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''elliptic geometry''

One of the [[Non-Euclidean geometries|non-Euclidean geometries]], i.e. a geometrical theory based on axioms whose requirements are different from the requirements of the axioms of [[Euclidean geometry|Euclidean geometry]]. Unlike Euclidean geometry, elliptic geometry has one of the two possible negations of the axiom of parallelism in Euclidean geometry: In the plane, through a point that is not incident to a given straight line there is no straight line that does not intersect the given line; the other negation of the Euclidean axiom on parallelism occurs in [[Lobachevskii geometry|Lobachevskii geometry]]: In the plane, through a given point that is not incident to a given straight line there are at least two straight lines that do not intersect the given line. From now on  "line"  is understood to correspond to the concept  "straight line" .

A system of axioms of three-dimensional elliptic geometry can be constructed using the same concepts as in the [[Hilbert system of axioms|Hilbert system of axioms]] of Euclidean geometry: the basic concepts are  "point" ,  "line" ,  "plane" .  "Line"  and  "plane"  are regarded as certain classes of  "points" , and  "space"  is taken to be the set of all objects:  "points" ,  "lines"  and  "planes" .

The system of axioms is composed of four groups:

==Group I. Axioms of incidence.==
This group contains all the axioms comprising group 1 of Hilbert's system, plus one further axiom: Any two distinct lines in a plane have one and only one common point.

==Group II. Axioms of order or of position of points on a line.==
The axioms of this group describe the concept of  "separation of two pairs of points on a line" , with the aid of which the order of points on a line is determined.

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r0818901.png" />. Given three distinct points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r0818902.png" /> of an arbitrary line, there is on this line a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r0818903.png" /> such that the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r0818904.png" /> separates the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r0818905.png" /> (designated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r0818906.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r0818907.png" />, then all four points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r0818908.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r0818909.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189010.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189011.png" /> are distinct.

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189012.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189013.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189015.png" />.

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189016.png" />. Given four distinct points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189017.png" /> on a line, two separated pairs can always be constructed from them.

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189018.png" />. Let the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189022.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189023.png" /> lie on a line; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189025.png" />, then the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189026.png" /> does not separate the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189027.png" />.

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189028.png" />. If the pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189030.png" /> do not separate the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189031.png" />, then the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189032.png" /> also does not separate the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189033.png" /> (see <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189034.png" />).

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189035.png" />. If four distinct lines of a certain pencil are intersected by two distinct lines at points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189037.png" />, respectively, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189038.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189039.png" />.

==Group III. Axioms of congruence.==
These describe the relationship  "congruence"  of segments, angles, etc. A segment is taken to mean the set of points of a line determined by a pair of distinct points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189040.png" /> of this line in the following way. According to the axioms of group II, there is on the line a pair of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189041.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189042.png" />; the set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189043.png" /> that satisfy the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189044.png" /> form the class of interior points of the segment determined by the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189046.png" />; this is written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189047.png" />. The points of the line exterior to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189048.png" /> form the mutually complementary segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189049.png" />, the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189051.png" /> being called the ends of the segments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189053.png" />.

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189054.png" />. Each segment is congruent to itself.

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189055.png" />. If the first segment is congruent to a second, then the second segment is congruent to the first.

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189056.png" />. If the first segment is congruent to a second, and the second is congruent to a third, then the first is congruent to the third.

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189057.png" />. If two segments are congruent, then their mutually complementary segments are also congruent.

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189058.png" />. A segment is not congruent to a part of it. Congruent mutually complementary segments of a line are called half-lines; the ends of such segments are called orthogonal points of the line.

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189059.png" />. Each point on a line has an orthogonal point.

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189060.png" />. All half-lines are congruent to each other.

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189061.png" />. If a segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189062.png" /> is congruent to a segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189063.png" /> and a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189064.png" /> is an interior point of the first segment, then inside the second segment there is a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189065.png" /> such that the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189066.png" /> is congruent to the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189067.png" />.

On the sides of the angle formed by two lines there are orthogonal points relative to the vertex of the angle; the segment joining these two points and located inside the given angle is known as the segment associated with (measuring) the angle. Two angles are called congruent if the segments associated with them are congruent.

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189068.png" />. If in two triangles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189069.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189070.png" /> the side <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189071.png" /> is congruent to the side <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189072.png" /> and the side <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189073.png" /> is congruent to the side <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189074.png" />, then the angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189075.png" /> is congruent to the angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189076.png" /> if and only if the sides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189077.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189078.png" /> are congruent.

==Group IV. Axiom of continuity.==
Let the interior points of a segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189079.png" /> be divided into two classes such that: 1) each point of the segment falls into one of these classes; 2) each class is not empty; and 3) if a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189080.png" /> belongs to the first class and a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189081.png" /> to the second, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189082.png" /> is always an interior point of the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189083.png" />. Thus, in the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189084.png" /> there is a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189085.png" /> such that each interior point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189086.png" /> belongs to the first class and each interior point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189087.png" /> belongs to the second.

There are other systems of axioms for elliptic geometry, at the basis of which there are other basic concepts and relations (see, for example, [[#References|[3]]], [[#References|[5]]]).

The metrical properties of  "local"  elliptic geometry coincide with those of a certain hypersphere in the corresponding Euclidean space. For example, for any point in an elliptic plane there is a part of the plane containing this point which is isometric to a part of the sphere in three-dimensional Euclidean space; the radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189088.png" /> of this sphere is the same for all planes of a given elliptic space, and is called the radius of curvature of this space. The metrical properties of a three-dimensional elliptic space  "locally"  coincide with those of a hypersphere in four-dimensional Euclidean space, etc. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189089.png" /> is called the curvature of the elliptic space. For most purposes it is natural to assume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189090.png" />.

The basic facts of elliptic geometry for a (straight) line, a plane and a three-dimensional space are given below.

The elliptic line is a closed finite line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189091.png" />. A circle of unit radius with diametrically-opposite points identified in the Euclidean plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189092.png" /> can serve as a model of this line. Two distinct points of the line divide it into two parts. The mutual arrangement of points on the line is determined using the concept of  "separation of two pairs of points" . The distance between two points of the line is defined in two ways: the shorter does not exceed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189093.png" />, the longer exceeds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189094.png" />. The length of the entire line is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189095.png" />. Two points which are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189096.png" /> apart are called orthogonal; each point of the line has a corresponding unique point orthogonal to it.

The elliptic plane is a closed finite one-sided surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189097.png" /> homeomorphic to a Möbius strip whose boundary is glued to a circle (i.e. the elliptic plane is homeomorphic to the real projective plane). A sphere of unit radius with antipoles identified in three-dimensional Euclidean space can serve as a model for the plane of elliptic geometry with curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189098.png" />. A line does not divide the plane into two regions.

Any two lines in the plane have a common perpendicular of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r08189099.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890100.png" /> is the angle between the lines. Two distinct lines divide the plane into two regions, known as angles. A trilateral arrangement (i.e. three lines having three intersections) divides the entire plane into four regions, known as triangles.

The metrical relationships in a triangle in the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890101.png" /> are expressed by the corresponding relationships of [[Spherical trigonometry|spherical trigonometry]] on a sphere of unit radius in the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890102.png" />. Generally speaking, the trigonometric formulas in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890103.png" /> of elliptic geometry are the same as the formulas of spherical trigonometry.

The set of points of a plane located at a distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890104.png" /> from a given point (the pole) is a line, the polar of the pole. Any line is uniquely determined by its pole, and conversely determines its pole. Poles of lines which pass through a given point are situated on the polar of this point, and polars of points lying on a line intersect at the pole of this line. Mutually polar triangles have as vertices the poles of corresponding sides. For two mutually polar triangles Chasles' theorem holds: The three straight lines joining the corresponding vertices of these triangles intersect at one point. If the vertices of the triangle are the poles of its sides, then the triangle is called a self-polar triangle. The sum of the angles of a triangle is greater than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890105.png" />; its area <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890106.png" /> is equal to the angular excess <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890107.png" />.

A circle in the elliptic plane is the set of points located at the same distance from a certain point (its centre). The radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890108.png" /> of a circle may be taken to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890109.png" />. A circle is equidistant from a certain line (the axis of the circle). When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890110.png" />, the circle is a line — the polar of its centre. The length of a circle of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890111.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890112.png" /> and the area of the disc is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890113.png" />.

There are four and only four circles which pass through three given points that do not lie on one line. Two distinct circles can intersect at not more than four distinct points. The total area of the elliptic plane is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890114.png" />.

The duality principle is applicable in the elliptic geometry of a plane: In every true statement the terms  "point"  and  "line"  can be interchanged, and the result is a true statement.

Three-dimensional elliptic space is a closed finite two-sided (orientable) space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890115.png" />. A hypersphere of unit radius with diametrically-opposite points identified in the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890116.png" /> can serve as a model for the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890117.png" />. The volume of the elliptic space is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890118.png" />.

A plane does not divide the space into two regions. Two distinct planes in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890119.png" /> intersect along a line. A line which does not lie in a plane intersects it at one point.

The set of points of the space at distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890120.png" /> from a given point (the pole) is a plane — the polar of the pole. Any plane is uniquely determined by its pole, and conversely determines its pole. If three planes pass through one line, then their poles lie on one line, and, conversely, if the poles of three planes lie on one line, these planes intersect along one line. For each line there is a another line such that the poles of the planes which pass through the first line lie on the second, while the polar planes of the points which lie on the first line pass through the second line. These skew lines are called mutually polar. Two skew lines are called oblique if they are not mutually polar or if each of them does not intersect the polar of the other. Two oblique lines have two common perpendiculars which are mutually polar. If the two common perpendiculars of oblique lines are of different lengths, then the lengths of the common perpendiculars give the least and greatest distances of one line from the other. Two oblique lines with an infinite set of common perpendiculars of the same length are called Clifford parallels (equidistant or paratactic lines). Through each point of the space not belonging to either of two given polar lines, two Clifford parallels to the given lines can be drawn. The set of points at the same distance less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890121.png" /> from a given line is called a Clifford surface. The given line is called the axis and the distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890122.png" /> of the points from the axis is the radius of the Clifford surface. This surface has two mutually polar axes and two corresponding radii which complement each other up to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890123.png" />. A Clifford surface is a surface of revolution in two distinct ways. Through each point of a Clifford surface two lines equidistant from its axes and entirely belonging to the surface can be drawn. These lines are known as rectilinear generators of the Clifford surface. Any three lines equidistant from each other determine a Clifford surface for which they are generators. Each pair of generators of different families intersect at a constant angle. A Clifford surface is isometric to a Euclidean rhombus with an acute angle equal to the angle between the generators of the different families, with side-length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890124.png" /> and with opposite sides identified. In other words, a Clifford surface is locally Euclidean. The area of the Clifford surface with radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890125.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890126.png" />.

A sphere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890127.png" /> is the set of points located at the same distance from a given point (its centre). A sphere is equidistant from a plane (the axial plane of the sphere). The area of a sphere of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890128.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890129.png" />.

The duality principle is applicable in the elliptic geometry of space: The terms  "point"  and  "plane"  can be interchanged in every true statement, and a true statement is obtained.

The concept of elliptic geometry was apparently introduced by B. Riemann in his lecture [[#References|[1]]] (1854, published in 1867). In it, elliptic geometry was examined as a special case of [[Riemannian geometry|Riemannian geometry]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B. Riemann,   "Über die Hypothesen, welche der Geometrie zugrunde liegen" , ''Das Kontinuum und andere Monographien'' , Chelsea, reprint  (1973)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.V. Efimov,   "Higher geometry" , MIR  (1980)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.A. Rozenfel'd,   "Non-Euclidean spaces" , Moscow  (1969)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.F. Kagan,   "Foundations of geometry" , '''2''' , Moscow-Leningrad  (1956)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S.A. Bogomolov,   "An introduction to Riemann's non-Euclidean geometry" , Leningrad-Moscow  (1934)  (In Russian)</TD></TR></table>



====Comments====
At the end of his celebrated habilitation address (1854, published in 1867), Riemann turned to spaces with the same curvature at each point and in each two-dimensional direction. For figures in such spaces he asserted the same free mobility as governs Euclidean space, although he did not mention explicitly their relation to non-Euclidean geometry. This was done in 1868 by H. Helmholtz, who, starting from the postulate of free mobility of figures in space, also arrived at the spaces of constant curvature — vanishing curvature for Euclidean space, positive for spheres, and negative for absolute geometry. This led to an extension of the term non-Euclidean geometry to spaces of positive constant curvature (Riemannian geometries), where great circles should play the role of straight lines, and great two-dimensional spheres that of planes. The disadvantage that two distinct lines in the same plane then intersect in two points (rather than in one) can be overcome by identifying antipodal points ( "elliptic geometry" ) with each other. The topology of such a space is the same as that of projective space of the same dimension. The metric relations are those of spherical geometry; spherical trigonometry applies in a natural way, but since straight lines are topological circles, one has to be as careful about distances as one is in Euclidean geometry about angles. Elliptic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890130.png" />-space is most easily studied by means of quaternions.

It was F. Klein who interpreted Cayley's metric within a conic in the projective plane as a model of absolute geometry, which he generalized in order to include non-Euclidean geometries of arbitrary dimension. Depending on the type of the basic hyperquadric in projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890131.png" />-space,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890132.png" /></td> </tr></table>

or

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890133.png" /></td> </tr></table>

he spoke of elliptic and hyperbolic geometries, which eventually became the usual terminology.

Many axiomatic approaches to elliptic geometry have been proposed. If one prefers to keep close to Hilbert's axiomatics of Euclidean geometry, one has to replace Hilbert's axioms on linear order by axioms on cyclic order: 1) On each line there are two (mutually opposite) cyclic orders distinguished; and 2) projections within a plane map distinguished orders on each other.

(Cyclic order is defined as follows. For a finite set a cyclic order is an equivalence class of linear orders, defined by the relations (for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890134.png" />)

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890135.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890136.png" /></td> </tr></table>

An arbitrary set is cyclically ordered by providing every finite subset with a cyclic order, such that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890137.png" /> the cyclic order on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890138.png" /> extends that on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890139.png" />. Replacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890140.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081890/r081890141.png" /> changes an order into its opposite.)

For a modern treatment of elliptic geometry see [[#References|[a3]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Helmholtz,   "Über die Tatsachen, die der Geometrie zum Grunde liegen" , ''Wissenschaftliche Abhandlungen'' , '''II'''  (1883)  pp. 618–639</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter,   "Non-Euclidean geometry" , Univ. Toronto Press  (1965)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  F. Bachmann,   "Aufbau der Geometrie aus dem Spiegelungbegriff" , Springer  (1959)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H.S.M. Coxeter,   "Introduction to geometry" , Wiley  (1989)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J. Gray,   "Ideas of space" , Oxford  (1989)  pp. Chapt. 14</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  H.P. Manning,   "Introductory non-Euclidean geometry" , New York  (1963)  pp. Chapt. III</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  O. Veblen,   J.W. Young,   "Projective geometry" , '''II''' , Blaisdell  (1946)  pp. Chapt. VII</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  M. Berger,   "Geometry" , '''II''' , Springer  (1987)  pp. Chapt. 19</TD></TR></table>