View source for Euclidean space over a field

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e1201901.png" /> be a (commutative) [[Field|field]] of characteristic not two. A Euclidean space is a [[Vector space|vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e1201902.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e1201903.png" /> equipped with a symmetric [[Bilinear form|bilinear form]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e1201904.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e1201905.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e1201906.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e1201907.png" />. The elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e1201908.png" /> are called points, and a set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e1201909.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019011.png" />) is called a line. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019012.png" />. Two pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019014.png" /> of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019018.png" /> are said to be congruent if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019019.png" />.

==Characterization of Euclidean planes.==
A fundamental problem is to characterize classes of Euclidean spaces by means of geometric structures, i.e. structures of abstract points and lines equipped with suitable relations on the objects (e.g., of order, congruence, parallelity, continuity). The classical case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019022.png" />, as well as some generalizations, were treated by D. Hilbert in [[#References|[a3]]] (see also [[#References|[a1]]] and [[#References|[a6]]]). There are many results concerning the fundamental problem (see [[#References|[a4]]]). An example of such a result is Schröder's theorem [[#References|[a5]]], which can be described in a fairly elementary way: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019023.png" /> be a set (no stipulation about the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019024.png" /> is made, except that they will be called points). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019025.png" /> be the set of all two-element sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019026.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019027.png" />. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019028.png" /> is some equivalence relation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019029.png" />. This structure will be written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019030.png" />. (Intuitively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019031.png" /> is the real plane and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019032.png" /> means that the segments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019034.png" /> are of equal length.) For two distinct points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019036.png" />, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019037.png" /> of all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019038.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019039.png" /> is called a line. Two lines are called parallel if they have no point in common or coincide. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019040.png" /> be the set of all lines of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019041.png" />. Such a structure is called a Euclidean plane if:

1) The structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019042.png" /> of points and lines of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019043.png" /> is an affine plane (i.e., for two distinct points there is exactly one line containing them; for a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019044.png" /> and a line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019045.png" /> there is exactly one line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019046.png" /> parallel to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019047.png" />; and there exist three distinct points not on a common line).

2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019051.png" /> be distinct points such that no three of them are collinear (i.e., are on a common line). If the line through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019053.png" /> is parallel to the line through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019055.png" /> and the line through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019056.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019057.png" /> is parallel to the line through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019059.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019060.png" />.

3) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019061.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019062.png" /> be distinct points. Then there is exactly one point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019063.png" /> on the line through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019064.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019065.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019067.png" />.

An example of a Euclidean plane can be constructed as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019068.png" /> be the set of nine (distinct) elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019069.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019070.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019071.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019074.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019076.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019077.png" />. Two such points are called related if, and only if, the letter of the points or their index are the same. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019078.png" />, for points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019079.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019080.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019081.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019082.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019083.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019084.png" />, precisely if the pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019085.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019086.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019087.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019088.png" /> of points are both related or both not related.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019089.png" /> is a Euclidean space of dimension two over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019090.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019091.png" /> is a Euclidean plane. Schröder's theorem states that, up to isomorphism, these are the only Euclidean planes.

==Historical remarks.==
A landmark in the development of the notion of a Euclidean space is the book [[#References|[a7]]] by M. Pasch (first published in 1882). In it, Pasch concentrated himself, in an in geometry at that time unusual rigorous manner, on understanding the basic notions of geometry and the relations between them. This was the time for clarifying the notions in mathematics: the notion of a real number was clarified by R. Dedekind in [[#References|[a8]]] (first published in 1872).

Pasch was aware of the gap left in the geometry of Euclid: Euclid had tacitly assumed that a line divides the plane in two parts, a fact that is not a consequence of the axioms (see also [[Euclidean geometry|Euclidean geometry]]). It led Pasch to the discovery of his famous axioms of order (see [[Pasch axiom|Pasch axiom]]). Note that already C.F. Gauss [[#References|[a9]]] noted that the word  "between"  was not properly defined in mathematics. Pasch now realized that a theory of between-ness was important in the systematic approach to geometry.

Nevertheless, Pasch was thinking of a single geometry,  "the"  geometry. Of course, there was the problem of the axiom of parallelity (cf. also [[Fifth postulate|Fifth postulate]]; [[Lobachevskii geometry|Lobachevskii geometry]]): Is it true or not in reality? Pasch finished his considerations on the basis of so-called [[Absolute geometry|absolute geometry]]. So, he left his theory open for development in the direction of either Euclidean or non–Euclidean geometry. But, again, it was  "the"  geometry he was concerned with. In his understanding, geometry was part of natural science.

When D. Hilbert published his book [[#References|[a3]]] in 1899, the discipline was no longer part of natural science, but had become a part of pure mathematics. Categorical structures like the real Euclidean plane were, and still are, fundamental. (A structure is called categoric if it is uniquely defined up to isomorphism, i.e. up to the notations of the objects and relations involved.) However, the so-called polymorphic structures became more and more important in geometry. The theory of space geometry gained very much by Hilbert's systematic transition from categorical to polymorphic structures.

==Distance spaces.==
Of special interest are the Euclidean spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019092.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019093.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019094.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019095.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019096.png" />, the so-called pre-Hilbert spaces (cf. also [[Pre-Hilbert space|Pre-Hilbert space]]). Of course, in this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019097.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019098.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e12019099.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190100.png" />. Such Euclidean spaces are the so-called real Euclidean spaces. To be able to present the fundamental geometric objects of these, very important, spaces, the notion of a real distance space is introduced. This is a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190101.png" /> together with a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190102.png" />. Note that for a pre-Hilbert space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190103.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190104.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190105.png" /> be a real distance space. Its points are the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190106.png" />. The real number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190107.png" /> is called the distance between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190108.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190109.png" /> (in this order). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190110.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190111.png" />, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190112.png" /> of all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190113.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190114.png" /> is called the hypersphere with centre at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190115.png" /> and radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190116.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190117.png" /> are points, then the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190118.png" /> of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190119.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190120.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190121.png" /> intersect precisely in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190122.png" />, is said to be a line. The segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190123.png" />, for points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190124.png" />, is defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190125.png" /></td> </tr></table>

A subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190126.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190127.png" /> is called a linear subspace, respectively a convex set, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190128.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190129.png" />, respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190130.png" />, whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190131.png" />. Intersections of hyperspheres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190132.png" /> with linear subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190133.png" /> are called spherical subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190134.png" />. A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190135.png" /> is an isometry of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190136.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190137.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190138.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190139.png" />. Motions are bijective isometries. The set of motions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190140.png" /> is a [[Group|group]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190141.png" />, with respect to the composition product. The image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190142.png" /> (respectively, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190143.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190144.png" />) is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190145.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190146.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190147.png" />) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190148.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190149.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190150.png" /> are distinct points, then the union of the segments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190151.png" /> is called the half-line with starting point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190152.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190153.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190154.png" /> be distinct half-lines with the same starting point, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190155.png" /> be the intersection of all linear subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190156.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190157.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190158.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190159.png" /> is not a line, then the angle-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190160.png" /> is the union of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190161.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190162.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190163.png" /> is on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190164.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190165.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190166.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190167.png" /> is the angle-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190168.png" />. The ordered quadruple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190169.png" /> is called an angle if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190170.png" /> is one of the angle-spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190171.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190172.png" />. Two angles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190173.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190174.png" /> are said to be equal if there exists a motion mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190175.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190176.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190177.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190178.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190179.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190180.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190181.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190182.png" />, respectively. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190183.png" /> is the set of all angles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190184.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190185.png" />, then a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190186.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190187.png" /> into the set of all non-negative real numbers is called an angle-measure if:

a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190188.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190189.png" /> imply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190190.png" />;

b) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190191.png" /> is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190192.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190193.png" /> is a half-line with starting point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190194.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190195.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190196.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190197.png" /> are angles in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190198.png" /> with corresponding angle-spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190199.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190200.png" />.

For a real Euclidean space with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190201.png" /> one obtains the classical notions. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190202.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120190/e120190203.png" /> one obtains the objects of hyperbolic geometry [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Benz,   "Grundlagen der Geometrie"  ''Dokumente Geschichte der Math.'' , '''6'''  (1990)  pp. 231–267</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Benz,   "Reelle Abstandsräume und hyperbolische Geometrie"  ''Results in Math.'' , '''34'''  (1998)  pp. 56–68</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Hilbert,   "Grundlagen der Geometrie" , Teubner  (1972)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Karzel,   H.-J. Kroll,   "Geschichte der Geometrie seit Hilbert" , Wiss. Buchgesell. Darmstadt  (1988)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Schröder,   "Geometrie euklidischer Ebenen" , Schöningh Paderborn  (1985)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  M.M. Toepell,   "Uber die Entstehung von David Hilberts Grundlagen der Geometrie" , Vandenhoeck&amp;Ruprecht  (1986)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  M. Pasch,   "Vorlesungen über neuere Geometrie" , Teubner  (1882)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  R. Dedekind,   "Stetigkeit und irrationale Zahlen" , Braunschweig  (1872)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  C.F. Gauss,   "Werke" , '''VIII''' , Teubner  (1900)  pp. 222</TD></TR></table>