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 . ${\displaystyle {\vec {A}}}$ ${\displaystyle {\vec {B}}}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle \lambda }$

See also

• Division of pages