Connecting line

Line g connecting two points P and Q

In mathematics, a connecting line is a straight line that runs through two specified points . Connecting lines are specifically considered in Euclidean geometry and more generally in incidence geometries . The existence and uniqueness of the straight line connecting two different given points is axiomatically required as a connecting axiom in geometry .

Euclidean geometry

definition

If and are two different points in the Euclidean plane or in Euclidean space , then the straight line that contains these two points is called the “straight line connecting the points and ” and with ${\ displaystyle P}$${\ displaystyle Q}$${\ displaystyle g}$${\ displaystyle P}$${\ displaystyle Q}$

${\ displaystyle g = (PQ)}$   or   ${\ displaystyle g = PQ}$

designated.

calculation

After choosing a Cartesian coordinate system , points in the Euclidean plane can be described by number pairs and . The straight line connecting two points can then be specified using a straight line equation . The two-point form of the straight line equation is in this case ${\ displaystyle P = (x_ {P}, y_ {P})}$${\ displaystyle Q = (x_ {Q}, y_ {Q})}$

${\ displaystyle (y-y_ {P}) \ cdot (x_ {Q} -x_ {P}) = (x-x_ {P}) \ cdot (y_ {Q} -y_ {P})}$.

A parametric form of the straight line equation is as a starting point and as a direction vector${\ displaystyle P}$${\ displaystyle {\ overrightarrow {PQ}}}$

${\ displaystyle {\ begin {pmatrix} x \\ y \ end {pmatrix}} = {\ begin {pmatrix} x_ {P} \\ y_ {P} \ end {pmatrix}} + s \, {\ begin { pmatrix} x_ {Q} -x_ {P} \\ y_ {Q} -y_ {P} \ end {pmatrix}}}$   with   .${\ displaystyle s \ in \ mathbb {R}}$

In barycentric coordinates , the straight line equation of the connecting straight line reads accordingly

${\ displaystyle {\ begin {pmatrix} x \\ y \ end {pmatrix}} = s \, {\ begin {pmatrix} x_ {P} \\ y_ {P} \ end {pmatrix}} + t \, { \ begin {pmatrix} x_ {Q} \\ y_ {Q} \ end {pmatrix}}}$   with   .${\ displaystyle s, t \ in \ mathbb {R}, \, s + t = 1}$

The two vector representations also apply analogously in three- and higher-dimensional spaces.

Axiomatics

In an axiomatic approach to Euclidean geometry, the existence and uniqueness of the straight line connecting two given points must be explicitly required. Euclid demands the existence of the connecting line in two steps. The first two postulates in his work The Elements are as follows:

1. You can draw the route from any point to any point.
2. You can extend a limited straight line continuously.

This means that there is always a straight line connecting two different points. These postulates are to be seen constructively, that is, for two given points, the associated straight line connecting can always be constructed with a compass and ruler .

In Hilbert's system of axioms of Euclidean geometry , the existence and uniqueness of the connecting lines are defined as axioms I1. and I2. listed within axiom group I: axioms of connection . Hilbert formulates the axioms I1. and I2. as follows:

I1. For two different points there is always a straight line on which the two points lie.${\ displaystyle P, \, Q}$${\ displaystyle g}$
I2. Two different points on a straight line clearly define this straight line.${\ displaystyle P, \, Q}$${\ displaystyle g}$

Incidence Geometry

definition

If, in general, is an incidence space and there are two different points in this space, then a straight line connecting these two points is called if the following two conditions apply: ${\ displaystyle ({\ mathfrak {P}}, G, I)}$${\ displaystyle P_ {1}, P_ {2} \ in {\ mathfrak {P}}}$${\ displaystyle g \ in G}$

(V1) ${\ displaystyle P_ {1} Ig \ land P_ {2} Ig}$
(V2) ${\ displaystyle \ operatorname {card} (\ {h \ in G \ colon P_ {1} Ih \ land P_ {2} Ih \}) \ leq 1}$

Notation and ways of speaking

If the two points and the straight line fulfill the conditions (V1) and (V2), one often writes

${\ displaystyle g = \ langle P_ {1}, P_ {2} \ rangle}$

or

${\ displaystyle g = P_ {1} \ vee P_ {2}}$

or briefly

${\ displaystyle g = P_ {1} P_ {2}}$.

In the usual parlance for this , one also says

• ${\ displaystyle g}$connects the points and .${\ displaystyle P_ {1}}$${\ displaystyle P_ {2}}$
• ${\ displaystyle g}$belongs with the points and together.${\ displaystyle P_ {1}}$${\ displaystyle P_ {2}}$
• The points and lie on .${\ displaystyle P_ {1}}$${\ displaystyle P_ {2}}$${\ displaystyle g}$
• ${\ displaystyle g}$goes through the dots and .${\ displaystyle P_ {1}}$${\ displaystyle P_ {2}}$
• The points and incise with .${\ displaystyle P_ {1}}$${\ displaystyle P_ {2}}$${\ displaystyle g}$
• ${\ displaystyle g}$incised with the dots and .${\ displaystyle P_ {1}}$${\ displaystyle P_ {2}}$

or similar.

Using this usage, the above conditions (V1) and (V2) can be put into words as follows:

(V1 ') The points and are connected by the straight line.${\ displaystyle P_ {1}}$${\ displaystyle P_ {2}}$${\ displaystyle g}$
(V2 ') For the points and there is at most one straight line connecting them.${\ displaystyle P_ {1}}$${\ displaystyle P_ {2}}$

Connection axiom

In the incidence spaces that are particularly important for geometry, i.e. in particular in the Euclidean spaces , in all affine spaces and in all projective spaces , the following basic condition (V) applies consistently to points and connecting lines:

(V) There is always a straight connecting line for two different points in the given incidence space, i.e. a straight line such that (V1) and (V2) are fulfilled.

This condition is called the axiom of connection.

In another formulation, the axiom of connection can also be expressed as follows:

(V ') For every two different points of the given incidence space there is exactly one straight line connecting these two points.

Subspaces and envelope system

The incidence spaces mainly dealt with in geometry - such as affine and projective spaces, but also many other linear spaces such as e.g. B. the block plans  - it is common that the incidence relation comes from the element relation and thus the straight lines are subsets of the corresponding point set . ${\ displaystyle g}$ ${\ displaystyle {\ mathfrak {P}}}$

Thus, it is then the set of lines , a subset of the power set of , thus the relationship given. In this case, the incidence space is briefly described in terms of shape rather than shape . ${\ displaystyle {\ mathfrak {P}}}$${\ displaystyle G \ subseteq 2 ^ {\ mathfrak {P}}}$${\ displaystyle ({\ mathfrak {P}}, G, I)}$${\ displaystyle ({\ mathfrak {P}}, G)}$${\ displaystyle ({\ mathfrak {P}}, G, {\ in})}$

Under these circumstances, it is called a subset of a subspace , when with two different points always their connecting line in is included, so this is always true. ${\ displaystyle {\ mathfrak {T}} \ subseteq {\ mathfrak {P}}}$${\ displaystyle ({\ mathfrak {P}}, G)}$${\ displaystyle P_ {1}, P_ {2} \ in {\ mathfrak {T}}}$${\ displaystyle \ langle P_ {1}, P_ {2} \ rangle}$${\ displaystyle {\ mathfrak {T}}}$${\ displaystyle \ langle P_ {1}, P_ {2} \ rangle \ subseteq {\ mathfrak {T}}}$

The set of subspaces from forms a shell system . ${\ displaystyle \ tau}$${\ displaystyle ({\ mathfrak {P}}, G)}$

Corresponding shell operator

The associated shell operator can be formed for the shell system in the usual way . This is often written as . For so true ${\ displaystyle \ tau}$${\ displaystyle \ langle \; \ rangle}$${\ displaystyle {\ mathfrak {P}} _ {0} \ subseteq {\ mathfrak {P}}}$

${\ displaystyle \ langle {\ mathfrak {P}} _ {0} \ rangle = \ bigcap \ {{\ mathfrak {T}} \ in {\ mathcal {T}} \ colon {\ mathfrak {T}} \ supseteq {\ mathfrak {P}} _ {0} \}}$.

That means:

${\ displaystyle \ langle {\ mathfrak {P}} _ {0} \ rangle}$is the smallest subspace of which includes.${\ displaystyle ({\ mathfrak {P}}, G)}$${\ displaystyle {\ mathfrak {P}} _ {0}}$

In the case that there is a finite set of points, for example , one also writes ${\ displaystyle {\ mathfrak {P}} _ {0}}$${\ displaystyle {\ mathfrak {P}} _ {0} = \ {P_ {1}, \ dotsc, P_ {m} \} \; (m \ in \ mathbb {N})}$

${\ displaystyle \ langle {\ mathfrak {P}} _ {0} \ rangle = \ langle P_ {1}, \ dotsc, P_ {m} \ rangle}$

or

${\ displaystyle \ langle {\ mathfrak {P}} _ {0} \ rangle = P_ {1} \ vee \ dotsc \ vee P_ {m}}$.

For and one has , thus again the connecting line from and . ${\ displaystyle m = 2}$${\ displaystyle P_ {1} \ neq P_ {2}}$${\ displaystyle \ langle {\ mathfrak {P}} _ {0} \ rangle = \ langle P_ {1}, P_ {2} \ rangle}$${\ displaystyle P_ {1}}$${\ displaystyle P_ {2}}$

Example of the coordinate plane

The coordinate plane over a commutative body gives a standard example of an incidence space in which the connection axiom applies. Here is the point set ${\ displaystyle K ^ {2}}$ ${\ displaystyle K}$${\ displaystyle ({\ mathfrak {P}}, G)}$

${\ displaystyle {\ mathfrak {P}} = K ^ {2}}$

and the set of lines

${\ displaystyle G = \ {a + Ku \ colon a \ in K ^ {2} \ land u \ in K ^ {2} \ setminus \ {\ mathbf {0} \} \}}$.

The set of lines is thus obtained by allowing all possible cosets to all in located subspaces of dimension 1 is formed. If you have two different points here , the connecting line can be represented in the following way: ${\ displaystyle G}$${\ displaystyle K ^ {2}}$${\ displaystyle P_ {1}, P_ {2} \ in K ^ {2}}$

${\ displaystyle \ langle P_ {1}, P_ {2} \ rangle = \ {\ alpha P_ {1} + \ beta P_ {2} \ colon \ alpha, \ beta \ in K \ land \ alpha + \ beta = 1\}}$

The standard example of this concept is provided by the straight lines connecting two points of the Euclidean plane .

literature

• Gerhard Hessenberg , Justus Diller: Fundamentals of geometry . 2nd Edition. Walter de Gruyter Verlag, Berlin 1967, p. 20, 220 .
• David Hilbert : Fundamentals of Geometry . With supplements from Dr. Paul Bernays (=  Teubner Study Books: Mathematics ). 11th edition. Teubner Verlag, Stuttgart 1972, ISBN 3-519-12020-8 , pp. 3 ff . ( MR1109913 ).
• Helmut Karzel , Kay Sörensen, Dirk Windelberg: Introduction to Geometry (=  Uni-Taschenbücher . Volume 184 ). Vandenhoeck & Ruprecht, Göttingen 1973, ISBN 3-525-03406-7 , pp. 11 ff .
• Max Koecher , Aloys Krieg : level geometry (=  Springer textbook ). 2nd, revised and expanded edition. Springer Verlag, Berlin (among others) 2000, ISBN 3-540-67643-0 , p. 7, 48 ff., 52, 212 .
• Herbert Meschkowski : The way of thinking of great mathematicians . A path to the history of mathematics (=  documents on the history of mathematics ). 3. Edition. Vieweg Verlag, Braunschweig 1990, ISBN 3-528-28179-0 ( MR1086172 ).
• Eberhard M. Schröder: Lectures on geometry . 2. Affine and projective geometry. BI Wissenschaftsverlag, Mannheim / Vienna / Zurich 1991, ISBN 3-411-15301-6 , p. 2 ff . ( MR1166803 ).
• Otto Kerner, Joseph Maurer, Jutta Steffens, Thomas Thode, Rudolf Voller (arr.): Vieweg-Mathematik-Lexikon . Terms, definitions, sentences, examples for basic studies. Vieweg Verlag, Braunschweig / Wiesbaden 1988, ISBN 3-528-06308-4 , p. 311 .