# Route (geometry)

Distance [AB] between the two points A and B

A line segment (also straight line segment or straight line segment ) is a straight line that is bounded by two points ; it is the shortest connection between its two endpoints. The delimitation of a line by these points distinguishes it from straight lines , which are unlimited on both sides, and half-straight lines , which are only limited on one side.

## Euclidean geometry

Historical illustration of the construction of routes (1699)

### definition

A segment is a straight line in the Euclidean plane or in Euclidean space that is bounded by two points . If and are two given points in the plane or in space, then the distance between these two points is called the connecting distance from and and is denoted by. ${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle [AB]}$

Lines can also be defined with the help of the intermediate relation ("... lies between ... and ..."): the line then consists of all points of the connecting straight lines that lie between points and . Depending on whether the points and are included or not, a distinction is made between the following cases: ${\ displaystyle [AB]}$ ${\ displaystyle AB}$${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle A}$${\ displaystyle B}$

• completed route : both endpoints are included${\ displaystyle [AB]}$
• open route : both endpoints are excluded${\ displaystyle (AB)}$
• half-open route or : one of the end points is included, the other excluded${\ displaystyle [AB)}$${\ displaystyle (AB]}$

If the order of the points and an orientation of the route are given, one speaks of a directed route (also arrow or bound vector ) . ${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle {\ overrightarrow {AB}}}$

### Special cases

A route is called:

• Page - if the two end points of the mutually adjacent vertices of a polygon are
• Edge - if the two end points adjacent vertices of a polyhedron are
• Diagonal - when the two end points are the non-adjacent corner points of a polygon
• Chord - when the two endpoints are on a curve , such as As a circle , are

### Parametric representation

In analytic geometry , points in the Euclidean plane or in Euclidean space are described by their position vectors . If and are the position vectors of the points and , then the segment consists of those points in the plane or in space whose position vectors are of the form ${\ displaystyle {\ vec {a}}}$${\ displaystyle {\ vec {b}}}$${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle [AB]}$${\ displaystyle {\ vec {x}}}$

${\ displaystyle {\ vec {x}} = {\ vec {a}} + t \, ({\ vec {b}} - {\ vec {a}})}$   With   ${\ displaystyle 0 \ leq t \ leq 1}$

are. In this parameter representation of a route there is a real parameter that can be freely selected in the parameter area. The open route here consists of the points in the parameter area , while the semi-open routes and through the areas and are parameterized. In barycentric coordinates , the parametric representation of a route reads accordingly ${\ displaystyle t}$${\ displaystyle (AB)}$${\ displaystyle 0 ${\ displaystyle [AB)}$${\ displaystyle (AB]}$${\ displaystyle 0 \ leq t <1}$${\ displaystyle 0 ${\ displaystyle [AB]}$

${\ displaystyle {\ vec {x}} = s \, {\ vec {a}} + t \, {\ vec {b}}}$   with   .${\ displaystyle s, t \ geq 0, s + t = 1}$

Here and are two real parameters which, however , cannot be selected independently of one another due to the condition . The open route consists of the points with the parameters , while the semi-open routes and are represented by the parameter areas and . ${\ displaystyle s}$${\ displaystyle t}$${\ displaystyle s + t = 1}$${\ displaystyle (AB)}$${\ displaystyle s, t> 0}$${\ displaystyle [AB)}$${\ displaystyle (AB]}$${\ displaystyle s> 0, t \ geq 0}$${\ displaystyle s \ geq 0, t> 0}$

### properties

When specifying a closed or open route, the order of the end points is irrelevant, so it applies

${\ displaystyle [AB] = [BA]}$   and   .${\ displaystyle (AB) = (BA)}$

The length of the route is the distance between its two end points. This route length is often referred to as , sometimes also as, or . The line connecting two points and can thus be characterized as a set of those points for which the sum of the distances ${\ displaystyle {\ overline {AB}}}$${\ displaystyle | AB |}$${\ displaystyle | {\ overline {AB}} |}$${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle X}$

${\ displaystyle {\ overline {XA}} + {\ overline {XB}}}$

is minimal. Since an ellipse is characterized by the fact that the sum of the distances to two given points (the focal points of the ellipse) is constant, a line segment is thus a special (degenerate) ellipse. A route can also be viewed as a special curve . Of all the curves connecting two given points, the line connecting these points has the shortest arc length .

## Linear Algebra

### definition

If a vector space is over the real or complex numbers , then a subset is called a (closed) segment if it goes through ${\ displaystyle V}$ ${\ displaystyle S \ subseteq V}$

${\ displaystyle S = \ {\ mathbf {u} + t \, (\ mathbf {v} - \ mathbf {u}) \ mid t \ in [0,1] \}}$

can be parameterized. Here, with two vectors , the endpoint of the segment represent. Alternatively, a closed path can also be created using the convex combination${\ displaystyle \ mathbf {u}, \ mathbf {v} \ in V}$${\ displaystyle \ mathbf {u} \ neq \ mathbf {v}}$${\ displaystyle S}$

${\ displaystyle S = \ {s \, \ mathbf {u} + t \, \ mathbf {v} \ mid s, t \ geq 0, s + t = 1 \}}$

are represented as a convex hull of their endpoints. In both representations, open and semi-open routes are also described by restricting the parameter range accordingly.

### properties

• A route is always a “non- empty set ”.
• If is a topological vector space , then every closed segment contained therein is a connected compact and in particular a topologically closed subset of .${\ displaystyle V}$ ${\ displaystyle V}$
• Note that an open segment of is generally not an open subset. An open line is open if and only if it is one-dimensional and therefore homeomorphic to .${\ displaystyle V}$${\ displaystyle V}$ ${\ displaystyle V}$${\ displaystyle \ mathbb {R}}$

## Incidence Geometry

### Axioms of lines

Essential characteristics of the Euclidean geometry concept of a segment can be formulated in a very general framework that allows this concept to be represented in abstract incidence geometries completely independently of topological or metric considerations. This was u. a. shown by Ernst Kunz in his textbook level geometry . An incidence geometry is used as a basis, which consists of a set of points and a set of straight lines and which satisfies the following conditions: ${\ displaystyle ({\ mathfrak {E}}, G)}$ ${\ displaystyle {\ mathfrak {E}}}$${\ displaystyle G \ subseteq 2 ^ {\ mathfrak {E}}}$

(A1) Every two points are connected by at least one straight line.
(A2) For every two different points there is at most one straight line connecting both.
(A3) There are at least two different points on every straight line.
(A4) There are at least three points that are not on a straight line.

The two conditions (A1) and (A2) mean that the incidence geometry fulfills the connection axiom, while (A3) and (A4) ensure that it meets certain richness requirements.

Kunz briefly calls an incidence geometry which fulfills these four conditions a plane . ${\ displaystyle ({\ mathfrak {E}}, G)}$

### Axioms of Range

On a level understood in this sense, the concept of a route can be grasped by the following route axioms: ${\ displaystyle ({\ mathfrak {E}}, G)}$

(B0) A subset is assigned to every two (not necessary) different points , which is called the distance from to .${\ displaystyle A, B \ in {\ mathfrak {E}}}$ ${\ displaystyle [AB] \ subseteq {\ mathfrak {E}}}$${\ displaystyle A}$${\ displaystyle B}$
(B1) It is for every route .${\ displaystyle A \ in [AB]}$${\ displaystyle [AB]}$
(B2) If is a straight line and are , then is .${\ displaystyle g}$${\ displaystyle A, B \ in g}$${\ displaystyle [AB] \ subseteq g}$
(B3) For everyone is always .${\ displaystyle A, B \ in {\ mathfrak {E}}}$${\ displaystyle [AB] = [BA]}$
(B4) For all there is an with and .${\ displaystyle A, B \ in {\ mathfrak {E}}}$${\ displaystyle C \ in {\ mathfrak {E}}}$${\ displaystyle C \ neq B}$${\ displaystyle B \ in [AC]}$
(B5) Is and , so is .${\ displaystyle C \ in [AB]}$${\ displaystyle C \ neq B}$${\ displaystyle B \ notin [AC]}$
(B6) If there are three points that do not lie on a straight line and if a straight line that does not contain any of the three points, then it follows that or is.${\ displaystyle A_ {1}, A_ {2}, A_ {3} \ in {\ mathfrak {E}}}$${\ displaystyle g \ in G}$${\ displaystyle g \ cap [A_ {1} A_ {2}] \ neq \ emptyset}$${\ displaystyle g \ cap [A_ {2} A_ {3}] \ neq \ emptyset}$${\ displaystyle g \ cap [A_ {1} A_ {3}] \ neq \ emptyset}$

Ernst Kunz calls a plane that also meets the conditions (B0) to (B6) a plane with lines . The plausibility of these conditions is easy to understand if one takes the Euclidean level as a basis. All these conditions are met here. ${\ displaystyle ({\ mathfrak {E}}, G)}$

The condition (B6) is called Pasch's axiom by Kunz according to the conditions in the Euclidean plane . There it clearly states that a straight line which “penetrates” a triangle must also leave it somewhere. The name of the axiom refers to the mathematician Moritz Pasch (1843–1930), who was the first to recognize that, within the framework of an axiomatic foundation of Euclidean geometry, the facts presented in the axiom cannot be inferred from the other axioms, but specifically required must become.

As can be shown, the system of line axioms is equivalent to that of Hilbert's arrangement axioms - assuming the incidence axioms . The connection to the intermediate relation results from the following definition:

If three points are different in pairs , then the point lies between the points and , if this applies.${\ displaystyle P, X, Y}$${\ displaystyle P}$ ${\ displaystyle X}$${\ displaystyle Y}$${\ displaystyle P \ in [XY]}$

If the mentioned condition is fulfilled for three pairs of different points , one also says: ${\ displaystyle P, X, Y}$

The point is the inner point of the route .${\ displaystyle P}$${\ displaystyle [XY]}$