Line axiom: Two points determine a line. There exist three points which do not lie in a line.

I will define a line in the plane to be a set L which can be labelled with real numbers so that the distance between two points on L is the same as the difference of the corresponding real numbers.

In precise mathematical terms, this labelling is a function from the real number line to L.

Definition: A line L is a set of points such that there is a function onto L,

P : R → L ,

such that for all a and b,

d(P(a),P(b)) = |a - b|.

(Here |a - b| deotes the absolute value of a - b.)

In other words, for all real numbers a and b, if P(a) = A and P(b) = B,

d(A,B) = |a - b|.

This says the distance between the points A and B of L is the same as the distance between their "labels" a and b on the real number line.

The function P is called a line parametrization function, or just a line function. If the point A is an element of the line L, we say that A lies on L, or that L passes through A. If a set of points lies on a single line, we say that this set of points is collinear.

Line axiom

1. For every pair of points A, B with A not equal to B, there exists a unique line L such that A and B lie on L.

2. There exist three points A, B, C which are not collinear.

Here are some related definitions and consequences of the definition of a line:

Definition: A ray with endpoint A is a set P([a, ∞)) where P is a line function with P(a) = A. (Here [a, ∞) denotes the interval of all real numbers greater than or equal to a.)

For every pair of distinct points A and B there is a unique ray R such that R has endpoint A and R contains B.

Definition: A segment with endpoints A and B is a set P([a,b]) where P is a line function with P(a) = A and P(b) = B. (Here [a, b] denotes the set of all real numbers greater than or equal to a and less than or equal to b.)

For every pair of distinct points A and B there is a unique segment S such that A and B are the endpoints of S.

Notation: AB will often be used to denote the distance between A and B, and AB will also be used to denote the line segment between A and B.

In GSP, the distance between A and B is denoted AB, the line segment between A and B is denoted AB with a bar over it, and the length of this line segment is denoted m(AB) with a bar over AB.

For the line through A and B we may write Line(AB), and for the ray from A containing B we may write Ray(AB).

Comment on Birkhoff's Ruler Axiom

My definition of a line is a equivalent to Birkhoff's Ruler Axiom, in which the real number line R is thought of as an infinite "ruler." Birkhoff's axiom is the following:

For every line L there is a function r from L onto the set of real numbers, such that for all points A, B on L,

d(A,B) = |r(A) - r(B)|.

The function r is called a ruler function.

(Note that a ruler function is the inverse of a line function.)

In his high school geometry book (Basic Geometry, by George David Birkhoff and Ralph Beatley, first published in 1940), Birkhoff states this axiom as follows:

"The points on any straight line can be numbered so that number differences measure distances."

C. McCrory 9/17/04