points of the line AB.

Given three points P, Q, and S in the plane, the lines SP and SQ can be defined.
The two lines have the point S in common. If the point Q is also on the line SP, then SQ and SP denote one and the same line. If the point Q does not belong to the line SP, the line SQ is different from the line SP. Then, the two lines have only the point S in common. The point S is called an intersection point.
If any two lines g and h do not have any points in common, the smallest distance between points on g and h, respectively, is called the distance between the lines g and h. Hence, g and h do not have any point in common if they have a distance larger than 0. Two lines are called parallel if every point on one of the two lines has the same distance from the other line.
A single line can be described by the distance of two points M and M' as well: The set of all points with the same distance from two points M and M' is a line.
In geometry, it is a typical approach to define new objects by means of certain properties such as the distance. In this way, a circle can also be described very easily.
Circle 5.1.3
Let a point M and a positive real number r be given.
Then, the set of all points at distance r from point M is a circle around M with radius r.