# Beam (geometry)

In geometry , a ray or a half-line is - to put it clearly - a straight line that is limited on one side, but extends to infinity on the other side.

• A half line is a geometric object that is created when a point divides a straight line on which it lies. The point is either part of the half-line or not.
• A ray has an orientation : It starts from a starting point .

Rays and half-straight lines must therefore be distinguished from straight lines that are unlimited on both sides and lines that are limited on both sides.

## Geometric representation

The notation used in the sketch expresses that it is a subset of the straight line that is bounded by the point , but extends beyond the point . ${\ displaystyle [AB}$${\ displaystyle AB}$ ${\ displaystyle A}$${\ displaystyle B}$

With the help of the intermediate relation ("... lies between ... and ...") the half-line can be defined as the set of all points on the line for which is not between and . ${\ displaystyle [AB}$${\ displaystyle X}$${\ displaystyle AB}$${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle X}$

Considering a straight line and an arbitrary point on , as the two blank half-lines defined thereby , and characterized as a non-empty subsets of which fulfill the following conditions: ${\ displaystyle g}$${\ displaystyle P}$${\ displaystyle g}$${\ displaystyle h_ {1}}$${\ displaystyle h_ {2}}$${\ displaystyle g}$

• Every point on the straight line that does not coincide with belongs to exactly one of the two subsets or .${\ displaystyle g}$${\ displaystyle P}$${\ displaystyle h_ {1}}$${\ displaystyle h_ {2}}$
• If any point is from and any point from , then lies between and .${\ displaystyle P_ {1}}$${\ displaystyle h_ {1}}$${\ displaystyle P_ {2}}$${\ displaystyle h_ {2}}$${\ displaystyle P}$${\ displaystyle P_ {1}}$${\ displaystyle P_ {2}}$

This means that the half-line is closely related to the term interval : An interval can be defined as the intersection of two half-lines.

## Analytical representation

In analytic geometry , the half-line corresponds to the set of all points whose position vector is given by ${\ displaystyle [AB}$${\ displaystyle X}$ ${\ displaystyle {\ vec {X}}}$

${\ displaystyle {\ vec {X}} = {\ vec {A}} + \ lambda ({\ vec {B}} - {\ vec {A}})}$with .${\ displaystyle \ lambda \ geq 0}$

Where and are the position vectors of the endpoints and . is the ( real ) parameter of this parametric equation . ${\ displaystyle {\ vec {A}}}$${\ displaystyle {\ vec {B}}}$${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle \ lambda}$