4084:
3621:
4173:
4079:{\displaystyle {\begin{aligned}|BC|^{2}&={\overrightarrow {BC}}\cdot {\overrightarrow {BC}}{\vphantom {\frac {(}{}}}\\&={\Bigl (}{\overrightarrow {BA}}+{\overrightarrow {AC}}{\Bigr )}\cdot {\Bigl (}{\overrightarrow {BA}}+{\overrightarrow {AC}}{\Bigr )}\\&={\overrightarrow {BA}}\cdot {\overrightarrow {BA}}+{\overrightarrow {AC}}\cdot {\overrightarrow {AC}}-2{\overrightarrow {AB}}\cdot {\overrightarrow {AC}}\\&={\overrightarrow {AB}}\cdot {\overrightarrow {AB}}+{\overrightarrow {AC}}\cdot {\overrightarrow {AC}}\\&=|AB|^{2}+|AC|^{2}.\end{aligned}}}
8670:
31:
8190:, many sorts of spaces have been considered, about which one can do geometric reasoning in the same way as with Euclidean spaces. In general, they share some properties with Euclidean spaces, but may also have properties that could appear as rather strange. Some of these spaces use Euclidean geometry for their definition, or can be modeled as subspaces of a Euclidean space of higher dimension. When such a space is defined by geometrical
8153:
5798:
9412:
4599:
6986:
4458:
8462:, which are the "shortest paths" between two points. This allows defining distances, which are measured along geodesics, and angles between geodesics, which are the angle of their tangents in the tangent space at their intersection. So, Riemannian manifolds behave locally like a Euclidean space that has been bent.
2177:
644:
5957:, coordinates may sometimes be defined as the limit of coordinates of neighbour points, but these coordinates may be not uniquely defined, and may be not continuous in the neighborhood of the point. For example, for the spherical coordinate system, the longitude is not defined at the pole, and on the
8011:
has proved that all these definitions of a
Euclidean space are equivalent. It is rather easy to prove that all definitions of Euclidean spaces satisfy Hilbert's axioms, and that those involving real numbers (including the above given definition) are equivalent. The difficult part of Artin's proof is
2713:
169:
has been shown to be equivalent to the axiomatic definition. It is this definition that is more commonly used in modern mathematics, and detailed in this article. In all definitions, Euclidean spaces consist of points, which are defined only by the properties that they must have for forming a
7148:
6834:
371:
allowed their use in defining
Euclidean spaces with a purely algebraic definition. This new definition has been shown to be equivalent to the classical definition in terms of geometric axioms. It is this algebraic definition that is now most often used for introducing Euclidean spaces.
2428:
2024:
1455:
6615:
It follows from the preceding results that an isometry of
Euclidean spaces maps lines to lines, and, more generally Euclidean subspaces to Euclidean subspaces of the same dimension, and that the restriction of the isometry on these subspaces are isometries of these subspaces.
7936:'s one. In reality, Euclid did not define formally the space, because it was thought as a description of the physical world that exists independently of human mind. The need of a formal definition appeared only at the end of 19th century, with the introduction of
3369:
Two lines, and more generally two
Euclidean subspaces (A line can be considered as one Euclidean subspace.) are orthogonal if their directions (the associated vector spaces of the Euclidean subspaces) are orthogonal. Two orthogonal lines that intersect are said
6608:
2059:
7517:
1801:
4594:{\displaystyle \operatorname {angle} (\lambda x,\mu y)={\begin{cases}\operatorname {angle} (x,y)\qquad \qquad {\text{if }}\lambda {\text{ and }}\mu {\text{ have the same sign}}\\\pi -\operatorname {angle} (x,y)\qquad {\text{otherwise}}.\end{cases}}}
2271:
2617:
6995:
4156:
3045:
6478:
6394:
1646:
6300:
3626:
8319:, that is, there is no property of the space that allows distinguishing between two points or two lines. Therefore, a more isotropic definition is commonly used, which consists as defining a projective space as the set of the
2321:
1939:
1359:
6720:
4303:
5662:
1916:) has two sorts of subspaces: its Euclidean subspaces and its linear subspaces. Linear subspaces are Euclidean subspaces and a Euclidean subspace is a linear subspace if and only if it contains the zero vector.
309:
all geometry by starting from a few very basic properties, which are abstracted from the physical world, and cannot be mathematically proved because of the lack of more basic tools. These properties are called
7404:
5788:
4720:
1073:
6981:{\displaystyle {\begin{aligned}E&\to \mathbb {R} ^{n}\\P&\mapsto {\Bigl (}e_{1}\cdot {\overrightarrow {OP}},\dots ,e_{n}\cdot {\overrightarrow {OP}}{\Bigr )},{\vphantom {\frac {(}{}}}\end{aligned}}}
4671:
3496:
3450:
3365:
are orthogonal if every nonzero vector of the first one is perpendicular to every nonzero vector of the second one. This implies that the intersection of the linear subspaces is reduced to the zero vector.
1093:
For any vector space, the addition acts freely and transitively on the vector space itself. Thus a
Euclidean vector space can be viewed as a Euclidean space that has itself as the associated vector space.
8024:
of a segment as its equivalence class. One must thus prove that this length satisfies properties that characterize nonnegative real numbers. Artin proved this with axioms equivalent to those of
Hilbert.
8821:
2789:. For this reason, and for historical reasons, the dot notation is more commonly used than the bracket notation for the inner product of Euclidean spaces. This article will follow this usage; that is
419:
In order to make all of this mathematically precise, the theory must clearly define what is a
Euclidean space, and the related notions of distance, angle, translation, and rotation. Even when used in
8823:. For that, it suffices to prove that the square of the norm of the left-hand side is zero. Using the bilinearity of the inner product, this squared norm can be expanded into a linear combination of
7419:
6517:
5573:
9298:
Schläfli ... discovered them before 1853 -- a time when Cayley, Grassman and Möbius were the only other people who had ever conceived of the possibility of geometry in more than three dimensions.
7000:
6839:
2622:
7787:, which are rigid transformations that fix a hyperplane and are not the identity. They are also the transformations consisting in changing the sign of one coordinate over some Euclidean frame.
2564:
The concept of parallel subspaces has been extended to subspaces of different dimensions: two subspaces are parallel if the direction of one of them is contained in the direction to the other.
1724:
2197:
3608:
7747:
7558:
3208:
2887:
8611:
6182:
2548:
6825:
5279:
5209:
1717:
1495:
795:
7337:
6653:
5344:
5310:
5050:
4220:
3363:
3288:
2936:
2599:
1914:
1887:
1857:
1830:
1683:
1553:
1526:
1325:
1179:
841:
9016:
6065:
5166:
5114:
4893:
396:
around a fixed point in the plane, in which all points in the plane turn around that fixed point through the same angle. One of the basic tenets of
Euclidean geometry is that two
5476:
5397:
2819:
2751:
4982:
2966:
7869:
5913:
1215:
591:
7184:
1250:
1124:
957:
707:
678:
620:
556:
499:
273:
233:
204:
8957:
7653:
7583:. The restriction to this stabilizer of above group homomorphism is an isomorphism. So the isometries that fix a given point form a group isomorphic to the orthogonal group.
6119:
717:
manner (that is, without choosing a preferred basis and a preferred origin). Another reason is that there is no standard origin nor any standard basis in the physical world.
6510:
6341:
8910:
8866:
6763:
5729:
1579:
6191:
2172:{\displaystyle {\Bigl \{}O+(1-\lambda ){\overrightarrow {OP}}+\lambda {\overrightarrow {OQ}}\mathrel {\Big |} \lambda \in \mathbb {R} {\Bigr \}},{\vphantom {\frac {(}{}}}}
7692:
5693:
4936:
4750:
are arbitrary points, one on each half-line. Although this is less used, one can define similarly the angle of segments or half-lines that do not share an initial point.
3334:
9045:
7289:
5514:
2494:
4091:
3116:
6430:
6346:
5417:
8731:
If the condition of being a bijection is removed, a function preserving the distance is necessarily injective, and is an isometry from its domain to its image.
6670:
4227:
2708:{\displaystyle {\begin{aligned}{\overrightarrow {E}}\times {\overrightarrow {E}}&\to \mathbb {R} \\(x,y)&\mapsto \langle x,y\rangle \end{aligned}}}
7143:{\displaystyle {\begin{aligned}\mathbb {R} ^{n}&\to E\\(x_{1}\dots ,x_{n})&\mapsto \left(O+x_{1}e_{1}+\dots +x_{n}e_{n}\right).\end{aligned}}}
3616:. Its proof is easy in this context, as, expressing this in terms of the inner product, one has, using bilinearity and symmetry of the inner product:
384:
satisfying certain relationships, expressible in terms of distance and angles. For example, there are two fundamental operations (referred to as
446:
The standard way to mathematically define a
Euclidean space, as carried out in the remainder of this article, is as a set of points on which a
1934:
is a
Euclidean subspace of dimension one. Since a vector space of dimension one is spanned by any nonzero vector, a line is a set of the form
2455:
if they have the same direction (i.e., the same associated vector space). Equivalently, they are parallel, if there is a translation vector
1077:
As previously explained, some of the basic properties of Euclidean spaces result from the structure of affine space. They are described in
5578:
2423:{\displaystyle PQ=QP={\Bigl \{}P+\lambda {\overrightarrow {PQ}}\mathrel {\Big |} 0\leq \lambda \leq 1{\Bigr \}}.{\vphantom {\frac {(}{}}}}
2019:{\displaystyle {\Bigl \{}P+\lambda {\overrightarrow {PQ}}\mathrel {\Big |} \lambda \in \mathbb {R} {\Bigr \}},{\vphantom {\frac {(}{}}}}
1450:{\displaystyle {\overrightarrow {F}}={\Bigl \{}{\overrightarrow {PQ}}\mathrel {\Big |} P\in F,Q\in F{\Bigr \}}{\vphantom {\frac {(}{}}}}
9343:
7360:
5744:
4676:
1029:
4630:
3455:
3409:
6396:
of the associated Euclidean vector spaces. This implies that two isometric Euclidean spaces have the same dimension. Conversely, if
392:, which means a shifting of the plane so that every point is shifted in the same direction and by the same distance. The other is
3518:
8743:
333:, and showed that these allow reducing geometric problems to algebraic computations with numbers. This reduction of geometry to
3133:
2847:
8547:
8367:, where this sum is less than 180°. Their introduction in the second half of 19th century, and the proof that their theory is
6128:
8171:
5519:
5993:
2961:) between two points of a Euclidean space is the norm of the translation vector that maps one point to the other; that is
7568:. The kernel of this homomorphism is the translation group, showing that it is a normal subgroup of the Euclidean group.
972:
denotes an action of a vector on a point. This notation is not ambiguous, as, to distinguish between the two meanings of
173:
There is essentially only one Euclidean space of each dimension; that is, all Euclidean spaces of a given dimension are
8372:
8057:. It is also widely used in all technical areas that are concerned with shapes, figure, location and position, such as
897:
9446:
9396:
9278:
9256:
9223:
9195:
8651:
6074:
5819:
on it, which is the same as a Euclidean frame, except that the basis is not required to be orthonormal. This define
431:, measurement instruments, and so on. A purely mathematical definition of Euclidean space also ignores questions of
8407:
is a space that in the neighborhood of each point resembles a Euclidean space. In technical terms, a manifold is a
6733:
367:, the definition of Euclidean space remained unchanged until the end of 19th century. The introduction of abstract
17:
8498:
7512:{\displaystyle {\overrightarrow {f}}{\Bigl (}{\overrightarrow {OP}}{\Bigr )}=f(P)-f(O).{\vphantom {\frac {(}{}}}}
4757:
is the angle of two segments, one on each line, the angle of any two other segments, one on each line, is either
8315:
lines meet in exactly one point". Projective space share with Euclidean and affine spaces the property of being
7719:
7530:
3307:
9336:
9287:
7259:
7897:
of a Euclidean space equals its dimension. This implies that Euclidean spaces of different dimensions are not
6603:{\displaystyle f(P)=O'+{\overrightarrow {f}}{\Bigl (}{\overrightarrow {OP}}{\Bigr )}{\vphantom {\frac {(}{}}}}
2517:
2276:
1134:. The importance of this particular example of Euclidean space lies in the fact that every Euclidean space is
9315:
8089:
5947:
4320:
8243:
have always two intersection points (possibly not distinct) in the complex affine space. Therefore, most of
6772:
5226:
5184:
2464:
1692:
1470:
770:
9650:
8465:
Euclidean spaces are trivially Riemannian manifolds. An example illustrating this well is the surface of a
8136:
7315:
6631:
6069:
In the case of a Euclidean vector space, an isometry that maps the origin to the origin preserves the norm
5322:
5288:
5000:
4198:
3341:
3266:
2914:
2577:
1892:
1865:
1835:
1808:
1796:{\displaystyle P+{\overrightarrow {V}}={\Bigl \{}P+v\mathrel {\Big |} v\in {\overrightarrow {V}}{\Bigr \}}}
1661:
1531:
1504:
1303:
1157:
819:
8966:
8231:
As soon as non-linear questions are considered, it is generally useful to consider affine spaces over the
5119:
5067:
4846:
9645:
9635:
9431:
9310:
8450:
at the points of the manifold (these tangent spaces are thus Euclidean vector spaces). This results in a
5943:
5422:
5353:
4831:
2792:
2778:
2724:
756:
such that the associated vector space is a Euclidean vector space. Euclidean spaces are sometimes called
236:
4941:
111:
8643:
8248:
7842:
7709:
5964:
This way of defining coordinates extends easily to other mathematical structures, and in particular to
5886:
4361:
1282:
and include the concepts of lines, subspaces, and parallelism, which are detailed in next subsections.
1188:
564:
450:
8311:" to Euclidean spaces, and, more generally to affine spaces, in order to make true the assertion "two
7160:
2266:{\displaystyle {\bigl \{}(1-\lambda )P+\lambda Q\mathrel {\big |} \lambda \in \mathbb {R} {\bigr \}}.}
1226:
1100:
683:
654:
596:
532:
475:
249:
209:
180:
9329:
9305:
8915:
8517:
7790:
As the special Euclidean group is a subgroup of index two of the Euclidean group, given a reflection
7616:
6123:
since the norm of a vector is its distance from the zero vector. It preserves also the inner product
8502:
7932:
The definition of Euclidean spaces that has been described in this article differs fundamentally of
6483:
6314:
4497:
9366:
9291:
9206:
8870:
8826:
8690:
8482:
8371:(if Euclidean geometry is not contradictory) is one of the paradoxes that are at the origin of the
8271:
8252:
8187:
8116:
8003:
7937:
7784:
7705:
5698:
5053:
864:
802:
409:
154:
8439:. However, none of these types of "resemblance" respect distances and angles, even approximately.
8423:
of a Euclidean space. Manifolds can be classified by increasing degree of this "resemblance" into
7909:) if and only if it is homeomorphic to an open subset of a Euclidean space of the same dimension.
7822:
6728:
to the zero vector and has the identity as associated linear map. The inverse isometry is the map
344:
Euclidean geometry was not applied in spaces of dimension more than three until the 19th century.
9640:
9574:
9569:
9549:
8428:
8224:. Affine spaces have many other uses in mathematics. In particular, as they are defined over any
8108:
7662:
7521:
It is straightforward to prove that this is a linear map that does not depend from the choice of
7153:
This means that, up to an isomorphism, there is exactly one Euclidean space of a given dimension.
5939:
5846:
5671:
4902:
2610:
992:
73:
57:
8251:. The shapes that are studied in algebraic geometry in these affine spaces are therefore called
9559:
9554:
9534:
9021:
8506:
8347:
8263:
7967:
7252:
4772:
4722:
As the multiplication of vectors by positive numbers do not change the angle, the angle of two
1274:
Some basic properties of Euclidean spaces depend only on the fact that a Euclidean space is an
860:
651:
A reason for introducing such an abstract definition of Euclidean spaces, and for working with
466:
393:
389:
9564:
9544:
9539:
8013:
7951:. The presentation of Euclidean spaces given in this article, is essentially issued from his
7894:
5481:
4840:(in fact, infinitely many in dimension higher than one, and two in dimension one), that is a
4434:
4151:{\displaystyle {\overrightarrow {AB}}\cdot {\overrightarrow {AC}}=0{\vphantom {\frac {(}{}}}}
3234:
405:
330:
767:
is a Euclidean space, its associated vector space (Euclidean vector space) is often denoted
114:
introduced Euclidean space for modeling the physical space. Their work was collected by the
9233:
9187:
8424:
8221:
8017:
7991:
7902:
7713:
7225:
7207:
4995:
4841:
3122:
3040:{\displaystyle d(P,Q)={\Bigl \|}{\overrightarrow {PQ}}{\Bigr \|}.{\vphantom {\frac {(}{}}}}
561:
It follows that everything that can be said about a Euclidean space can also be said about
436:
408:) if one can be transformed into the other by some sequence of translations, rotations and
318:
in modern language. This way of defining Euclidean space is still in use under the name of
297:
50:
6473:{\displaystyle {\overrightarrow {f}}\colon {\overrightarrow {E}}\to {\overrightarrow {F}}}
6389:{\displaystyle {\overrightarrow {f}}\colon {\overrightarrow {E}}\to {\overrightarrow {F}}}
3226:
is smaller than the sum of the lengths of the other edges. This is the origin of the term
3088:
2561:
It follows that in a Euclidean plane, two lines either meet in one point or are parallel.
8:
9441:
9436:
8694:
8655:
8451:
8398:
8364:
8331:
8225:
8161:
7963:
7576:
3613:
3126:
2606:
2555:
2438:
734:
458:
305:
100:
8084:
Space of dimensions higher than three occurs in several modern theories of physics; see
9615:
9456:
9411:
9245:
8675:
8639:
8536:
8356:
8335:
8308:
8244:
8203:
8100:
7995:
7971:
7836:
7697:
7561:
6305:
5820:
5402:
4387:
4172:
2948:
2900:
428:
319:
147:
128:
61:
1641:{\displaystyle F={\Bigl \{}P+v\mathrel {\Big |} v\in {\overrightarrow {F}}{\Bigr \}}.}
1081:
and its subsections. The properties resulting from the inner product are explained in
345:
9451:
9274:
9252:
9219:
9191:
8669:
8436:
8408:
8360:
8078:
8074:
8034:
7906:
7832:
7769:
6295:{\displaystyle x\cdot y={\tfrac {1}{2}}\left(\|x+y\|^{2}-\|x\|^{2}-\|y\|^{2}\right).}
5806:
4837:
2786:
2718:
1151:
979:
The fact that the action is free and transitive means that, for every pair of points
518:
447:
397:
385:
364:
293:
115:
8202:
of its definition, or, more precisely for proving that its theory is consistent, if
7802:
is an example of a rigid transformation that is not a rigid motion or a reflection.
326:
177:. Therefore it is usually possible to work with a specific Euclidean space, denoted
9381:
9240:
9211:
8722:
It may depend on the context or the author whether a subspace is parallel to itself
8376:
8302:
8085:
7952:
7872:
7799:
7565:
845:
158:
8135:
not of a geometrical nature. An example among many is the usual representation of
9426:
9229:
8532:
8524:
8443:
8432:
8259:
8240:
8166:
8093:
7917:
7913:
7887:
7810:
7298:
7203:
5177:
4312:
1925:
1686:
1464:
1291:
1279:
96:
34:
A point in three-dimensional Euclidean space can be located by three coordinates.
9210:, Wiley Classics Library, New York: John Wiley & Sons Inc., pp. x+214,
8278:
of degree higher than two has no point in the affine plane over the rationals."
7293:
They are in bijective correspondence with vectors. This is a reason for calling
352:, using both synthetic and algebraic methods, and discovered all of the regular
9508:
9493:
8699:
8540:
8513:
8510:
8286:
8232:
8131:. It is common to represent in a Euclidean space mathematical objects that are
8124:
7979:
5869:
4723:
4366:
432:
357:
166:
99:. The qualifier "Euclidean" is used to distinguish Euclidean spaces from other
92:
46:
7749:
is a normal subgroup of index two of the Euclidean group, which is called the
2275:
A standard convention allows using this formula in every Euclidean space, see
296:
as an abstraction of our physical space. Their great innovation, appearing in
9629:
9498:
9266:
8683:
8647:
8494:
8455:
8447:
8282:
8267:
8104:
7983:
7959:
5865:
3250:
3246:
2041:
there is exactly one line that passes through (contains) two distinct points.
1131:
143:
142:, which either were considered as evident (for example, there is exactly one
9518:
9483:
9376:
8470:
8416:
8412:
8388:
8324:
8290:
8275:
8215:
8058:
7898:
7828:
5981:
5958:
5873:
5831:
5816:
5812:
4398:
2892:
2781:
has been chosen, as, in this case, the inner product of two vectors is the
2284:
1275:
1269:
868:
749:
462:
368:
240:
162:
157:, the old postulates were re-formalized to define Euclidean spaces through
9215:
7794:, every rigid transformation that is not a rigid motion is the product of
7297:
the vector space associated to a Euclidean space. The translations form a
9603:
9386:
8420:
8368:
8320:
8199:
7987:
7975:
7944:
7924:(that is, contained in a ball). In particular, closed balls are compact.
7921:
7783:
Typical examples of rigid transformations that are not rigid motions are
7716:
two of the orthogonal group. Its inverse image by the group homomorphism
7564:
from the Euclidean group onto the group of linear isometries, called the
6715:{\displaystyle P\mapsto {\overrightarrow {OP}},{\vphantom {\frac {(}{}}}}
5876:
5061:
4896:
4298:{\displaystyle \theta =\arccos \left({\frac {x\cdot y}{|x|\,|y|}}\right)}
3402:
2782:
1135:
1127:
851:
753:
738:
522:
506:
424:
338:
276:
174:
65:
8509:
is an affine space with an associated real vector space equipped with a
8442:
Distances and angles can be defined on a smooth manifold by providing a
7955:, with the emphasis given on the groups of translations and isometries.
643:
9598:
9478:
8528:
8478:
8474:
8375:
of the beginning of 20th century, and motivated the systematization of
8103:, Euclidean spaces are also widely used in other areas of mathematics.
8070:
8066:
8008:
7948:
7777:
7344:
5845:
points that are not contained in a hyperplane. An affine basis define
2036:
are two distinct points of the Euclidean space as a part of the line.
9579:
9488:
9401:
9352:
8539:. It is a four-dimensional space, where the metric is defined by the
8316:
8195:
8128:
8120:
8054:
8050:
7883:
7806:
7224:. The rigid transformations of a Euclidean space form a group (under
5985:
4316:
2043:
This implies that two distinct lines intersect in at most one point.
381:
311:
138:
7920:. That is, a closed subset of a Euclidean space is compact if it is
5827:
for emphasizing that the basis vectors are not pairwise orthogonal.
5657:{\displaystyle v=\alpha _{1}e_{1}+\alpha _{2}e_{2}+\alpha _{3}e_{3}}
4353:
Angles are not useful in a Euclidean line, as they can be only 0 or
465:, and this allows defining lines, planes, subspaces, dimension, and
30:
9503:
9466:
9391:
9153:
8687:
8459:
8458:
do not exist in a Riemannian manifold, but their role is played by
8404:
8394:
8363:, where the sum of the angles of a triangle is more than 180°, and
8312:
8112:
7879:
7773:
5977:
5965:
5852:
Many other coordinates systems can be defined on a Euclidean space
3223:
2896:
2188:
In a Euclidean vector space, the zero vector is usually chosen for
353:
42:
8115:
is a space that is locally approximated by Euclidean spaces. Most
6990:
which is an isometry of Euclidean spaces. The inverse isometry is
3015:
2993:
9513:
8703:
8635:
8062:
8046:
8042:
4615:
are three points in a Euclidean space, the angle of the segments
420:
334:
133:
104:
7399:{\displaystyle {\overrightarrow {PQ}},{\vphantom {\frac {(}{}}}}
5797:
5783:{\displaystyle {\overrightarrow {OP}}.{\vphantom {\frac {(}{}}}}
4715:{\displaystyle {\overrightarrow {AC}}.{\vphantom {\frac {(}{}}}}
1068:{\displaystyle {\overrightarrow {PQ}}{\vphantom {\frac {)}{}}}.}
9116:
9114:
8466:
8236:
7933:
7886:
around each of their points. In other words, open balls form a
4666:{\displaystyle {\overrightarrow {AB}}{\vphantom {\frac {(}{}}}}
3491:{\displaystyle {\overrightarrow {AC}}{\vphantom {\frac {)}{}}}}
3445:{\displaystyle {\overrightarrow {AB}}{\vphantom {\frac {)}{}}}}
816:, and are commonly denoted by capital letters. The elements of
440:
401:
119:
8654:
of this manifold at a point is a function of the value of the
8307:
Originally, projective spaces have been introduced by adding "
7708:
is the normal subgroup of the orthogonal group that preserves
593:
Therefore, many authors, especially at elementary level, call
363:
Despite the wide use of Descartes' approach, which was called
9470:
9131:
9129:
8330:
As for affine spaces, projective spaces are defined over any
8198:
the space in a Euclidean space is a standard way for proving
8191:
8038:
4167:
2891:
The inner product and the norm allows expressing and proving
315:
9321:
9111:
7905:
asserts that a subset of a Euclidean space is open (for the
5961:, the longitude passes discontinuously from –180° to +180°.
4386:. In this case, the angle of two vectors can have any value
2046:
A more symmetric representation of the line passing through
521:
of the space of translations is equivalent with defining an
146:
passing through two points), or seemed impossible to prove (
4587:
3125:, as it is positive definite, symmetric, and satisfies the
91:
equal to one or two, they are commonly called respectively
9126:
8816:{\displaystyle f(\lambda x+\mu y)-\lambda f(x)-\mu f(y)=0}
8123:
in a Euclidean space of higher dimension. For example, an
2903:. The next subsection describe the most fundamental ones.
976:, it suffices to look at the nature of its left argument.
9065:
9063:
8247:
is built in complex affine spaces and affine spaces over
7805:
All groups that have been considered in this section are
7212:
An isometry from a Euclidean space onto itself is called
3222:. This inequality means that the length of any edge of a
469:. The inner product allows defining distance and angles.
337:
was a major change in point of view, as, until then, the
153:
After the introduction at the end of the 19th century of
7776:(the rigid motions that fix at least a point), and also
3233:
With the Euclidean distance, every Euclidean space is a
2554:
is a line (subspace of dimension one), this property is
136:, by starting from a few fundamental properties, called
8186:
Since the introduction, at the end of 19th century, of
5950:
coordinate systems (dimension 3) are defined this way.
5516:
as a convention) in a 3-dimensional Euclidean space is
5181:
is a set of data consisting of an orthonormal basis of
2763:
The inner product of a Euclidean space is often called
968:
in the left-hand side is a vector addition; each other
9060:
6619:
6208:
3212:
Moreover, the equality is true if and only if a point
2277:
Affine space § Affine combinations and barycenter
375:
348:
generalized Euclidean geometry to spaces of dimension
9024:
8969:
8918:
8873:
8829:
8746:
8550:
8220:
A Euclidean space is an affine space equipped with a
7845:
7722:
7665:
7619:
7533:
7422:
7363:
7318:
7262:
7163:
6998:
6837:
6775:
6736:
6673:
6634:
6520:
6486:
6433:
6349:
6317:
6194:
6131:
6077:
5996:
5889:
5747:
5701:
5674:
5581:
5568:{\displaystyle (\alpha _{1},\alpha _{2},\alpha _{3})}
5522:
5484:
5425:
5405:
5356:
5325:
5291:
5229:
5187:
5122:
5070:
5003:
4944:
4905:
4849:
4679:
4633:
4461:
4433:
The angle of two vectors does not change if they are
4230:
4201:
4094:
3624:
3521:
3458:
3412:
3344:
3310:
3269:
3136:
3091:
2969:
2917:
2850:
2795:
2727:
2620:
2580:
2520:
2467:
2324:
2200:
2192:; this allows simplifying the preceding formula into
2062:
1942:
1895:
1868:
1838:
1811:
1727:
1695:
1664:
1582:
1534:
1528:
as the associated vector space. This linear subspace
1507:
1473:
1362:
1306:
1229:
1191:
1160:
1103:
1032:
900:
822:
773:
686:
657:
599:
567:
535:
478:
252:
212:
183:
161:. Another definition of Euclidean spaces by means of
9075:
8665:
8235:
as an extension of Euclidean spaces. For example, a
6655:
can be considered as a Euclidean space. Every point
8963:is an isometry, this gives a linear combination of
5399:For example, the Cartesian coordinates of a vector
5056:computes an orthonormal basis such that, for every
4323:, the argument of the arccosine is in the interval
2185:is an arbitrary point (not necessary on the line).
1832:. (The associated vector space of this subspace is
360:) that exist in Euclidean spaces of any dimension.
9244:
9099:
9039:
9010:
8951:
8904:
8860:
8815:
8605:
7863:
7741:
7700:of the translation group and the orthogonal group.
7686:
7647:
7552:
7511:
7398:
7331:
7283:
7178:
7142:
6980:
6819:
6757:
6714:
6647:
6628:is a Euclidean space, its associated vector space
6602:
6504:
6472:
6388:
6335:
6294:
6176:
6113:
6059:
5907:
5782:
5723:
5687:
5656:
5567:
5508:
5470:
5411:
5391:
5338:
5304:
5273:
5203:
5160:
5108:
5044:
4976:
4930:
4887:
4714:
4665:
4593:
4370:of two vectors. The oriented angle of two vectors
4297:
4214:
4176:Positive and negative angles on the oriented plane
4150:
4078:
3602:
3490:
3444:
3357:
3328:
3290:(the associated vector space of a Euclidean space
3282:
3202:
3110:
3039:
2930:
2881:
2813:
2745:
2707:
2593:
2542:
2506:, there exists exactly one subspace that contains
2488:
2422:
2265:
2171:
2018:
1908:
1881:
1851:
1824:
1795:
1711:
1677:
1640:
1547:
1520:
1489:
1449:
1319:
1244:
1209:
1173:
1118:
1067:
951:
835:
789:
760:to distinguish them from Euclidean vector spaces.
701:
672:
614:
585:
550:
493:
427:detached from actual physical locations, specific
267:
227:
198:
9165:
9087:
8266:provide a link between (algebraic) geometry and
7827:The Euclidean distance makes a Euclidean space a
7457:
7435:
6952:
6880:
6581:
6559:
4753:The angle of two lines is defined as follows. If
3808:
3768:
3758:
3718:
2398:
2376:
2345:
2147:
2129:
2065:
1994:
1976:
1945:
1788:
1765:
1749:
1630:
1607:
1591:
1428:
1400:
1378:
9627:
4730:can be defined: it is the angle of the segments
4158:is used since these two vectors are orthogonal.
1181:defines an isomorphism of Euclidean spaces from
461:. The action of translations makes the space an
404:) of the plane should be considered equivalent (
380:One way to think of the Euclidean plane is as a
341:were defined in terms of lengths and distances.
8355:refers usually to geometrical spaces where the
8228:, they allow doing geometry in other contexts.
8111:are Euclidean vector spaces. More generally, a
8041:in the physical world. It is thus used in many
5281:allows defining Cartesian coordinates for both
2451:of the same dimension in a Euclidean space are
1138:to it. More precisely, given a Euclidean space
647:Origin-free illustration of the Euclidean plane
513:. Conversely, the choice of a point called the
443:, not something expressed in inches or metres.
87:when one wants to specify their dimension. For
4378:is then the opposite of the oriented angle of
871:of a Euclidean vector on the Euclidean space.
439:: the distance in a "mathematical" space is a
9337:
9247:A Short Account of the History of Mathematics
8523:A fundamental example of such a space is the
8037:, Euclidean space has been used for modeling
7947:suggested to define geometries through their
2255:
2237:
2203:
1889:(that is, a Euclidean space that is equal to
8996:
8989:
8977:
8970:
8890:
8874:
8846:
8830:
6304:An isometry of Euclidean vector spaces is a
6275:
6268:
6256:
6249:
6237:
6224:
6105:
6099:
6093:
6078:
5741:are the Cartesian coordinates of the vector
4919:
4906:
4798:of the two lines is the one in the interval
2857:
2851:
2808:
2796:
2740:
2728:
2698:
2686:
1097:A typical case of Euclidean vector space is
709:is that it is often preferable to work in a
287:
9303:
9069:
8341:
8285:has also been widely studied. For example,
8160:It has been suggested that this section be
7986:proposed simpler sets of axioms, which use
7974:, as they do not involve any definition of
7823:Real n-space § Topological properties
7251:The simplest Euclidean transformations are
3603:{\displaystyle |BC|^{2}=|AB|^{2}+|AC|^{2}.}
1126:viewed as a vector space equipped with the
9344:
9330:
7742:{\displaystyle f\to {\overrightarrow {f}}}
7553:{\displaystyle f\to {\overrightarrow {f}}}
5953:For points that are outside the domain of
2911:denotes an arbitrary Euclidean space, and
505:-tuples of real numbers equipped with the
8488:
8485:can be realized as Riemannian manifolds.
8142:
7943:Two different approaches have been used.
7848:
7343:, it is meant an isometry that is also a
7166:
7157:This justifies that many authors talk of
7005:
6854:
5892:
4274:
3203:{\displaystyle d(P,Q)\leq d(P,R)+d(R,Q).}
2938:denotes its vector space of translations.
2882:{\displaystyle \|x\|={\sqrt {x\cdot x}}.}
2656:
2308:in the preceding formulas. It is denoted
2249:
2141:
1988:
1232:
1223:is isomorphic to it, the Euclidean space
1194:
1106:
689:
660:
602:
570:
538:
481:
255:
215:
8606:{\displaystyle x^{2}+y^{2}+z^{2}-t^{2},}
8206:is consistent (which cannot be proved).
7927:
6665:defines an isometry of Euclidean spaces
6343:of Euclidean spaces defines an isometry
6177:{\displaystyle f(x)\cdot f(y)=x\cdot y,}
5796:
4825:
4437:by positive numbers. More precisely, if
4171:
2543:{\displaystyle P+{\overrightarrow {S}}.}
1088:
642:
29:
9286:
9105:
8289:over finite fields are widely used in
7579:of the Euclidean group with respect to
3054:
720:
525:between a Euclidean space of dimension
14:
9628:
9265:
9159:
9135:
9120:
9093:
7571:The isometries that fix a given point
6820:{\displaystyle (O,e_{1},\dots ,e_{n})}
5315:The Cartesian coordinates of a vector
5274:{\displaystyle (O,e_{1},\dots ,e_{n})}
5204:{\displaystyle {\overrightarrow {E}},}
4806:. In an oriented Euclidean plane, the
2942:
1712:{\displaystyle {\overrightarrow {E}},}
1490:{\displaystyle {\overrightarrow {E}}.}
1219:As every Euclidean space of dimension
790:{\displaystyle {\overrightarrow {E}}.}
9325:
9203:
9181:
9171:
9147:
8164:out into another article titled
8020:on segments. One can thus define the
7332:{\displaystyle {\overrightarrow {f}}}
6648:{\displaystyle {\overrightarrow {E}}}
5733:The Cartesian coordinates of a point
5339:{\displaystyle {\overrightarrow {E}}}
5305:{\displaystyle {\overrightarrow {E}}}
5045:{\displaystyle (b_{1},\dots ,b_{n}),}
4810:of two lines belongs to the interval
4215:{\displaystyle {\overrightarrow {E}}}
3358:{\displaystyle {\overrightarrow {E}}}
3283:{\displaystyle {\overrightarrow {E}}}
2931:{\displaystyle {\overrightarrow {E}}}
2594:{\displaystyle {\overrightarrow {E}}}
1919:
1909:{\displaystyle {\overrightarrow {E}}}
1882:{\displaystyle {\overrightarrow {E}}}
1852:{\displaystyle {\overrightarrow {V}}}
1825:{\displaystyle {\overrightarrow {V}}}
1805:is a Euclidean subspace of direction
1678:{\displaystyle {\overrightarrow {V}}}
1548:{\displaystyle {\overrightarrow {F}}}
1521:{\displaystyle {\overrightarrow {F}}}
1320:{\displaystyle {\overrightarrow {E}}}
1174:{\displaystyle {\overrightarrow {E}}}
1082:
1078:
836:{\displaystyle {\overrightarrow {E}}}
356:(higher-dimensional analogues of the
9251:(4th ed.). Dover Publications.
9239:
9081:
9011:{\displaystyle \|x\|^{2},\|y\|^{2},}
8619:) is temporal, and the other three (
8146:
8012:the following. In Hilbert's axioms,
7347:) in the following way: denoting by
6612:is an isometry of Euclidean spaces.
6060:{\displaystyle d(f(x),f(y))=d(x,y).}
5792:
5161:{\displaystyle (b_{1},\dots ,b_{i})}
5109:{\displaystyle (e_{1},\dots ,e_{i})}
4888:{\displaystyle (e_{1},\dots ,e_{n})}
4836:Every Euclidean vector space has an
4364:Euclidean plane, one can define the
4350:if angles are measured in degrees).
2777:. This is specially the case when a
413:
8296:
7339:of the associated vector space (by
6620:Isometry with prototypical examples
5664:. As the basis is orthonormal, the
5471:{\displaystyle (e_{1},e_{2},e_{3})}
5392:{\displaystyle e_{1},\dots ,e_{n}.}
2814:{\displaystyle \langle x,y\rangle }
2746:{\displaystyle \langle x,x\rangle }
2569:
1263:
1146:, the choice of a point, called an
376:Motivation of the modern definition
24:
8373:foundational crisis in mathematics
8119:can be modeled by a manifold, and
7197:
5834:of a Euclidean space of dimension
4977:{\displaystyle e_{i}\cdot e_{j}=0}
2831:in the remainder of this article.
2300:is the subset of points such that
1459:as the associated vector space of
509:is a Euclidean space of dimension
292:Euclidean space was introduced by
25:
9662:
9296:(3rd ed.). New York: Dover.
8281:Geometry in affine spaces over a
7882:are the subsets that contains an
7864:{\displaystyle \mathbb {R} ^{n},}
5988:preserving the distance, that is
5908:{\displaystyle \mathbb {R} ^{n}.}
1210:{\displaystyle \mathbb {R} ^{n}.}
882:provides a point that is denoted
859:, although, properly speaking, a
586:{\displaystyle \mathbb {R} ^{n}.}
235:, which can be represented using
9410:
8668:
8503:positive definite quadratic form
8481:. More generally, the spaces of
8382:
8334:, and are fundamental spaces of
8151:
7179:{\displaystyle \mathbb {R} ^{n}}
4938:) that are pairwise orthogonal (
4771:. One of these angles is in the
3302:if their inner product is zero:
3240:
2601:associated to a Euclidean space
1245:{\displaystyle \mathbb {R} ^{n}}
1119:{\displaystyle \mathbb {R} ^{n}}
952:{\displaystyle P+(v+w)=(P+v)+w.}
805:of its associated vector space.
702:{\displaystyle \mathbb {R} ^{n}}
673:{\displaystyle \mathbb {E} ^{n}}
615:{\displaystyle \mathbb {R} ^{n}}
551:{\displaystyle \mathbb {R} ^{n}}
494:{\displaystyle \mathbb {R} ^{n}}
423:theories, Euclidean space is an
268:{\displaystyle \mathbb {R} ^{n}}
228:{\displaystyle \mathbb {E} ^{n}}
199:{\displaystyle \mathbf {E} ^{n}}
186:
8952:{\displaystyle f(x)\cdot f(y).}
8734:
8499:positive definite bilinear form
8209:
7648:{\displaystyle g=t^{-1}\circ f}
6114:{\displaystyle \|f(x)\|=\|x\|,}
4575:
4522:
4521:
132:all properties of the space as
126:, with the great innovation of
9141:
8943:
8937:
8928:
8922:
8886:
8880:
8842:
8836:
8804:
8798:
8786:
8780:
8768:
8750:
8725:
8716:
8469:. In this case, geodesics are
7835:. This topology is called the
7726:
7537:
7500:
7489:
7483:
7474:
7468:
7387:
7266:
7065:
7058:
7029:
7019:
6965:
6875:
6849:
6814:
6776:
6740:
6703:
6677:
6591:
6530:
6524:
6505:{\displaystyle f\colon E\to F}
6496:
6457:
6373:
6336:{\displaystyle f\colon E\to F}
6327:
6156:
6150:
6141:
6135:
6090:
6084:
6051:
6039:
6030:
6027:
6021:
6012:
6006:
6000:
5801:3-dimensional skew coordinates
5771:
5562:
5523:
5503:
5485:
5465:
5426:
5268:
5230:
5155:
5123:
5103:
5071:
5036:
5004:
4882:
4850:
4703:
4654:
4572:
4560:
4518:
4506:
4486:
4468:
4284:
4276:
4270:
4262:
4139:
4059:
4047:
4033:
4021:
3697:
3642:
3630:
3587:
3575:
3561:
3549:
3535:
3523:
3479:
3433:
3194:
3182:
3173:
3161:
3152:
3140:
3104:
3093:
3028:
2985:
2973:
2683:
2676:
2664:
2652:
2432:
2411:
2220:
2208:
2160:
2088:
2076:
2007:
1438:
1053:
937:
925:
919:
907:
103:that were later considered in
13:
1:
9351:
9054:
8905:{\displaystyle \|f(y)\|^{2},}
8861:{\displaystyle \|f(x)\|^{2},}
8646:that has Minkowski spaces as
8411:, such that each point has a
7190:Euclidean space of dimension
6758:{\displaystyle v\mapsto O+v.}
6480:is an isometry, then the map
5971:
5724:{\displaystyle v\cdot e_{i}.}
4994:). More precisely, given any
3389:that share a common endpoint
1327:its associated vector space.
636:Euclidean space of dimension
558:viewed as a Euclidean space.
282:
27:Fundamental space of geometry
8709:
8501:, and so characterized by a
8497:of a real vector space is a
7497:
7384:
6962:
6700:
6588:
5860:, in the following way. Let
5768:
5695:is equal to the dot product
4700:
4651:
4627:is the angle of the vectors
4187:between two nonzero vectors
4136:
3694:
3513:form a right angle, one has
3476:
3430:
3025:
2459:that maps one to the other:
2408:
2157:
2004:
1435:
1285:
1050:
874:The action of a translation
801:of a Euclidean space is the
41:is the fundamental space of
7:
9311:Encyclopedia of Mathematics
9304:Solomentsev, E.D. (2001) ,
8740:Proof: one must prove that
8661:
8615:where the last coordinate (
8249:algebraically closed fields
7901:. Moreover, the theorem of
7816:
7696:the Euclidean group is the
7687:{\displaystyle f=t\circ g,}
5811:As a Euclidean space is an
5688:{\displaystyle \alpha _{i}}
4931:{\displaystyle \|e_{i}\|=1}
4832:Cartesian coordinate system
2779:Cartesian coordinate system
275:equipped with the standard
10:
9667:
9186:(5th ed.), New York:
8644:pseudo-Riemannian manifold
8392:
8386:
8345:
8300:
8253:affine algebraic varieties
8213:
7871:this topology is also the
7820:
7768:Rigid motions include the
7757:. Its elements are called
7598:the translation that maps
7312:defines a linear isometry
7201:
5804:
4829:
4414:equals the negative angle
4165:
3329:{\displaystyle u\cdot v=0}
3244:
2946:
2436:
1923:
1501:is a Euclidean space with
1289:
1267:
457:which is equipped with an
9612:
9591:
9527:
9465:
9419:
9408:
9359:
9184:Elementary Linear Algebra
9047:which simplifies to zero.
9040:{\displaystyle x\cdot y,}
7410:is an arbitrary point of
7284:{\displaystyle P\to P+v.}
5350:on the orthonormal basis
4782:, and the other being in
4321:Cauchy–Schwarz inequality
1862:A Euclidean vector space
1300:be a Euclidean space and
288:History of the definition
8483:non-Euclidean geometries
8446:Euclidean metric on the
8429:differentiable manifolds
8342:Non-Euclidean geometries
8262:and more generally over
8188:non-Euclidean geometries
8117:non-Euclidean geometries
8109:differentiable manifolds
8028:
7938:non-Euclidean geometries
7706:special orthogonal group
7301:of the Euclidean group.
7218:Euclidean transformation
6829:allows defining the map
5419:on an orthonormal basis
5346:are the coefficients of
5171:Given a Euclidean space
4541: have the same sign
4161:
3338:Two linear subspaces of
1930:In a Euclidean space, a
1254:standard Euclidean space
1252:is sometimes called the
892:. This action satisfies
865:geometric transformation
733:is a finite-dimensional
625:standard Euclidean space
155:non-Euclidean geometries
107:and modern mathematics.
72:of any positive integer
45:, intended to represent
8359:is false. They include
8327:of dimension one more.
8264:algebraic number fields
8258:Affine spaces over the
7751:special Euclidean group
5940:polar coordinate system
5847:barycentric coordinates
5509:{\displaystyle (x,y,z)}
4453:are real numbers, then
4390:an integer multiple of
3216:belongs to the segment
2753:is always positive for
2611:symmetric bilinear form
1083:§ Metric structure
1079:§ Affine structure
991:, there is exactly one
855:. They are also called
758:Euclidean affine spaces
400:(usually considered as
388:) on the plane. One is
58:three-dimensional space
9182:Anton, Howard (1987),
9041:
9012:
8953:
8906:
8862:
8817:
8686:, a generalization to
8607:
8507:pseudo-Euclidean space
8489:Pseudo-Euclidean space
8353:Non-Euclidean geometry
8348:Non-Euclidean geometry
8143:Other geometric spaces
7865:
7798:and a rigid motion. A
7743:
7712:. It is a subgroup of
7688:
7649:
7554:
7513:
7400:
7333:
7285:
7180:
7144:
6982:
6821:
6759:
6716:
6649:
6604:
6506:
6474:
6404:are Euclidean spaces,
6390:
6337:
6296:
6178:
6115:
6061:
5942:(dimension 2) and the
5927:are the components of
5909:
5815:, one can consider an
5802:
5784:
5725:
5689:
5658:
5569:
5510:
5478:(that may be named as
5472:
5413:
5393:
5340:
5312:in the following way.
5306:
5275:
5205:
5162:
5110:
5046:
4978:
4932:
4889:
4716:
4667:
4595:
4299:
4216:
4177:
4152:
4080:
3604:
3492:
3446:
3359:
3330:
3284:
3204:
3112:
3085:. It is often denoted
3077:between its endpoints
3041:
2932:
2883:
2815:
2747:
2709:
2595:
2544:
2490:
2489:{\displaystyle T=S+v.}
2424:
2267:
2173:
2020:
1910:
1883:
1853:
1826:
1797:
1713:
1679:
1642:
1549:
1522:
1491:
1451:
1321:
1246:
1211:
1175:
1120:
1069:
953:
837:
791:
729:Euclidean vector space
703:
674:
648:
616:
587:
552:
495:
269:
229:
200:
35:
9216:10.1002/9781118164518
9204:Artin, Emil (1988) ,
9042:
9013:
8954:
8907:
8863:
8818:
8608:
8425:topological manifolds
8393:Further information:
8272:Fermat's Last Theorem
8127:can be modeled by an
8088:. They occur also in
7928:Axiomatic definitions
7912:Euclidean spaces are
7895:topological dimension
7866:
7744:
7689:
7650:
7555:
7514:
7401:
7334:
7308:of a Euclidean space
7304:A Euclidean isometry
7295:space of translations
7286:
7181:
7145:
6983:
6822:
6760:
6717:
6650:
6605:
6507:
6475:
6391:
6338:
6297:
6179:
6116:
6062:
5910:
5883:to an open subset of
5800:
5785:
5726:
5690:
5659:
5570:
5511:
5473:
5414:
5394:
5341:
5307:
5276:
5206:
5163:
5111:
5047:
4979:
4933:
4890:
4826:Cartesian coordinates
4717:
4668:
4596:
4445:are two vectors, and
4300:
4217:
4175:
4153:
4081:
3605:
3493:
3447:
3360:
3331:
3285:
3235:complete metric space
3205:
3113:
3042:
2933:
2905:In these subsections,
2884:
2816:
2748:
2710:
2596:
2545:
2491:
2425:
2292:, joining the points
2268:
2174:
2021:
1911:
1884:
1854:
1827:
1798:
1714:
1680:
1643:
1550:
1523:
1497:A Euclidean subspace
1492:
1467:(vector subspace) of
1452:
1322:
1247:
1212:
1176:
1121:
1089:Prototypical examples
1085:and its subsections.
1070:
954:
838:
792:
704:
675:
646:
617:
588:
553:
496:
455:space of translations
331:Cartesian coordinates
270:
237:Cartesian coordinates
230:
201:
33:
9528:Dimensions by number
9273:, Berlin: Springer,
9162:, Proposition 9.1.3.
9022:
8967:
8916:
8871:
8827:
8744:
8702:, an application in
8548:
8471:arcs of great circle
8090:configuration spaces
8018:equivalence relation
7903:invariance of domain
7888:base of the topology
7843:
7720:
7663:
7617:
7531:
7505:
7420:
7392:
7361:
7316:
7260:
7222:rigid transformation
7208:Rigid transformation
7161:
6996:
6970:
6835:
6773:
6734:
6708:
6671:
6632:
6596:
6518:
6484:
6431:
6347:
6315:
6192:
6129:
6075:
5994:
5887:
5776:
5745:
5699:
5672:
5579:
5520:
5482:
5423:
5403:
5354:
5323:
5289:
5227:
5223:. A Cartesian frame
5185:
5120:
5068:
5054:Gram–Schmidt process
5001:
4942:
4903:
4847:
4708:
4677:
4659:
4631:
4459:
4228:
4199:
4144:
4092:
3702:
3622:
3519:
3484:
3456:
3438:
3410:
3342:
3308:
3267:
3255:Two nonzero vectors
3134:
3111:{\displaystyle |PQ|}
3089:
3033:
2967:
2957:(more precisely the
2915:
2848:
2793:
2725:
2618:
2578:
2518:
2465:
2416:
2322:
2198:
2165:
2060:
2012:
1940:
1893:
1866:
1836:
1809:
1725:
1693:
1662:
1580:
1532:
1505:
1471:
1443:
1360:
1304:
1227:
1189:
1158:
1101:
1058:
1030:
898:
820:
771:
721:Technical definition
684:
655:
597:
565:
533:
476:
250:
210:
181:
9651:Norms (mathematics)
9150:, pp. 209–215)
8695:functional analysis
8656:gravitational field
8473:, which are called
8452:Riemannian manifold
8399:Riemannian manifold
8365:hyperbolic geometry
8270:. For example, the
7968:Euclid's postulates
7958:On the other hand,
7577:stabilizer subgroup
7506:
7498:
7393:
7385:
6971:
6963:
6709:
6701:
6597:
6589:
5868:(or, more often, a
5823:, sometimes called
5777:
5769:
4726:with initial point
4709:
4701:
4660:
4652:
4397:. In particular, a
4180:The (non-oriented)
4145:
4137:
3703:
3695:
3614:Pythagorean theorem
3485:
3477:
3439:
3431:
3228:triangle inequality
3127:triangle inequality
3034:
3026:
2943:Distance and length
2607:inner product space
2510:and is parallel to
2439:Parallel (geometry)
2417:
2409:
2166:
2158:
2013:
2005:
1555:is also called the
1444:
1436:
1059:
1051:
993:displacement vector
867:resulting from the
735:inner product space
437:physical dimensions
79:, which are called
9646:Homogeneous spaces
9636:Euclidean geometry
9457:Degrees of freedom
9360:Dimensional spaces
9037:
9008:
8949:
8902:
8858:
8813:
8676:Mathematics portal
8640:general relativity
8603:
8537:special relativity
8477:in the context of
8437:analytic manifolds
8377:axiomatic theories
8357:parallel postulate
8336:algebraic geometry
8309:points at infinity
8245:algebraic geometry
8204:Euclidean geometry
8101:Euclidean geometry
7972:synthetic geometry
7962:proposed a set of
7861:
7837:Euclidean topology
7755:displacement group
7739:
7698:semidirect product
7684:
7645:
7562:group homomorphism
7550:
7509:
7396:
7329:
7281:
7232:and often denoted
7214:Euclidean isometry
7176:
7140:
7138:
6978:
6976:
6817:
6767:A Euclidean frame
6755:
6712:
6645:
6600:
6502:
6470:
6386:
6333:
6306:linear isomorphism
6292:
6217:
6174:
6111:
6057:
5905:
5821:affine coordinates
5803:
5780:
5721:
5685:
5654:
5565:
5506:
5468:
5409:
5389:
5336:
5302:
5271:
5219:and often denoted
5201:
5158:
5106:
5042:
4974:
4928:
4885:
4796:non-oriented angle
4712:
4663:
4591:
4586:
4295:
4212:
4178:
4148:
4076:
4074:
3600:
3488:
3442:
3355:
3326:
3280:
3200:
3121:The distance is a
3108:
3037:
2959:Euclidean distance
2949:Euclidean distance
2928:
2901:Euclidean geometry
2879:
2811:
2787:coordinate vectors
2743:
2705:
2703:
2591:
2550:In the case where
2540:
2486:
2420:
2263:
2169:
2016:
1920:Lines and segments
1906:
1879:
1849:
1822:
1793:
1709:
1675:
1638:
1545:
1518:
1487:
1447:
1336:Euclidean subspace
1317:
1278:. They are called
1242:
1207:
1171:
1116:
1065:
949:
833:
787:
699:
670:
649:
612:
583:
548:
491:
320:synthetic geometry
265:
225:
196:
148:parallel postulate
62:Euclidean geometry
36:
9623:
9622:
9432:Lebesgue covering
9397:Algebraic variety
9306:"Euclidean space"
9293:Regular Polytopes
9207:Geometric Algebra
9084:, pp. 50–62.
8409:topological space
8361:elliptic geometry
8274:can be stated "a
8184:
8183:
8179:
8079:technical drawing
8075:industrial design
8004:Geometric Algebra
7992:Birkhoff's axioms
7970:. They belong to
7907:subspace topology
7839:. In the case of
7833:topological space
7737:
7594:an isometry, and
7548:
7504:
7503:
7453:
7431:
7391:
7390:
7377:
7327:
6969:
6968:
6948:
6911:
6707:
6706:
6693:
6643:
6595:
6594:
6577:
6555:
6468:
6455:
6442:
6384:
6371:
6358:
6216:
5849:for every point.
5807:Coordinate system
5793:Other coordinates
5775:
5774:
5761:
5412:{\displaystyle v}
5334:
5300:
5196:
4838:orthonormal basis
4707:
4706:
4693:
4658:
4657:
4647:
4579:
4542:
4534:
4526:
4289:
4210:
4143:
4142:
4126:
4108:
4008:
3990:
3972:
3954:
3929:
3911:
3890:
3872:
3854:
3836:
3804:
3786:
3754:
3736:
3701:
3700:
3690:
3672:
3483:
3482:
3472:
3437:
3436:
3426:
3353:
3278:
3032:
3031:
3011:
2926:
2874:
2719:positive definite
2646:
2633:
2609:. This implies a
2589:
2574:The vector space
2535:
2415:
2414:
2372:
2164:
2163:
2125:
2104:
2011:
2010:
1972:
1904:
1877:
1847:
1820:
1784:
1742:
1704:
1673:
1626:
1543:
1516:
1482:
1442:
1441:
1396:
1371:
1315:
1280:affine properties
1169:
1152:orthonormal basis
1057:
1056:
1046:
846:Euclidean vectors
831:
782:
519:orthonormal basis
448:real vector space
365:analytic geometry
303:was to build and
170:Euclidean space.
49:. Originally, in
16:(Redirected from
9658:
9420:Other dimensions
9414:
9382:Projective space
9346:
9339:
9332:
9323:
9322:
9318:
9300:
9283:
9262:
9250:
9241:Ball, W.W. Rouse
9236:
9200:
9175:
9169:
9163:
9157:
9151:
9145:
9139:
9133:
9124:
9118:
9109:
9103:
9097:
9091:
9085:
9079:
9073:
9070:Solomentsev 2001
9067:
9048:
9046:
9044:
9043:
9038:
9017:
9015:
9014:
9009:
9004:
9003:
8985:
8984:
8962:
8958:
8956:
8955:
8950:
8911:
8909:
8908:
8903:
8898:
8897:
8867:
8865:
8864:
8859:
8854:
8853:
8822:
8820:
8819:
8814:
8738:
8732:
8729:
8723:
8720:
8678:
8673:
8672:
8612:
8610:
8609:
8604:
8599:
8598:
8586:
8585:
8573:
8572:
8560:
8559:
8444:smoothly varying
8433:smooth manifolds
8379:in mathematics.
8303:Projective space
8297:Projective space
8260:rational numbers
8175:
8155:
8154:
8147:
8094:physical systems
8086:Higher dimension
7953:Erlangen program
7873:product topology
7870:
7868:
7867:
7862:
7857:
7856:
7851:
7811:algebraic groups
7800:glide reflection
7797:
7793:
7772:, translations,
7748:
7746:
7745:
7740:
7738:
7730:
7693:
7691:
7690:
7685:
7658:
7654:
7652:
7651:
7646:
7638:
7637:
7612:
7601:
7597:
7593:
7589:
7582:
7574:
7566:orthogonal group
7559:
7557:
7556:
7551:
7549:
7541:
7524:
7518:
7516:
7515:
7510:
7508:
7507:
7499:
7461:
7460:
7454:
7449:
7441:
7439:
7438:
7432:
7424:
7413:
7409:
7405:
7403:
7402:
7397:
7395:
7394:
7386:
7378:
7373:
7365:
7356:
7338:
7336:
7335:
7330:
7328:
7320:
7311:
7307:
7290:
7288:
7287:
7282:
7247:
7239:
7198:Euclidean group
7193:
7185:
7183:
7182:
7177:
7175:
7174:
7169:
7149:
7147:
7146:
7141:
7139:
7132:
7128:
7127:
7126:
7117:
7116:
7098:
7097:
7088:
7087:
7057:
7056:
7041:
7040:
7014:
7013:
7008:
6987:
6985:
6984:
6979:
6977:
6973:
6972:
6964:
6956:
6955:
6949:
6944:
6936:
6931:
6930:
6912:
6907:
6899:
6894:
6893:
6884:
6883:
6863:
6862:
6857:
6828:
6826:
6824:
6823:
6818:
6813:
6812:
6794:
6793:
6764:
6762:
6761:
6756:
6727:
6721:
6719:
6718:
6713:
6711:
6710:
6702:
6694:
6689:
6681:
6664:
6654:
6652:
6651:
6646:
6644:
6636:
6627:
6609:
6607:
6606:
6601:
6599:
6598:
6590:
6585:
6584:
6578:
6573:
6565:
6563:
6562:
6556:
6548:
6543:
6511:
6509:
6508:
6503:
6479:
6477:
6476:
6471:
6469:
6461:
6456:
6448:
6443:
6435:
6426:
6421:
6413:
6403:
6399:
6395:
6393:
6392:
6387:
6385:
6377:
6372:
6364:
6359:
6351:
6342:
6340:
6339:
6334:
6301:
6299:
6298:
6293:
6288:
6284:
6283:
6282:
6264:
6263:
6245:
6244:
6218:
6209:
6183:
6181:
6180:
6175:
6120:
6118:
6117:
6112:
6066:
6064:
6063:
6058:
5956:
5937:
5926:
5922:
5914:
5912:
5911:
5906:
5901:
5900:
5895:
5882:
5863:
5859:
5855:
5844:
5837:
5825:skew coordinates
5789:
5787:
5786:
5781:
5779:
5778:
5770:
5762:
5757:
5749:
5740:
5736:
5730:
5728:
5727:
5722:
5717:
5716:
5694:
5692:
5691:
5686:
5684:
5683:
5668:-th coefficient
5667:
5663:
5661:
5660:
5655:
5653:
5652:
5643:
5642:
5630:
5629:
5620:
5619:
5607:
5606:
5597:
5596:
5574:
5572:
5571:
5566:
5561:
5560:
5548:
5547:
5535:
5534:
5515:
5513:
5512:
5507:
5477:
5475:
5474:
5469:
5464:
5463:
5451:
5450:
5438:
5437:
5418:
5416:
5415:
5410:
5398:
5396:
5395:
5390:
5385:
5384:
5366:
5365:
5349:
5345:
5343:
5342:
5337:
5335:
5327:
5318:
5311:
5309:
5308:
5303:
5301:
5293:
5284:
5280:
5278:
5277:
5272:
5267:
5266:
5248:
5247:
5222:
5214:
5210:
5208:
5207:
5202:
5197:
5189:
5174:
5167:
5165:
5164:
5159:
5154:
5153:
5135:
5134:
5115:
5113:
5112:
5107:
5102:
5101:
5083:
5082:
5059:
5051:
5049:
5048:
5043:
5035:
5034:
5016:
5015:
4993:
4983:
4981:
4980:
4975:
4967:
4966:
4954:
4953:
4937:
4935:
4934:
4929:
4918:
4917:
4894:
4892:
4891:
4886:
4881:
4880:
4862:
4861:
4821:
4805:
4793:
4781:
4770:
4760:
4756:
4749:
4745:
4741:
4735:
4729:
4721:
4719:
4718:
4713:
4711:
4710:
4702:
4694:
4689:
4681:
4672:
4670:
4669:
4664:
4662:
4661:
4653:
4648:
4643:
4635:
4626:
4620:
4614:
4610:
4606:
4600:
4598:
4597:
4592:
4590:
4589:
4580:
4577:
4543:
4540:
4535:
4532:
4527:
4524:
4452:
4448:
4444:
4440:
4429:
4413:
4396:
4385:
4381:
4377:
4373:
4356:
4349:
4341:
4330:
4326:
4310:
4304:
4302:
4301:
4296:
4294:
4290:
4288:
4287:
4279:
4273:
4265:
4259:
4248:
4221:
4219:
4218:
4213:
4211:
4203:
4194:
4190:
4186:
4157:
4155:
4154:
4149:
4147:
4146:
4138:
4127:
4122:
4114:
4109:
4104:
4096:
4085:
4083:
4082:
4077:
4075:
4068:
4067:
4062:
4050:
4042:
4041:
4036:
4024:
4013:
4009:
4004:
3996:
3991:
3986:
3978:
3973:
3968:
3960:
3955:
3950:
3942:
3934:
3930:
3925:
3917:
3912:
3907:
3899:
3891:
3886:
3878:
3873:
3868:
3860:
3855:
3850:
3842:
3837:
3832:
3824:
3816:
3812:
3811:
3805:
3800:
3792:
3787:
3782:
3774:
3772:
3771:
3762:
3761:
3755:
3750:
3742:
3737:
3732:
3724:
3722:
3721:
3709:
3705:
3704:
3696:
3691:
3686:
3678:
3673:
3668:
3660:
3651:
3650:
3645:
3633:
3609:
3607:
3606:
3601:
3596:
3595:
3590:
3578:
3570:
3569:
3564:
3552:
3544:
3543:
3538:
3526:
3512:
3506:
3498:are orthogonal.
3497:
3495:
3494:
3489:
3487:
3486:
3478:
3473:
3468:
3460:
3451:
3449:
3448:
3443:
3441:
3440:
3432:
3427:
3422:
3414:
3394:
3388:
3382:
3364:
3362:
3361:
3356:
3354:
3346:
3335:
3333:
3332:
3327:
3293:
3289:
3287:
3286:
3281:
3279:
3271:
3262:
3258:
3221:
3215:
3209:
3207:
3206:
3201:
3117:
3115:
3114:
3109:
3107:
3096:
3076:
3062:is the distance
3061:
3046:
3044:
3043:
3038:
3036:
3035:
3027:
3019:
3018:
3012:
3007:
2999:
2997:
2996:
2937:
2935:
2934:
2929:
2927:
2919:
2909:
2888:
2886:
2885:
2880:
2875:
2864:
2841:
2830:
2821:will be denoted
2820:
2818:
2817:
2812:
2776:
2759:
2752:
2750:
2749:
2744:
2714:
2712:
2711:
2706:
2704:
2659:
2647:
2639:
2634:
2626:
2604:
2600:
2598:
2597:
2592:
2590:
2582:
2570:Metric structure
2556:Playfair's axiom
2553:
2549:
2547:
2546:
2541:
2536:
2528:
2513:
2509:
2505:
2501:
2495:
2493:
2492:
2487:
2458:
2450:
2446:
2429:
2427:
2426:
2421:
2419:
2418:
2410:
2402:
2401:
2380:
2379:
2373:
2368:
2360:
2349:
2348:
2315:
2311:
2307:
2299:
2295:
2272:
2270:
2269:
2264:
2259:
2258:
2252:
2241:
2240:
2207:
2206:
2191:
2184:
2178:
2176:
2175:
2170:
2168:
2167:
2159:
2151:
2150:
2144:
2133:
2132:
2126:
2121:
2113:
2105:
2100:
2092:
2069:
2068:
2053:
2049:
2039:It follows that
2035:
2031:
2025:
2023:
2022:
2017:
2015:
2014:
2006:
1998:
1997:
1991:
1980:
1979:
1973:
1968:
1960:
1949:
1948:
1915:
1913:
1912:
1907:
1905:
1897:
1888:
1886:
1885:
1880:
1878:
1870:
1858:
1856:
1855:
1850:
1848:
1840:
1831:
1829:
1828:
1823:
1821:
1813:
1802:
1800:
1799:
1794:
1792:
1791:
1785:
1777:
1769:
1768:
1753:
1752:
1743:
1735:
1718:
1716:
1715:
1710:
1705:
1697:
1684:
1682:
1681:
1676:
1674:
1666:
1657:
1653:
1647:
1645:
1644:
1639:
1634:
1633:
1627:
1619:
1611:
1610:
1595:
1594:
1573:
1569:
1562:
1554:
1552:
1551:
1546:
1544:
1536:
1527:
1525:
1524:
1519:
1517:
1509:
1500:
1496:
1494:
1493:
1488:
1483:
1475:
1462:
1456:
1454:
1453:
1448:
1446:
1445:
1437:
1432:
1431:
1404:
1403:
1397:
1392:
1384:
1382:
1381:
1372:
1364:
1353:
1349:
1345:
1326:
1324:
1323:
1318:
1316:
1308:
1299:
1264:Affine structure
1259:
1251:
1249:
1248:
1243:
1241:
1240:
1235:
1222:
1216:
1214:
1213:
1208:
1203:
1202:
1197:
1184:
1180:
1178:
1177:
1172:
1170:
1162:
1145:
1141:
1125:
1123:
1122:
1117:
1115:
1114:
1109:
1074:
1072:
1071:
1066:
1061:
1060:
1052:
1047:
1042:
1034:
1025:
1015:
1011:
997:
990:
975:
971:
967:
958:
956:
955:
950:
891:
881:
877:
842:
840:
839:
834:
832:
824:
811:
808:The elements of
796:
794:
793:
788:
783:
775:
766:
731:
730:
708:
706:
705:
700:
698:
697:
692:
679:
677:
676:
671:
669:
668:
663:
639:
631:
621:
619:
618:
613:
611:
610:
605:
592:
590:
589:
584:
579:
578:
573:
557:
555:
554:
549:
547:
546:
541:
528:
512:
504:
500:
498:
497:
492:
490:
489:
484:
429:reference frames
351:
274:
272:
271:
266:
264:
263:
258:
244:
234:
232:
231:
226:
224:
223:
218:
205:
203:
202:
197:
195:
194:
189:
159:axiomatic theory
97:Euclidean planes
70:Euclidean spaces
64:, but in modern
21:
18:Euclidean spaces
9666:
9665:
9661:
9660:
9659:
9657:
9656:
9655:
9626:
9625:
9624:
9619:
9608:
9587:
9523:
9461:
9415:
9406:
9372:Euclidean space
9355:
9350:
9288:Coxeter, H.S.M.
9281:
9259:
9226:
9198:
9178:
9170:
9166:
9158:
9154:
9146:
9142:
9134:
9127:
9119:
9112:
9104:
9100:
9092:
9088:
9080:
9076:
9068:
9061:
9057:
9052:
9051:
9023:
9020:
9019:
8999:
8995:
8980:
8976:
8968:
8965:
8964:
8960:
8917:
8914:
8913:
8893:
8889:
8872:
8869:
8868:
8849:
8845:
8828:
8825:
8824:
8745:
8742:
8741:
8739:
8735:
8730:
8726:
8721:
8717:
8712:
8674:
8667:
8664:
8658:at this point.
8631:) are spatial.
8594:
8590:
8581:
8577:
8568:
8564:
8555:
8551:
8549:
8546:
8545:
8527:, which is the
8525:Minkowski space
8491:
8401:
8391:
8385:
8350:
8344:
8305:
8299:
8287:elliptic curves
8233:complex numbers
8218:
8212:
8180:
8167:Geometric space
8156:
8152:
8145:
8031:
7996:Tarski's axioms
7930:
7918:locally compact
7852:
7847:
7846:
7844:
7841:
7840:
7825:
7819:
7795:
7791:
7729:
7721:
7718:
7717:
7664:
7661:
7660:
7656:
7630:
7626:
7618:
7615:
7614:
7613:. The isometry
7603:
7599:
7595:
7591:
7587:
7580:
7572:
7540:
7532:
7529:
7528:
7522:
7496:
7495:
7456:
7455:
7442:
7440:
7434:
7433:
7423:
7421:
7418:
7417:
7411:
7407:
7383:
7382:
7366:
7364:
7362:
7359:
7358:
7348:
7341:linear isometry
7319:
7317:
7314:
7313:
7309:
7305:
7299:normal subgroup
7261:
7258:
7257:
7241:
7233:
7230:Euclidean group
7210:
7204:Euclidean group
7202:Main articles:
7200:
7191:
7170:
7165:
7164:
7162:
7159:
7158:
7137:
7136:
7122:
7118:
7112:
7108:
7093:
7089:
7083:
7079:
7072:
7068:
7061:
7052:
7048:
7036:
7032:
7026:
7025:
7015:
7009:
7004:
7003:
6999:
6997:
6994:
6993:
6975:
6974:
6961:
6960:
6951:
6950:
6937:
6935:
6926:
6922:
6900:
6898:
6889:
6885:
6879:
6878:
6871:
6865:
6864:
6858:
6853:
6852:
6845:
6838:
6836:
6833:
6832:
6808:
6804:
6789:
6785:
6774:
6771:
6770:
6768:
6735:
6732:
6731:
6725:
6699:
6698:
6682:
6680:
6672:
6669:
6668:
6656:
6635:
6633:
6630:
6629:
6625:
6622:
6587:
6586:
6580:
6579:
6566:
6564:
6558:
6557:
6547:
6536:
6519:
6516:
6515:
6485:
6482:
6481:
6460:
6447:
6434:
6432:
6429:
6428:
6419:
6415:
6405:
6401:
6397:
6376:
6363:
6350:
6348:
6345:
6344:
6316:
6313:
6312:
6278:
6274:
6259:
6255:
6240:
6236:
6223:
6219:
6207:
6193:
6190:
6189:
6130:
6127:
6126:
6076:
6073:
6072:
5995:
5992:
5991:
5974:
5954:
5928:
5924:
5920:
5896:
5891:
5890:
5888:
5885:
5884:
5880:
5861:
5857:
5853:
5839:
5835:
5809:
5795:
5767:
5766:
5750:
5748:
5746:
5743:
5742:
5738:
5734:
5712:
5708:
5700:
5697:
5696:
5679:
5675:
5673:
5670:
5669:
5665:
5648:
5644:
5638:
5634:
5625:
5621:
5615:
5611:
5602:
5598:
5592:
5588:
5580:
5577:
5576:
5556:
5552:
5543:
5539:
5530:
5526:
5521:
5518:
5517:
5483:
5480:
5479:
5459:
5455:
5446:
5442:
5433:
5429:
5424:
5421:
5420:
5404:
5401:
5400:
5380:
5376:
5361:
5357:
5355:
5352:
5351:
5347:
5326:
5324:
5321:
5320:
5316:
5292:
5290:
5287:
5286:
5282:
5262:
5258:
5243:
5239:
5228:
5225:
5224:
5220:
5212:
5211:and a point of
5188:
5186:
5183:
5182:
5178:Cartesian frame
5172:
5149:
5145:
5130:
5126:
5121:
5118:
5117:
5097:
5093:
5078:
5074:
5069:
5066:
5065:
5057:
5030:
5026:
5011:
5007:
5002:
4999:
4998:
4985:
4962:
4958:
4949:
4945:
4943:
4940:
4939:
4913:
4909:
4904:
4901:
4900:
4876:
4872:
4857:
4853:
4848:
4845:
4844:
4834:
4828:
4811:
4799:
4783:
4775:
4762:
4758:
4754:
4747:
4743:
4737:
4731:
4727:
4699:
4698:
4682:
4680:
4678:
4675:
4674:
4650:
4649:
4636:
4634:
4632:
4629:
4628:
4622:
4616:
4612:
4608:
4604:
4585:
4584:
4576:
4545:
4544:
4539:
4533: and
4531:
4523:
4493:
4492:
4460:
4457:
4456:
4450:
4446:
4442:
4438:
4415:
4401:
4391:
4383:
4379:
4375:
4371:
4354:
4343:
4332:
4328:
4325:[−1, 1]
4324:
4313:principal value
4308:
4283:
4275:
4269:
4261:
4260:
4249:
4247:
4243:
4229:
4226:
4225:
4202:
4200:
4197:
4196:
4192:
4188:
4184:
4170:
4164:
4135:
4134:
4115:
4113:
4097:
4095:
4093:
4090:
4089:
4073:
4072:
4063:
4058:
4057:
4046:
4037:
4032:
4031:
4020:
4011:
4010:
3997:
3995:
3979:
3977:
3961:
3959:
3943:
3941:
3932:
3931:
3918:
3916:
3900:
3898:
3879:
3877:
3861:
3859:
3843:
3841:
3825:
3823:
3814:
3813:
3807:
3806:
3793:
3791:
3775:
3773:
3767:
3766:
3757:
3756:
3743:
3741:
3725:
3723:
3717:
3716:
3707:
3706:
3693:
3692:
3679:
3677:
3661:
3659:
3652:
3646:
3641:
3640:
3629:
3625:
3623:
3620:
3619:
3591:
3586:
3585:
3574:
3565:
3560:
3559:
3548:
3539:
3534:
3533:
3522:
3520:
3517:
3516:
3508:
3502:
3475:
3474:
3461:
3459:
3457:
3454:
3453:
3429:
3428:
3415:
3413:
3411:
3408:
3407:
3406:if the vectors
3390:
3384:
3378:
3345:
3343:
3340:
3339:
3309:
3306:
3305:
3291:
3270:
3268:
3265:
3264:
3260:
3256:
3253:
3245:Main articles:
3243:
3217:
3213:
3135:
3132:
3131:
3103:
3092:
3090:
3087:
3086:
3063:
3057:
3024:
3023:
3014:
3013:
3000:
2998:
2992:
2991:
2968:
2965:
2964:
2951:
2945:
2918:
2916:
2913:
2912:
2907:
2863:
2849:
2846:
2845:
2839:
2822:
2794:
2791:
2790:
2768:
2754:
2726:
2723:
2722:
2702:
2701:
2679:
2661:
2660:
2655:
2648:
2638:
2625:
2621:
2619:
2616:
2615:
2602:
2581:
2579:
2576:
2575:
2572:
2551:
2527:
2519:
2516:
2515:
2511:
2507:
2503:
2502:and a subspace
2499:
2466:
2463:
2462:
2456:
2448:
2444:
2441:
2435:
2407:
2406:
2397:
2396:
2375:
2374:
2361:
2359:
2344:
2343:
2323:
2320:
2319:
2313:
2309:
2301:
2297:
2293:
2254:
2253:
2248:
2236:
2235:
2202:
2201:
2199:
2196:
2195:
2189:
2182:
2156:
2155:
2146:
2145:
2140:
2128:
2127:
2114:
2112:
2093:
2091:
2064:
2063:
2061:
2058:
2057:
2051:
2047:
2033:
2029:
2003:
2002:
1993:
1992:
1987:
1975:
1974:
1961:
1959:
1944:
1943:
1941:
1938:
1937:
1928:
1926:Line (geometry)
1922:
1896:
1894:
1891:
1890:
1869:
1867:
1864:
1863:
1839:
1837:
1834:
1833:
1812:
1810:
1807:
1806:
1787:
1786:
1776:
1764:
1763:
1748:
1747:
1734:
1726:
1723:
1722:
1696:
1694:
1691:
1690:
1687:linear subspace
1665:
1663:
1660:
1659:
1655:
1651:
1650:Conversely, if
1629:
1628:
1618:
1606:
1605:
1590:
1589:
1581:
1578:
1577:
1571:
1567:
1560:
1535:
1533:
1530:
1529:
1508:
1506:
1503:
1502:
1498:
1474:
1472:
1469:
1468:
1465:linear subspace
1460:
1434:
1433:
1427:
1426:
1399:
1398:
1385:
1383:
1377:
1376:
1363:
1361:
1358:
1357:
1351:
1347:
1343:
1340:affine subspace
1307:
1305:
1302:
1301:
1297:
1294:
1292:Flat (geometry)
1288:
1272:
1266:
1257:
1236:
1231:
1230:
1228:
1225:
1224:
1220:
1198:
1193:
1192:
1190:
1187:
1186:
1182:
1161:
1159:
1156:
1155:
1143:
1139:
1110:
1105:
1104:
1102:
1099:
1098:
1091:
1049:
1048:
1035:
1033:
1031:
1028:
1027:
1017:
1013:
999:
995:
980:
973:
969:
965:
899:
896:
895:
883:
879:
875:
823:
821:
818:
817:
809:
774:
772:
769:
768:
764:
746:Euclidean space
728:
727:
723:
711:coordinate-free
693:
688:
687:
685:
682:
681:
664:
659:
658:
656:
653:
652:
637:
629:
606:
601:
600:
598:
595:
594:
574:
569:
568:
566:
563:
562:
542:
537:
536:
534:
531:
530:
526:
510:
502:
485:
480:
479:
477:
474:
473:
433:units of length
378:
358:Platonic solids
349:
346:Ludwig Schläfli
290:
285:
259:
254:
253:
251:
248:
247:
242:
219:
214:
213:
211:
208:
207:
190:
185:
184:
182:
179:
178:
112:Greek geometers
93:Euclidean lines
39:Euclidean space
28:
23:
22:
15:
12:
11:
5:
9664:
9654:
9653:
9648:
9643:
9641:Linear algebra
9638:
9621:
9620:
9613:
9610:
9609:
9607:
9606:
9601:
9595:
9593:
9589:
9588:
9586:
9585:
9577:
9572:
9567:
9562:
9557:
9552:
9547:
9542:
9537:
9531:
9529:
9525:
9524:
9522:
9521:
9516:
9511:
9509:Cross-polytope
9506:
9501:
9496:
9494:Hyperrectangle
9491:
9486:
9481:
9475:
9473:
9463:
9462:
9460:
9459:
9454:
9449:
9444:
9439:
9434:
9429:
9423:
9421:
9417:
9416:
9409:
9407:
9405:
9404:
9399:
9394:
9389:
9384:
9379:
9374:
9369:
9363:
9361:
9357:
9356:
9349:
9348:
9341:
9334:
9326:
9320:
9319:
9301:
9284:
9279:
9267:Berger, Marcel
9263:
9257:
9237:
9224:
9201:
9196:
9177:
9176:
9164:
9152:
9140:
9125:
9123:, Section 9.1.
9110:
9098:
9086:
9074:
9058:
9056:
9053:
9050:
9049:
9036:
9033:
9030:
9027:
9007:
9002:
8998:
8994:
8991:
8988:
8983:
8979:
8975:
8972:
8948:
8945:
8942:
8939:
8936:
8933:
8930:
8927:
8924:
8921:
8901:
8896:
8892:
8888:
8885:
8882:
8879:
8876:
8857:
8852:
8848:
8844:
8841:
8838:
8835:
8832:
8812:
8809:
8806:
8803:
8800:
8797:
8794:
8791:
8788:
8785:
8782:
8779:
8776:
8773:
8770:
8767:
8764:
8761:
8758:
8755:
8752:
8749:
8733:
8724:
8714:
8713:
8711:
8708:
8707:
8706:
8700:Position space
8697:
8680:
8679:
8663:
8660:
8648:tangent spaces
8638:into account,
8602:
8597:
8593:
8589:
8584:
8580:
8576:
8571:
8567:
8563:
8558:
8554:
8541:quadratic form
8514:quadratic form
8511:non-degenerate
8490:
8487:
8456:straight lines
8448:tangent spaces
8387:Main article:
8384:
8381:
8346:Main article:
8343:
8340:
8301:Main article:
8298:
8295:
8214:Main article:
8211:
8208:
8182:
8181:
8159:
8157:
8150:
8144:
8141:
8125:elliptic space
8105:Tangent spaces
8035:ancient Greeks
8030:
8027:
7980:G. D. Birkhoff
7966:, inspired by
7929:
7926:
7860:
7855:
7850:
7821:Main article:
7818:
7815:
7736:
7733:
7728:
7725:
7683:
7680:
7677:
7674:
7671:
7668:
7644:
7641:
7636:
7633:
7629:
7625:
7622:
7547:
7544:
7539:
7536:
7502:
7494:
7491:
7488:
7485:
7482:
7479:
7476:
7473:
7470:
7467:
7464:
7459:
7452:
7448:
7445:
7437:
7430:
7427:
7389:
7381:
7376:
7372:
7369:
7326:
7323:
7280:
7277:
7274:
7271:
7268:
7265:
7228:), called the
7199:
7196:
7173:
7168:
7135:
7131:
7125:
7121:
7115:
7111:
7107:
7104:
7101:
7096:
7092:
7086:
7082:
7078:
7075:
7071:
7067:
7064:
7062:
7060:
7055:
7051:
7047:
7044:
7039:
7035:
7031:
7028:
7027:
7024:
7021:
7018:
7016:
7012:
7007:
7002:
7001:
6967:
6959:
6954:
6947:
6943:
6940:
6934:
6929:
6925:
6921:
6918:
6915:
6910:
6906:
6903:
6897:
6892:
6888:
6882:
6877:
6874:
6872:
6870:
6867:
6866:
6861:
6856:
6851:
6848:
6846:
6844:
6841:
6840:
6816:
6811:
6807:
6803:
6800:
6797:
6792:
6788:
6784:
6781:
6778:
6754:
6751:
6748:
6745:
6742:
6739:
6705:
6697:
6692:
6688:
6685:
6679:
6676:
6642:
6639:
6621:
6618:
6593:
6583:
6576:
6572:
6569:
6561:
6554:
6551:
6546:
6542:
6539:
6535:
6532:
6529:
6526:
6523:
6501:
6498:
6495:
6492:
6489:
6467:
6464:
6459:
6454:
6451:
6446:
6441:
6438:
6383:
6380:
6375:
6370:
6367:
6362:
6357:
6354:
6332:
6329:
6326:
6323:
6320:
6291:
6287:
6281:
6277:
6273:
6270:
6267:
6262:
6258:
6254:
6251:
6248:
6243:
6239:
6235:
6232:
6229:
6226:
6222:
6215:
6212:
6206:
6203:
6200:
6197:
6173:
6170:
6167:
6164:
6161:
6158:
6155:
6152:
6149:
6146:
6143:
6140:
6137:
6134:
6110:
6107:
6104:
6101:
6098:
6095:
6092:
6089:
6086:
6083:
6080:
6056:
6053:
6050:
6047:
6044:
6041:
6038:
6035:
6032:
6029:
6026:
6023:
6020:
6017:
6014:
6011:
6008:
6005:
6002:
5999:
5973:
5970:
5904:
5899:
5894:
5870:diffeomorphism
5805:Main article:
5794:
5791:
5773:
5765:
5760:
5756:
5753:
5720:
5715:
5711:
5707:
5704:
5682:
5678:
5651:
5647:
5641:
5637:
5633:
5628:
5624:
5618:
5614:
5610:
5605:
5601:
5595:
5591:
5587:
5584:
5564:
5559:
5555:
5551:
5546:
5542:
5538:
5533:
5529:
5525:
5505:
5502:
5499:
5496:
5493:
5490:
5487:
5467:
5462:
5458:
5454:
5449:
5445:
5441:
5436:
5432:
5428:
5408:
5388:
5383:
5379:
5375:
5372:
5369:
5364:
5360:
5333:
5330:
5299:
5296:
5270:
5265:
5261:
5257:
5254:
5251:
5246:
5242:
5238:
5235:
5232:
5200:
5195:
5192:
5157:
5152:
5148:
5144:
5141:
5138:
5133:
5129:
5125:
5105:
5100:
5096:
5092:
5089:
5086:
5081:
5077:
5073:
5041:
5038:
5033:
5029:
5025:
5022:
5019:
5014:
5010:
5006:
4973:
4970:
4965:
4961:
4957:
4952:
4948:
4927:
4924:
4921:
4916:
4912:
4908:
4884:
4879:
4875:
4871:
4868:
4865:
4860:
4856:
4852:
4827:
4824:
4808:oriented angle
4705:
4697:
4692:
4688:
4685:
4656:
4646:
4642:
4639:
4588:
4583:
4574:
4571:
4568:
4565:
4562:
4559:
4556:
4553:
4550:
4547:
4546:
4538:
4530:
4520:
4517:
4514:
4511:
4508:
4505:
4502:
4499:
4498:
4496:
4491:
4488:
4485:
4482:
4479:
4476:
4473:
4470:
4467:
4464:
4367:oriented angle
4293:
4286:
4282:
4278:
4272:
4268:
4264:
4258:
4255:
4252:
4246:
4242:
4239:
4236:
4233:
4209:
4206:
4166:Main article:
4163:
4160:
4141:
4133:
4130:
4125:
4121:
4118:
4112:
4107:
4103:
4100:
4071:
4066:
4061:
4056:
4053:
4049:
4045:
4040:
4035:
4030:
4027:
4023:
4019:
4016:
4014:
4012:
4007:
4003:
4000:
3994:
3989:
3985:
3982:
3976:
3971:
3967:
3964:
3958:
3953:
3949:
3946:
3940:
3937:
3935:
3933:
3928:
3924:
3921:
3915:
3910:
3906:
3903:
3897:
3894:
3889:
3885:
3882:
3876:
3871:
3867:
3864:
3858:
3853:
3849:
3846:
3840:
3835:
3831:
3828:
3822:
3819:
3817:
3815:
3810:
3803:
3799:
3796:
3790:
3785:
3781:
3778:
3770:
3765:
3760:
3753:
3749:
3746:
3740:
3735:
3731:
3728:
3720:
3715:
3712:
3710:
3708:
3699:
3689:
3685:
3682:
3676:
3671:
3667:
3664:
3658:
3655:
3653:
3649:
3644:
3639:
3636:
3632:
3628:
3627:
3599:
3594:
3589:
3584:
3581:
3577:
3573:
3568:
3563:
3558:
3555:
3551:
3547:
3542:
3537:
3532:
3529:
3525:
3481:
3471:
3467:
3464:
3435:
3425:
3421:
3418:
3352:
3349:
3325:
3322:
3319:
3316:
3313:
3277:
3274:
3242:
3239:
3199:
3196:
3193:
3190:
3187:
3184:
3181:
3178:
3175:
3172:
3169:
3166:
3163:
3160:
3157:
3154:
3151:
3148:
3145:
3142:
3139:
3106:
3102:
3099:
3095:
3030:
3022:
3017:
3010:
3006:
3003:
2995:
2990:
2987:
2984:
2981:
2978:
2975:
2972:
2947:Main article:
2944:
2941:
2925:
2922:
2899:properties of
2878:
2873:
2870:
2867:
2862:
2859:
2856:
2853:
2836:Euclidean norm
2810:
2807:
2804:
2801:
2798:
2742:
2739:
2736:
2733:
2730:
2700:
2697:
2694:
2691:
2688:
2685:
2682:
2680:
2678:
2675:
2672:
2669:
2666:
2663:
2662:
2658:
2654:
2651:
2649:
2645:
2642:
2637:
2632:
2629:
2624:
2623:
2588:
2585:
2571:
2568:
2539:
2534:
2531:
2526:
2523:
2498:Given a point
2485:
2482:
2479:
2476:
2473:
2470:
2443:Two subspaces
2437:Main article:
2434:
2431:
2413:
2405:
2400:
2395:
2392:
2389:
2386:
2383:
2378:
2371:
2367:
2364:
2358:
2355:
2352:
2347:
2342:
2339:
2336:
2333:
2330:
2327:
2262:
2257:
2251:
2247:
2244:
2239:
2234:
2231:
2228:
2225:
2222:
2219:
2216:
2213:
2210:
2205:
2162:
2154:
2149:
2143:
2139:
2136:
2131:
2124:
2120:
2117:
2111:
2108:
2103:
2099:
2096:
2090:
2087:
2084:
2081:
2078:
2075:
2072:
2067:
2009:
2001:
1996:
1990:
1986:
1983:
1978:
1971:
1967:
1964:
1958:
1955:
1952:
1947:
1924:Main article:
1921:
1918:
1903:
1900:
1876:
1873:
1846:
1843:
1819:
1816:
1790:
1783:
1780:
1775:
1772:
1767:
1762:
1759:
1756:
1751:
1746:
1741:
1738:
1733:
1730:
1708:
1703:
1700:
1672:
1669:
1654:is a point of
1637:
1632:
1625:
1622:
1617:
1614:
1609:
1604:
1601:
1598:
1593:
1588:
1585:
1570:is a point of
1542:
1539:
1515:
1512:
1486:
1481:
1478:
1440:
1430:
1425:
1422:
1419:
1416:
1413:
1410:
1407:
1402:
1395:
1391:
1388:
1380:
1375:
1370:
1367:
1314:
1311:
1290:Main article:
1287:
1284:
1268:Main article:
1265:
1262:
1239:
1234:
1206:
1201:
1196:
1168:
1165:
1113:
1108:
1090:
1087:
1064:
1055:
1045:
1041:
1038:
1012:. This vector
948:
945:
942:
939:
936:
933:
930:
927:
924:
921:
918:
915:
912:
909:
906:
903:
830:
827:
786:
781:
778:
722:
719:
696:
691:
667:
662:
609:
604:
582:
577:
572:
545:
540:
488:
483:
377:
374:
327:René Descartes
294:ancient Greeks
289:
286:
284:
281:
262:
257:
222:
217:
193:
188:
167:linear algebra
118:mathematician
47:physical space
26:
9:
6:
4:
3:
2:
9663:
9652:
9649:
9647:
9644:
9642:
9639:
9637:
9634:
9633:
9631:
9618:
9617:
9611:
9605:
9602:
9600:
9597:
9596:
9594:
9590:
9584:
9582:
9578:
9576:
9573:
9571:
9568:
9566:
9563:
9561:
9558:
9556:
9553:
9551:
9548:
9546:
9543:
9541:
9538:
9536:
9533:
9532:
9530:
9526:
9520:
9517:
9515:
9512:
9510:
9507:
9505:
9502:
9500:
9499:Demihypercube
9497:
9495:
9492:
9490:
9487:
9485:
9482:
9480:
9477:
9476:
9474:
9472:
9468:
9464:
9458:
9455:
9453:
9450:
9448:
9445:
9443:
9440:
9438:
9435:
9433:
9430:
9428:
9425:
9424:
9422:
9418:
9413:
9403:
9400:
9398:
9395:
9393:
9390:
9388:
9385:
9383:
9380:
9378:
9375:
9373:
9370:
9368:
9365:
9364:
9362:
9358:
9354:
9347:
9342:
9340:
9335:
9333:
9328:
9327:
9324:
9317:
9313:
9312:
9307:
9302:
9299:
9295:
9294:
9289:
9285:
9282:
9280:3-540-11658-3
9276:
9272:
9268:
9264:
9260:
9258:0-486-20630-0
9254:
9249:
9248:
9242:
9238:
9235:
9231:
9227:
9225:0-471-60839-4
9221:
9217:
9213:
9209:
9208:
9202:
9199:
9197:0-471-84819-0
9193:
9189:
9185:
9180:
9179:
9173:
9168:
9161:
9156:
9149:
9144:
9137:
9132:
9130:
9122:
9117:
9115:
9107:
9102:
9095:
9090:
9083:
9078:
9071:
9066:
9064:
9059:
9034:
9031:
9028:
9025:
9005:
9000:
8992:
8986:
8981:
8973:
8946:
8940:
8934:
8931:
8925:
8919:
8899:
8894:
8883:
8877:
8855:
8850:
8839:
8833:
8810:
8807:
8801:
8795:
8792:
8789:
8783:
8777:
8774:
8771:
8765:
8762:
8759:
8756:
8753:
8747:
8737:
8728:
8719:
8715:
8705:
8701:
8698:
8696:
8692:
8689:
8685:
8684:Hilbert space
8682:
8681:
8677:
8671:
8666:
8659:
8657:
8653:
8649:
8645:
8641:
8637:
8632:
8630:
8626:
8622:
8618:
8613:
8600:
8595:
8591:
8587:
8582:
8578:
8574:
8569:
8565:
8561:
8556:
8552:
8543:
8542:
8538:
8534:
8530:
8526:
8521:
8519:
8516:(that may be
8515:
8512:
8508:
8504:
8500:
8496:
8495:inner product
8486:
8484:
8480:
8476:
8472:
8468:
8463:
8461:
8457:
8454:. Generally,
8453:
8449:
8445:
8440:
8438:
8434:
8430:
8426:
8422:
8418:
8414:
8410:
8406:
8400:
8396:
8390:
8383:Curved spaces
8380:
8378:
8374:
8370:
8366:
8362:
8358:
8354:
8349:
8339:
8337:
8333:
8328:
8326:
8322:
8318:
8314:
8310:
8304:
8294:
8292:
8288:
8284:
8283:finite fields
8279:
8277:
8273:
8269:
8268:number theory
8265:
8261:
8256:
8254:
8250:
8246:
8242:
8238:
8234:
8229:
8227:
8223:
8217:
8207:
8205:
8201:
8197:
8193:
8189:
8178:
8173:
8169:
8168:
8163:
8158:
8149:
8148:
8140:
8138:
8134:
8130:
8126:
8122:
8118:
8114:
8110:
8106:
8102:
8097:
8095:
8091:
8087:
8082:
8080:
8076:
8072:
8068:
8064:
8060:
8056:
8052:
8048:
8044:
8040:
8036:
8026:
8023:
8019:
8015:
8010:
8006:
8005:
7999:
7997:
7993:
7989:
7985:
7984:Alfred Tarski
7981:
7977:
7973:
7969:
7965:
7961:
7960:David Hilbert
7956:
7954:
7950:
7946:
7941:
7939:
7935:
7925:
7923:
7919:
7915:
7910:
7908:
7904:
7900:
7896:
7891:
7889:
7885:
7881:
7876:
7874:
7858:
7853:
7838:
7834:
7831:, and thus a
7830:
7824:
7814:
7812:
7808:
7803:
7801:
7788:
7786:
7781:
7779:
7778:screw motions
7775:
7771:
7766:
7764:
7763:displacements
7760:
7759:rigid motions
7756:
7752:
7734:
7731:
7723:
7715:
7711:
7707:
7702:
7701:
7699:
7681:
7678:
7675:
7672:
7669:
7666:
7642:
7639:
7634:
7631:
7627:
7623:
7620:
7610:
7606:
7584:
7578:
7569:
7567:
7563:
7545:
7542:
7534:
7525:
7519:
7492:
7486:
7480:
7477:
7471:
7465:
7462:
7450:
7446:
7443:
7428:
7425:
7415:
7379:
7374:
7370:
7367:
7355:
7351:
7346:
7342:
7324:
7321:
7302:
7300:
7296:
7291:
7278:
7275:
7272:
7269:
7263:
7255:
7254:
7249:
7245:
7237:
7231:
7227:
7223:
7219:
7215:
7209:
7205:
7195:
7189:
7171:
7155:
7154:
7150:
7133:
7129:
7123:
7119:
7113:
7109:
7105:
7102:
7099:
7094:
7090:
7084:
7080:
7076:
7073:
7069:
7063:
7053:
7049:
7045:
7042:
7037:
7033:
7022:
7017:
7010:
6991:
6988:
6957:
6945:
6941:
6938:
6932:
6927:
6923:
6919:
6916:
6913:
6908:
6904:
6901:
6895:
6890:
6886:
6873:
6868:
6859:
6847:
6842:
6830:
6809:
6805:
6801:
6798:
6795:
6790:
6786:
6782:
6779:
6765:
6752:
6749:
6746:
6743:
6737:
6729:
6722:
6695:
6690:
6686:
6683:
6674:
6666:
6663:
6659:
6640:
6637:
6617:
6613:
6610:
6574:
6570:
6567:
6552:
6549:
6544:
6540:
6537:
6533:
6527:
6521:
6513:
6499:
6493:
6490:
6487:
6465:
6462:
6452:
6449:
6444:
6439:
6436:
6425:
6418:
6412:
6408:
6381:
6378:
6368:
6365:
6360:
6355:
6352:
6330:
6324:
6321:
6318:
6309:
6307:
6302:
6289:
6285:
6279:
6271:
6265:
6260:
6252:
6246:
6241:
6233:
6230:
6227:
6220:
6213:
6210:
6204:
6201:
6198:
6195:
6187:
6184:
6171:
6168:
6165:
6162:
6159:
6153:
6147:
6144:
6138:
6132:
6124:
6121:
6108:
6102:
6096:
6087:
6081:
6070:
6067:
6054:
6048:
6045:
6042:
6036:
6033:
6024:
6018:
6015:
6009:
6003:
5997:
5989:
5987:
5983:
5982:metric spaces
5979:
5969:
5967:
5962:
5960:
5951:
5949:
5945:
5941:
5935:
5931:
5918:
5902:
5897:
5878:
5875:
5871:
5867:
5866:homeomorphism
5856:of dimension
5850:
5848:
5842:
5833:
5828:
5826:
5822:
5818:
5814:
5808:
5799:
5790:
5763:
5758:
5754:
5751:
5731:
5718:
5713:
5709:
5705:
5702:
5680:
5676:
5649:
5645:
5639:
5635:
5631:
5626:
5622:
5616:
5612:
5608:
5603:
5599:
5593:
5589:
5585:
5582:
5557:
5553:
5549:
5544:
5540:
5536:
5531:
5527:
5500:
5497:
5494:
5491:
5488:
5460:
5456:
5452:
5447:
5443:
5439:
5434:
5430:
5406:
5386:
5381:
5377:
5373:
5370:
5367:
5362:
5358:
5331:
5328:
5313:
5297:
5294:
5263:
5259:
5255:
5252:
5249:
5244:
5240:
5236:
5233:
5218:
5215:, called the
5198:
5193:
5190:
5180:
5179:
5169:
5150:
5146:
5142:
5139:
5136:
5131:
5127:
5098:
5094:
5090:
5087:
5084:
5079:
5075:
5063:
5055:
5039:
5031:
5027:
5023:
5020:
5017:
5012:
5008:
4997:
4992:
4988:
4971:
4968:
4963:
4959:
4955:
4950:
4946:
4925:
4922:
4914:
4910:
4898:
4877:
4873:
4869:
4866:
4863:
4858:
4854:
4843:
4839:
4833:
4823:
4819:
4815:
4809:
4803:
4797:
4791:
4787:
4779:
4774:
4769:
4765:
4751:
4740:
4734:
4725:
4695:
4690:
4686:
4683:
4644:
4640:
4637:
4625:
4619:
4601:
4581:
4569:
4566:
4563:
4557:
4554:
4551:
4548:
4536:
4528:
4515:
4512:
4509:
4503:
4500:
4494:
4489:
4483:
4480:
4477:
4474:
4471:
4465:
4462:
4454:
4436:
4431:
4427:
4423:
4419:
4412:
4408:
4404:
4400:
4395:
4389:
4369:
4368:
4363:
4358:
4351:
4347:
4340:
4336:
4331:is real, and
4322:
4319:function. By
4318:
4314:
4305:
4291:
4280:
4266:
4256:
4253:
4250:
4244:
4240:
4237:
4234:
4231:
4223:
4207:
4204:
4183:
4174:
4169:
4159:
4131:
4128:
4123:
4119:
4116:
4110:
4105:
4101:
4098:
4086:
4069:
4064:
4054:
4051:
4043:
4038:
4028:
4025:
4017:
4015:
4005:
4001:
3998:
3992:
3987:
3983:
3980:
3974:
3969:
3965:
3962:
3956:
3951:
3947:
3944:
3938:
3936:
3926:
3922:
3919:
3913:
3908:
3904:
3901:
3895:
3892:
3887:
3883:
3880:
3874:
3869:
3865:
3862:
3856:
3851:
3847:
3844:
3838:
3833:
3829:
3826:
3820:
3818:
3801:
3797:
3794:
3788:
3783:
3779:
3776:
3763:
3751:
3747:
3744:
3738:
3733:
3729:
3726:
3713:
3711:
3687:
3683:
3680:
3674:
3669:
3665:
3662:
3656:
3654:
3647:
3637:
3634:
3617:
3615:
3610:
3597:
3592:
3582:
3579:
3571:
3566:
3556:
3553:
3545:
3540:
3530:
3527:
3514:
3511:
3505:
3499:
3469:
3465:
3462:
3423:
3419:
3416:
3405:
3404:
3398:
3397:perpendicular
3393:
3387:
3381:
3377:Two segments
3375:
3373:
3372:perpendicular
3367:
3350:
3347:
3336:
3323:
3320:
3317:
3314:
3311:
3303:
3301:
3297:
3296:perpendicular
3275:
3272:
3252:
3251:Orthogonality
3248:
3247:Perpendicular
3241:Orthogonality
3238:
3236:
3231:
3229:
3225:
3220:
3210:
3197:
3191:
3188:
3185:
3179:
3176:
3170:
3167:
3164:
3158:
3155:
3149:
3146:
3143:
3137:
3129:
3128:
3124:
3119:
3100:
3097:
3084:
3080:
3074:
3070:
3066:
3060:
3056:
3052:
3047:
3020:
3008:
3004:
3001:
2988:
2982:
2979:
2976:
2970:
2962:
2960:
2956:
2950:
2940:
2939:
2923:
2920:
2906:
2902:
2898:
2894:
2889:
2876:
2871:
2868:
2865:
2860:
2854:
2843:
2837:
2832:
2829:
2825:
2805:
2802:
2799:
2788:
2784:
2780:
2775:
2771:
2766:
2761:
2757:
2737:
2734:
2731:
2720:
2715:
2695:
2692:
2689:
2681:
2673:
2670:
2667:
2650:
2643:
2640:
2635:
2630:
2627:
2613:
2612:
2608:
2586:
2583:
2567:
2565:
2562:
2559:
2557:
2537:
2532:
2529:
2524:
2521:
2496:
2483:
2480:
2477:
2474:
2471:
2468:
2460:
2454:
2440:
2430:
2403:
2393:
2390:
2387:
2384:
2381:
2369:
2365:
2362:
2356:
2353:
2350:
2340:
2337:
2334:
2331:
2328:
2325:
2317:
2305:
2291:
2287:
2286:
2280:
2278:
2273:
2260:
2245:
2242:
2232:
2229:
2226:
2223:
2217:
2214:
2211:
2193:
2186:
2179:
2152:
2137:
2134:
2122:
2118:
2115:
2109:
2106:
2101:
2097:
2094:
2085:
2082:
2079:
2073:
2070:
2055:
2044:
2042:
2037:
2026:
1999:
1984:
1981:
1969:
1965:
1962:
1956:
1953:
1950:
1935:
1933:
1927:
1917:
1901:
1898:
1874:
1871:
1860:
1844:
1841:
1817:
1814:
1803:
1781:
1778:
1773:
1770:
1760:
1757:
1754:
1744:
1739:
1736:
1731:
1728:
1720:
1706:
1701:
1698:
1688:
1670:
1667:
1648:
1635:
1623:
1620:
1615:
1612:
1602:
1599:
1596:
1586:
1583:
1575:
1564:
1558:
1540:
1537:
1513:
1510:
1484:
1479:
1476:
1466:
1457:
1423:
1420:
1417:
1414:
1411:
1408:
1405:
1393:
1389:
1386:
1373:
1368:
1365:
1355:
1341:
1337:
1333:
1328:
1312:
1309:
1293:
1283:
1281:
1277:
1271:
1261:
1256:of dimension
1255:
1237:
1217:
1204:
1199:
1166:
1163:
1153:
1149:
1142:of dimension
1137:
1133:
1132:inner product
1129:
1111:
1095:
1086:
1084:
1080:
1075:
1062:
1043:
1039:
1036:
1024:
1020:
1010:
1006:
1002:
994:
988:
984:
977:
963:
959:
946:
943:
940:
934:
931:
928:
922:
916:
913:
910:
904:
901:
893:
890:
886:
872:
870:
866:
862:
858:
854:
853:
848:
847:
828:
825:
815:
806:
804:
800:
784:
779:
776:
761:
759:
755:
751:
747:
742:
740:
736:
732:
718:
716:
712:
694:
665:
645:
641:
635:
628:of dimension
627:
626:
607:
580:
575:
559:
543:
524:
520:
516:
508:
486:
470:
468:
464:
460:
459:inner product
456:
452:
449:
444:
442:
438:
434:
430:
426:
422:
417:
415:
411:
407:
403:
399:
395:
391:
387:
383:
382:set of points
373:
370:
369:vector spaces
366:
361:
359:
355:
347:
342:
340:
336:
332:
328:
323:
321:
317:
313:
308:
307:
302:
301:
295:
280:
278:
260:
246:
238:
220:
191:
176:
171:
168:
164:
163:vector spaces
160:
156:
151:
149:
145:
144:straight line
141:
140:
135:
131:
130:
125:
121:
117:
116:ancient Greek
113:
108:
106:
102:
98:
94:
90:
86:
84:
78:
75:
71:
67:
63:
59:
56:, it was the
55:
54:
48:
44:
40:
32:
19:
9614:
9580:
9519:Hyperpyramid
9484:Hypersurface
9377:Affine space
9371:
9367:Vector space
9309:
9297:
9292:
9270:
9246:
9205:
9183:
9167:
9155:
9143:
9138:, Chapter 9.
9106:Coxeter 1973
9101:
9089:
9077:
8736:
8727:
8718:
8633:
8628:
8624:
8620:
8616:
8614:
8544:
8522:
8492:
8464:
8441:
8417:homeomorphic
8413:neighborhood
8402:
8389:Curved space
8352:
8351:
8329:
8325:vector space
8321:vector lines
8306:
8291:cryptography
8280:
8276:Fermat curve
8257:
8230:
8219:
8216:Affine space
8210:Affine space
8185:
8177:(March 2023)
8176:
8165:
8132:
8098:
8083:
8059:architecture
8032:
8021:
8002:
8000:
7988:real numbers
7976:real numbers
7957:
7942:
7931:
7911:
7899:homeomorphic
7892:
7877:
7829:metric space
7826:
7804:
7789:
7782:
7767:
7762:
7758:
7754:
7750:
7703:
7695:
7608:
7604:
7590:be a point,
7585:
7570:
7526:
7520:
7416:
7353:
7349:
7340:
7303:
7294:
7292:
7256:
7253:translations
7250:
7243:
7235:
7229:
7221:
7217:
7213:
7211:
7187:
7156:
7152:
7151:
6992:
6989:
6831:
6766:
6730:
6723:
6667:
6661:
6657:
6623:
6614:
6611:
6514:
6423:
6416:
6410:
6406:
6311:An isometry
6310:
6303:
6188:
6185:
6125:
6122:
6071:
6068:
5990:
5980:between two
5975:
5963:
5959:antimeridian
5952:
5933:
5929:
5916:
5851:
5840:
5838:is a set of
5832:affine basis
5829:
5824:
5817:affine frame
5813:affine space
5810:
5732:
5314:
5216:
5176:
5170:
5062:linear spans
4990:
4986:
4897:unit vectors
4835:
4817:
4813:
4807:
4801:
4795:
4789:
4785:
4777:
4767:
4763:
4752:
4738:
4732:
4623:
4617:
4602:
4455:
4432:
4425:
4421:
4417:
4410:
4406:
4402:
4399:reflex angle
4393:
4365:
4359:
4352:
4345:
4338:
4334:
4327:. Therefore
4306:
4224:
4181:
4179:
4087:
3618:
3612:This is the
3611:
3515:
3509:
3503:
3500:
3400:
3396:
3391:
3385:
3379:
3376:
3371:
3368:
3337:
3304:
3299:
3295:
3254:
3232:
3227:
3218:
3211:
3130:
3120:
3082:
3078:
3072:
3068:
3064:
3058:
3050:
3048:
2963:
2958:
2954:
2952:
2910:
2904:
2890:
2844:
2838:of a vector
2835:
2833:
2827:
2823:
2773:
2769:
2767:and denoted
2764:
2762:
2755:
2716:
2614:
2573:
2566:
2563:
2560:
2497:
2461:
2452:
2442:
2318:
2303:
2289:
2288:, or simply
2285:line segment
2283:
2281:
2274:
2194:
2187:
2180:
2056:
2045:
2040:
2038:
2027:
1936:
1931:
1929:
1861:
1804:
1721:
1649:
1576:
1565:
1556:
1458:
1356:
1346:is a subset
1339:
1335:
1331:
1329:
1295:
1276:affine space
1273:
1270:Affine space
1253:
1218:
1147:
1096:
1092:
1076:
1022:
1018:
1008:
1004:
1000:
986:
982:
978:
961:
960:
894:
888:
884:
873:
857:translations
856:
852:free vectors
850:
844:
813:
807:
798:
762:
757:
750:affine space
745:
743:
739:real numbers
726:
724:
714:
710:
680:instead of
650:
633:
632:, or simply
624:
623:
560:
514:
471:
463:affine space
454:
445:
418:
379:
362:
343:
339:real numbers
324:
304:
299:
291:
172:
152:
137:
127:
123:
109:
88:
82:
80:
76:
69:
52:
38:
37:
9604:Codimension
9583:-dimensions
9504:Hypersphere
9387:Free module
9160:Berger 1987
9148:Anton (1987
9136:Berger 1987
9121:Berger 1987
9094:Berger 1987
8475:orthodromes
8421:open subset
8200:consistency
7945:Felix Klein
7785:reflections
7357:the vector
7226:composition
6724:which maps
6512:defined by
5948:cylindrical
5919:of a point
5917:coordinates
5877:open subset
5168:are equal.
3403:right angle
2897:topological
2783:dot product
2765:dot product
2514:, which is
2433:Parallelism
1128:dot product
1016:is denoted
964:The second
878:on a point
861:translation
843:are called
812:are called
715:origin-free
523:isomorphism
507:dot product
467:parallelism
425:abstraction
410:reflections
390:translation
329:introduced
277:dot product
66:mathematics
9630:Categories
9599:Hyperspace
9479:Hyperplane
9271:Geometry I
9172:Artin 1988
9055:References
8693:, used in
8529:space-time
8518:indefinite
8479:navigation
8369:consistent
8071:navigation
8067:topography
8045:, such as
8033:Since the
8014:congruence
8009:Emil Artin
7949:symmetries
7807:Lie groups
7710:handedness
7414:, one has
7345:linear map
5972:Isometries
4830:See also:
4724:half-lines
4435:multiplied
3300:orthogonal
2316:; that is
1354:such that
1136:isomorphic
998:such that
435:and other
312:postulates
283:Definition
175:isomorphic
139:postulates
81:Euclidean
68:there are
9489:Hypercube
9467:Polytopes
9447:Minkowski
9442:Hausdorff
9437:Inductive
9402:Spacetime
9353:Dimension
9316:EMS Press
9290:(1973) .
9243:(1960) .
9082:Ball 1960
9029:⋅
8997:‖
8990:‖
8978:‖
8971:‖
8932:⋅
8891:‖
8875:‖
8847:‖
8831:‖
8793:μ
8790:−
8775:λ
8772:−
8763:μ
8754:λ
8710:Footnotes
8691:dimension
8652:curvature
8588:−
8460:geodesics
8317:isotropic
8196:embedding
8129:ellipsoid
8055:astronomy
8051:mechanics
7884:open ball
7880:open sets
7774:rotations
7735:→
7727:→
7676:∘
7640:∘
7632:−
7575:form the
7546:→
7538:→
7478:−
7451:→
7429:→
7375:→
7325:→
7267:→
7103:⋯
7066:↦
7043:…
7020:→
6946:→
6933:⋅
6917:…
6909:→
6896:⋅
6876:↦
6850:→
6799:…
6741:↦
6691:→
6678:↦
6641:→
6575:→
6553:→
6497:→
6491::
6466:→
6458:→
6453:→
6445::
6440:→
6382:→
6374:→
6369:→
6361::
6356:→
6328:→
6322::
6276:‖
6269:‖
6266:−
6257:‖
6250:‖
6247:−
6238:‖
6225:‖
6199:⋅
6166:⋅
6145:⋅
6106:‖
6100:‖
6094:‖
6079:‖
5986:bijection
5966:manifolds
5944:spherical
5872:) from a
5759:→
5706:⋅
5677:α
5636:α
5613:α
5590:α
5554:α
5541:α
5528:α
5371:…
5332:→
5298:→
5253:…
5194:→
5140:…
5088:…
5021:…
4956:⋅
4920:‖
4907:‖
4867:…
4691:→
4645:→
4578:otherwise
4558:
4552:−
4549:π
4537:μ
4529:λ
4504:
4481:μ
4472:λ
4466:
4317:arccosine
4254:⋅
4241:
4232:θ
4208:→
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3993:⋅
3988:→
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3952:→
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3893:−
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3276:→
3156:≤
3009:→
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2858:‖
2852:‖
2809:⟩
2797:⟨
2785:of their
2741:⟩
2729:⟨
2721:(that is
2699:⟩
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2644:→
2636:×
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2391:≤
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2246:∈
2243:λ
2230:λ
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2110:λ
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1985:∈
1982:λ
1970:→
1957:λ
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1421:∈
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1286:Subspaces
1167:→
1044:→
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803:dimension
799:dimension
780:→
752:over the
737:over the
406:congruent
354:polytopes
325:In 1637,
298:Euclid's
74:dimension
51:Euclid's
9616:Category
9592:See also
9392:Manifold
9269:(1987),
8688:infinite
8662:See also
8634:To take
8533:Einstein
8415:that is
8405:manifold
8395:Manifold
8313:coplanar
8133:a priori
8121:embedded
8113:manifold
8043:sciences
7978:. Later
7914:complete
7817:Topology
7770:identity
7527:The map
6541:′
5978:isometry
4800:[0,
4776:[0,
4773:interval
4742:, where
4525:if
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3224:triangle
3016:‖
2994:‖
2955:distance
2717:that is
2453:parallel
472:The set
421:physical
394:rotation
300:Elements
134:theorems
124:Elements
110:Ancient
53:Elements
43:geometry
9514:Simplex
9452:Fractal
9234:1009557
8704:physics
8642:uses a
8636:gravity
8172:Discuss
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517:and an
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398:figures
386:motions
335:algebra
239:as the
129:proving
122:in his
105:physics
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814:points
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515:origin
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441:number
316:axioms
245:-space
120:Euclid
101:spaces
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9570:Seven
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932:+
929:P
926:(
923:=
920:)
917:w
914:+
911:v
908:(
905:+
902:P
889:v
885:P
880:P
876:v
826:E
810:E
785:.
777:E
765:E
695:n
690:R
666:n
661:E
638:n
630:n
608:n
603:R
581:.
576:n
571:R
544:n
539:R
527:n
511:n
503:n
487:n
482:R
350:n
261:n
256:R
243:n
221:n
216:E
192:n
187:E
89:n
83:n
77:n
20:)
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