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50-777: The park shares borders, largely laid out as straight lines , with the Sermersooq municipality in the south and with the Avannaata municipality in the west, partly along the 45° West meridian on the ice cap. The large interior of the park is part of the Greenland Ice Sheet , but there are also large ice-free areas along the coast and on Peary Land in the north. The park includes the King Frederick VIII Land and King Christian X Land geographical areas. Originally established on 22 May 1974 from

100-406: A set of points satisfying certain relationships, expressible in terms of distance and angles. For example, there are two fundamental operations (referred to as motions ) on the plane. One is translation , 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 rotation around a fixed point in the plane, in which all points in

150-402: A Euclidean space E of dimension n , the choice of a point, called an origin and an orthonormal basis of E → {\displaystyle {\overrightarrow {E}}} defines an isomorphism of Euclidean spaces from E to R n . {\displaystyle \mathbb {R} ^{n}.} As every Euclidean space of dimension n is isomorphic to it,

200-636: A Euclidean space and E → {\displaystyle {\overrightarrow {E}}} its associated vector space. A flat , Euclidean subspace or affine subspace of E is a subset F of E such that F → = { P Q → | P ∈ F , Q ∈ F } ( {\displaystyle {\overrightarrow {F}}={\Bigl \{}{\overrightarrow {PQ}}\mathrel {\Big |} P\in F,Q\in F{\Bigr \}}{\vphantom {\frac {(}{}}}} as

250-678: A Euclidean space can also be said about R n . {\displaystyle \mathbb {R} ^{n}.} Therefore, many authors, especially at elementary level, call R n {\displaystyle \mathbb {R} ^{n}} the standard Euclidean space of dimension n , or simply the Euclidean space of dimension n . A reason for introducing such an abstract definition of Euclidean spaces, and for working with E n {\displaystyle \mathbb {E} ^{n}} instead of R n {\displaystyle \mathbb {R} ^{n}}

300-444: A Euclidean space is the dimension of its associated vector space. The elements of E are called points , and are commonly denoted by capital letters. The elements of E → {\displaystyle {\overrightarrow {E}}} are called Euclidean vectors or free vectors . They are also called translations , although, properly speaking, a translation is the geometric transformation resulting from

350-419: A Euclidean space that has itself as the associated vector space. A typical case of Euclidean vector space is R n {\displaystyle \mathbb {R} ^{n}} viewed as a vector space equipped with the dot product as an inner product . The importance of this particular example of Euclidean space lies in the fact that every Euclidean space is isomorphic to it. More precisely, given

400-406: A Euclidean subspace of direction V → {\displaystyle {\overrightarrow {V}}} . (The associated vector space of this subspace is V → {\displaystyle {\overrightarrow {V}}} .) A Euclidean vector space E → {\displaystyle {\overrightarrow {E}}} (that is, a Euclidean space that

450-501: A diameter of the sphere, and therefore every great circle is concentric with the sphere and shares the same radius . Any other circle of the sphere is called a small circle , and is the intersection of the sphere with a plane not passing through its center. Small circles are the spherical-geometry analog of circles in Euclidean space. Every circle in Euclidean 3-space is a great circle of exactly one sphere. The disk bounded by

500-529: A few fundamental properties, called postulates , which either were considered as evident (for example, there is exactly one straight line passing through two points), or seemed impossible to prove ( parallel postulate ). After the introduction at the end of the 19th century of non-Euclidean geometries , the old postulates were re-formalized to define Euclidean spaces through axiomatic theory . Another definition of Euclidean spaces by means of vector spaces and linear algebra has been shown to be equivalent to

550-643: A function along all great circles of the sphere. Euclidean space Euclidean space is the fundamental space of geometry , intended to represent physical space . Originally, in Euclid's Elements , it was the three-dimensional space of Euclidean geometry , but in modern mathematics there are Euclidean spaces of any positive integer dimension n , which are called Euclidean n -spaces when one wants to specify their dimension. For n equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes . The qualifier "Euclidean"

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600-418: A great circle is called a great disk : it is the intersection of a ball and a plane passing through its center. In higher dimensions, the great circles on the n -sphere are the intersection of the n -sphere with 2-planes that pass through the origin in the Euclidean space R . Half of a great circle may be called a great semicircle (e.g., as in parts of a meridian in astronomy ). To prove that

650-471: A meridian of the sphere. In a Cartesian coordinate system , this is which is a plane through the origin, i.e., the center of the sphere. Some examples of great circles on the celestial sphere include the celestial horizon , the celestial equator , and the ecliptic . Great circles are also used as rather accurate approximations of geodesics on the Earth 's surface for air or sea navigation (although it

700-569: A vector on a point. This notation is not ambiguous, as, to distinguish between the two meanings of + , it suffices to look at the nature of its left argument. The fact that the action is free and transitive means that, for every pair of points ( P , Q ) , there is exactly one displacement vector v such that P + v = Q . This vector v is denoted Q − P or P Q → ) . {\displaystyle {\overrightarrow {PQ}}{\vphantom {\frac {)}{}}}.} As previously explained, some of

750-483: Is a functional of the curve given by According to the Euler–Lagrange equation , S [ γ ] {\displaystyle S[\gamma ]} is minimized if and only if where C {\displaystyle C} is a t {\displaystyle t} -independent constant, and From the first equation of these two, it can be obtained that Integrating both sides and considering

800-400: Is a linear subspace of E → , {\displaystyle {\overrightarrow {E}},} then P + V → = { P + v | v ∈ V → } {\displaystyle P+{\overrightarrow {V}}={\Bigl \{}P+v\mathrel {\Big |} v\in {\overrightarrow {V}}{\Bigr \}}} is

850-402: Is a Euclidean space, and the related notions of distance, angle, translation, and rotation. Even when used in physical theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames , measurement instruments, and so on. A purely mathematical definition of Euclidean space also ignores questions of units of length and other physical dimensions :

900-417: Is also called the direction of F . If P is a point of F then F = { P + v | v ∈ F → } . {\displaystyle F={\Bigl \{}P+v\mathrel {\Big |} v\in {\overrightarrow {F}}{\Bigr \}}.} Conversely, if P is a point of E and V → {\displaystyle {\overrightarrow {V}}}

950-642: Is an arbitrary point (not necessary on the line). In a Euclidean vector space, the zero vector is usually chosen for O ; this allows simplifying the preceding formula into { ( 1 − λ ) P + λ Q | λ ∈ R } . {\displaystyle {\bigl \{}(1-\lambda )P+\lambda Q\mathrel {\big |} \lambda \in \mathbb {R} {\bigr \}}.} A standard convention allows using this formula in every Euclidean space, see Affine space § Affine combinations and barycenter . The line segment , or simply segment , joining

1000-421: Is equal to E → {\displaystyle {\overrightarrow {E}}} ) 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. In a Euclidean space, a line is a Euclidean subspace of dimension one. Since a vector space of dimension one

1050-450: Is not a perfect sphere ), as well as on spheroidal celestial bodies . The equator of the idealized earth is a great circle and any meridian and its opposite meridian form a great circle. Another great circle is the one that divides the land and water hemispheres . A great circle divides the earth into two hemispheres and if a great circle passes through a point it must pass through its antipodal point . The Funk transform integrates

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1100-434: Is spanned by any nonzero vector, a line is a set of the form { P + λ P Q → | λ ∈ R } , ( {\displaystyle {\Bigl \{}P+\lambda {\overrightarrow {PQ}}\mathrel {\Big |} \lambda \in \mathbb {R} {\Bigr \}},{\vphantom {\frac {(}{}}}} where P and Q are two distinct points of

1150-457: Is still in use under the name of synthetic geometry . In 1637, René Descartes introduced Cartesian coordinates , and showed that these allow reducing geometric problems to algebraic computations with numbers. This reduction of geometry to algebra was a major change in point of view, as, until then, the real numbers were defined in terms of lengths and distances. Euclidean geometry was not applied in spaces of dimension more than three until

1200-406: Is that it is often preferable to work in a coordinate-free and origin-free 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. A Euclidean vector space is a finite-dimensional inner product space over the real numbers . A Euclidean space is an affine space over

1250-403: Is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the ancient Greek mathematician Euclid in his Elements , with the great innovation of proving all properties of the space as theorems , by starting from

1300-399: Is usually possible to work with a specific Euclidean space, denoted E n {\displaystyle \mathbf {E} ^{n}} or E n {\displaystyle \mathbb {E} ^{n}} , which can be represented using Cartesian coordinates as the real n -space R n {\displaystyle \mathbb {R} ^{n}} equipped with

1350-418: The action of a Euclidean vector on the Euclidean space. The action of a translation v on a point P provides a point that is denoted P + v . This action satisfies P + ( v + w ) = ( P + v ) + w . {\displaystyle P+(v+w)=(P+v)+w.} Note: The second + in the left-hand side is a vector addition; each other + denotes an action of

1400-418: The dot product is a Euclidean space of dimension n . Conversely, the choice of a point called the origin and an orthonormal basis of the space of translations is equivalent with defining an isomorphism between a Euclidean space of dimension n and R n {\displaystyle \mathbb {R} ^{n}} viewed as a Euclidean space. It follows that everything that can be said about

1450-406: The reals such that the associated vector space is a Euclidean vector space. Euclidean spaces are sometimes called Euclidean affine spaces to distinguish them from Euclidean vector spaces. If E is a Euclidean space, its associated vector space (Euclidean vector space) is often denoted E → . {\displaystyle {\overrightarrow {E}}.} The dimension of

1500-489: The 19th century. Ludwig Schläfli generalized Euclidean geometry to spaces of dimension n , using both synthetic and algebraic methods, and discovered all of the regular polytopes (higher-dimensional analogues of the Platonic solids ) that exist in Euclidean spaces of any dimension. Despite the wide use of Descartes' approach, which was called analytic geometry , the definition of Euclidean space remained unchanged until

1550-432: The Euclidean space R n {\displaystyle \mathbb {R} ^{n}} is sometimes called the standard Euclidean space of dimension n . Some basic properties of Euclidean spaces depend only on the fact that a Euclidean space is an affine space . They are called affine properties and include the concepts of lines, subspaces, and parallelism, which are detailed in next subsections. Let E be

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1600-739: The Euclidean space as a part of the line. It follows that there is exactly one line that passes through (contains) two distinct points. This implies that two distinct lines intersect in at most one point. A more symmetric representation of the line passing through P and Q is { O + ( 1 − λ ) O P → + λ O Q → | λ ∈ R } , ( {\displaystyle {\Bigl \{}O+(1-\lambda ){\overrightarrow {OP}}+\lambda {\overrightarrow {OQ}}\mathrel {\Big |} \lambda \in \mathbb {R} {\Bigr \}},{\vphantom {\frac {(}{}}}} where O

1650-765: The Greenland Department of Environment and Nature. The historical research camps on the ice sheet— Eismitte and North Ice —fall within the boundaries of the present-day park. The park has no permanent human population, although 400 sites see occasional summertime use. In 1986, the population of the park was 40, living at Mestersvig . These 40 were involved in cleanup and closeout operations at mining exploration sites and soon left. Since then, censuses have recorded zero permanent human population. In 2008, only 31 people and about 110 dogs were present over winter in North East Greenland, distributed among

1700-441: The associated vector space of F is a linear subspace (vector subspace) of E → . {\displaystyle {\overrightarrow {E}}.} A Euclidean subspace F is a Euclidean space with F → {\displaystyle {\overrightarrow {F}}} as the associated vector space. This linear subspace F → {\displaystyle {\overrightarrow {F}}}

1750-426: 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 Euclidean space. There is essentially only one Euclidean space of each dimension; that is, all Euclidean spaces of a given dimension are isomorphic . Therefore, it

1800-405: The basic properties of Euclidean spaces result from the structure of affine space. They are described in § Affine structure and its subsections. The properties resulting from the inner product are explained in § Metric structure and its subsections. For any vector space, the addition acts freely and transitively on the vector space itself. Thus a Euclidean vector space can be viewed as

1850-403: The boundary condition, the real solution of C {\displaystyle C} is zero. Thus, ϕ ′ = 0 {\displaystyle \phi '=0} and θ {\displaystyle \theta } can be any value between 0 and θ 0 {\displaystyle \theta _{0}} , indicating that the curve must lie on

1900-410: The coastal regions of the park. In 1993, this was estimated to be 40% of the world population of musk oxen. Other mammals include Arctic fox , stoat , collared lemming , Arctic hare and a small but important population of Greenland wolf . Other marine mammals include ringed seal , bearded seal , harp seal and hooded seal as well as narwhal and beluga whale . Species of birds which breed in

1950-674: The distance in a "mathematical" space is a number , not something expressed in inches or metres. 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 real vector space acts — the space of translations which is equipped with an inner product . The action of translations makes the space an affine space , and this allows defining lines, planes, subspaces, dimension, and parallelism . The inner product allows defining distance and angles. The set R n {\displaystyle \mathbb {R} ^{n}} of n -tuples of real numbers equipped with

2000-404: The end of 19th century. The introduction of abstract vector spaces 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. One way to think of the Euclidean plane is as

2050-470: The following stations (all on the coast, except Summit Camp ): During summer, scientists add to these numbers. The research station Zackenberg Ecological Research Operations (ZERO) 74°28′11″N 20°34′15″W  /  74.469725°N 20.570847°W  / 74.469725; -20.570847 can cater for over 20 scientists and station personnel. An estimated 5,000 to 15,000 musk oxen , as well as numerous polar bears and walrus , can be found near

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2100-416: The minor arc of a great circle is the shortest path connecting two points on the surface of a sphere, one can apply calculus of variations to it. Consider the class of all regular paths from a point p {\displaystyle p} to another point q {\displaystyle q} . Introduce spherical coordinates so that p {\displaystyle p} coincides with

2150-479: The north pole. Any curve on the sphere that does not intersect either pole, except possibly at the endpoints, can be parametrized by provided ϕ {\displaystyle \phi } is allowed to take on arbitrary real values. The infinitesimal arc length in these coordinates is So the length of a curve γ {\displaystyle \gamma } from p {\displaystyle p} to q {\displaystyle q}

2200-449: The northern, practically uninhabited part of the former Ittoqqortoormiit Municipality in Tunu (East Greenland), in 1988 the park was expanded by another 272,000 km (105,000 sq mi) to its present size, adding the northeastern part of the former county of Avannaa (North Greenland). In January 1977, it was designated an international biosphere reserve . The park is overseen by

2250-414: The park include great northern diver , barnacle goose , pink-footed goose , common eider , king eider , gyrfalcon , snowy owl , sanderling , ptarmigan and raven . Great Circle In mathematics , a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point . Any arc of a great circle is a geodesic of

2300-432: The plane turn around that fixed point through the same angle. One of the basic tenets of Euclidean geometry is that two figures (usually considered as subsets ) of the plane should be considered equivalent ( congruent ) if one can be transformed into the other by some sequence of translations, rotations and reflections (see below ). In order to make all of this mathematically precise, the theory must clearly define what

2350-537: The points P and Q is the subset of points such that 0 ≤ 𝜆 ≤ 1 in the preceding formulas. It is denoted PQ or QP ; that is P Q = Q P = { P + λ P Q → | 0 ≤ λ ≤ 1 } . ( {\displaystyle PQ=QP={\Bigl \{}P+\lambda {\overrightarrow {PQ}}\mathrel {\Big |} 0\leq \lambda \leq 1{\Bigr \}}.{\vphantom {\frac {(}{}}}} Two subspaces S and T of

2400-408: The sphere, so that great circles in spherical geometry are the natural analog of straight lines in Euclidean space . For any pair of distinct non- antipodal points on the sphere, there is a unique great circle passing through both. (Every great circle through any point also passes through its antipodal point, so there are infinitely many great circles through two antipodal points.) The shorter of

2450-500: The standard dot product . Euclidean space was introduced by ancient Greeks as an abstraction of our physical space. Their great innovation, appearing in Euclid's Elements was to build and prove 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 postulates , or axioms in modern language. This way of defining Euclidean space

2500-491: The two great-circle arcs between two distinct points on the sphere is called the minor arc , and is the shortest surface-path between them. Its arc length is the great-circle distance between the points (the intrinsic distance on a sphere), and is proportional to the measure of the central angle formed by the two points and the center of the sphere. A great circle is the largest circle that can be drawn on any given sphere. Any diameter of any great circle coincides with

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