Elliptic geometry

Last updated April 17, 2024

Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines are usually assumed to intersect at a single point (rather than two). Because of this, the elliptic geometry described in this article is sometimes referred to as single elliptic geometry whereas spherical geometry is sometimes referred to as double elliptic geometry.

Definitions

In elliptic geometry, two lines perpendicular to a given line must intersect. In fact, the perpendiculars on one side all intersect at a single point called the absolute pole of that line. The perpendiculars on the other side also intersect at a point. However, unlike in spherical geometry, the poles on either side are the same. This is because there are no antipodal points in elliptic geometry. For example, this is achieved in the hyperspherical model (described below) by making the "points" in our geometry actually be pairs of opposite points on a sphere. The reason for doing this is that it allows elliptic geometry to satisfy the axiom that there is a unique line passing through any two points.

Every point corresponds to an absolute polar line of which it is the absolute pole. Any point on this polar line forms an absolute conjugate pair with the pole. Such a pair of points is orthogonal, and the distance between them is a quadrant.^[1]^: 89

The distance between a pair of points is proportional to the angle between their absolute polars.^[1]^: 101

As explained by H. S. M. Coxeter:

The name "elliptic" is possibly misleading. It does not imply any direct connection with the curve called an ellipse, but only a rather far-fetched analogy. A central conic is called an ellipse or a hyperbola according as it has no asymptote or two asymptotes. Analogously, a non-Euclidean plane is said to be elliptic or hyperbolic according as each of its lines contains no point at infinity or two points at infinity.^[2]

Two dimensions

Elliptic plane

The elliptic plane is the real projective plane provided with a metric. Kepler and Desargues used the gnomonic projection to relate a plane σ to points on a hemisphere tangent to it. With O the center of the hemisphere, a point P in σ determines a line OP intersecting the hemisphere, and any line L ⊂ σ determines a plane OL which intersects the hemisphere in half of a great circle. The hemisphere is bounded by a plane through O and parallel to σ. No ordinary line of σ corresponds to this plane; instead a line at infinity is appended to σ. As any line in this extension of σ corresponds to a plane through O, and since any pair of such planes intersects in a line through O, one can conclude that any pair of lines in the extension intersect: the point of intersection lies where the plane intersection meets σ or the line at infinity. Thus the axiom of projective geometry, requiring all pairs of lines in a plane to intersect, is confirmed.^[3]

Given P and Q in σ, the elliptic distance between them is the measure of the angle POQ, usually taken in radians. Arthur Cayley initiated the study of elliptic geometry when he wrote "On the definition of distance".^[4]^: 82 This venture into abstraction in geometry was followed by Felix Klein and Bernhard Riemann leading to non-Euclidean geometry and Riemannian geometry.

Comparison with Euclidean geometry

In Euclidean geometry, a figure can be scaled up or scaled down indefinitely, and the resulting figures are similar, i.e., they have the same angles and the same internal proportions. In elliptic geometry, this is not the case. For example, in the spherical model we can see that the distance between any two points must be strictly less than half the circumference of the sphere (because antipodal points are identified). A line segment therefore cannot be scaled up indefinitely.

A great deal of Euclidean geometry carries over directly to elliptic geometry. For example, the first and fourth of Euclid's postulates, that there is a unique line between any two points and that all right angles are equal, hold in elliptic geometry. Postulate 3, that one can construct a circle with any given center and radius, fails if "any radius" is taken to mean "any real number", but holds if it is taken to mean "the length of any given line segment". Therefore any result in Euclidean geometry that follows from these three postulates will hold in elliptic geometry, such as proposition 1 from book I of the Elements, which states that given any line segment, an equilateral triangle can be constructed with the segment as its base.

Elliptic geometry is also like Euclidean geometry in that space is continuous, homogeneous, isotropic, and without boundaries. Isotropy is guaranteed by the fourth postulate, that all right angles are equal. For an example of homogeneity, note that Euclid's proposition I.1 implies that the same equilateral triangle can be constructed at any location, not just in locations that are special in some way. The lack of boundaries follows from the second postulate, extensibility of a line segment.

One way in which elliptic geometry differs from Euclidean geometry is that the sum of the interior angles of a triangle is greater than 180 degrees. In the spherical model, for example, a triangle can be constructed with vertices at the locations where the three positive Cartesian coordinate axes intersect the sphere, and all three of its internal angles are 90 degrees, summing to 270 degrees. For sufficiently small triangles, the excess over 180 degrees can be made arbitrarily small.

The Pythagorean theorem fails in elliptic geometry. In the 90°–90°–90° triangle described above, all three sides have the same length, and consequently do not satisfy $a^{2}+b^{2}=c^{2}$ . The Pythagorean result is recovered in the limit of small triangles.

The ratio of a circle's circumference to its area is smaller than in Euclidean geometry. In general, area and volume do not scale as the second and third powers of linear dimensions.

Elliptic space (the 3D case)

Note: This section uses the term "elliptic space" to refer specifically to 3-dimensional elliptic geometry. This is in contrast to the previous section, which was about 2-dimensional elliptic geometry. The quaternions are used to elucidate this space.

Elliptic space can be constructed in a way similar to the construction of three-dimensional vector space: with equivalence classes. One uses directed arcs on great circles of the sphere. As directed line segments are equipollent when they are parallel, of the same length, and similarly oriented, so directed arcs found on great circles are equipollent when they are of the same length, orientation, and great circle. These relations of equipollence produce 3D vector space and elliptic space, respectively.

Access to elliptic space structure is provided through the vector algebra of William Rowan Hamilton: he envisioned a sphere as a domain of square roots of minus one. Then Euler's formula $\exp(\theta r)=\cos \theta +r\sin \theta$ (where r is on the sphere) represents the great circle in the plane containing 1 and r. Opposite points r and –r correspond to oppositely directed circles. An arc between θ and φ is equipollent with one between 0 and φ – θ. In elliptic space, arc length is less than π, so arcs may be parametrized with θ in [0, π) or (–π/2, π/2].^[5]

For $z=\exp(\theta r),\ z^{*}=\exp(-\theta r)\implies zz^{*}=1.$ It is said that the modulus or norm of z is one (Hamilton called it the tensor of z). But since r ranges over a sphere in 3-space, exp(θ r) ranges over a sphere in 4-space, now called the 3-sphere, as its surface has three dimensions. Hamilton called his algebra quaternions and it quickly became a useful and celebrated tool of mathematics. Its space of four dimensions is evolved in polar co-ordinates $t\exp(\theta r),$ with t in the positive real numbers.

When doing trigonometry on Earth or the celestial sphere, the sides of the triangles are great circle arcs. The first success of quaternions was a rendering of spherical trigonometry to algebra.^[6] Hamilton called a quaternion of norm one a versor, and these are the points of elliptic space.

With $r$ fixed, the versors

e^{ar},\quad 0\leq a<\pi

form an elliptic line. The distance from $e^{ar}$ to 1 is $a$ . For an arbitrary versor $u$ , the distance will be that θ for which $cos θ = (u + u *)/2$ since this is the formula for the scalar part of any quaternion.

An elliptic motion is described by the quaternion mapping

q\mapsto uqv,

where

u

and

v

are fixed versors.

Distances between points are the same as between image points of an elliptic motion. In the case that $u$ and $v$ are quaternion conjugates of one another, the motion is a spatial rotation, and their vector part is the axis of rotation. In the case $u = 1$ the elliptic motion is called a right Clifford translation, or a parataxy. The case $v = 1$ corresponds to left Clifford translation.

Elliptic lines through versor $u$ may be of the form

\lbrace ue^{ar}:0\leq a<\pi \rbrace

or

\lbrace e^{ar}u:0\leq a<\pi \rbrace

for a fixed

r

.

They are the right and left Clifford translations of $u$ along an elliptic line through 1. The elliptic space is formed from $S 3$ by identifying antipodal points.^[7]

Elliptic space has special structures called Clifford parallels and Clifford surfaces.

The versor points of elliptic space are mapped by the Cayley transform to $\mathbb {R} ^{3}$ for an alternative representation of the space.

Higher-dimensional spaces

Hyperspherical model

The hyperspherical model is the generalization of the spherical model to higher dimensions. The points of n-dimensional elliptic space are the pairs of unit vectors $(x, - x)$ in Rⁿ⁺¹, that is, pairs of antipodal points on the surface of the unit ball in (n + 1)-dimensional space (the n-dimensional hypersphere). Lines in this model are great circles, i.e., intersections of the hypersphere with flat hypersurfaces of dimension n passing through the origin.

Projective elliptic geometry

In the projective model of elliptic geometry, the points of n-dimensional real projective space are used as points of the model. This models an abstract elliptic geometry that is also known as projective geometry.

The points of n-dimensional projective space can be identified with lines through the origin in (n + 1)-dimensional space, and can be represented non-uniquely by nonzero vectors in Rⁿ⁺¹, with the understanding that $u$ and $λ u$ , for any non-zero scalar $λ$ , represent the same point. Distance is defined using the metric

d(u,v)=\arccos \left({\frac {|u\cdot v|}{\|u\|\ \|v\|}}\right);

that is, the distance between two points is the angle between their corresponding lines in Rⁿ⁺¹. The distance formula is homogeneous in each variable, with $d (λ u, μ v) = d (u, v)$ if $λ$ and $μ$ are non-zero scalars, so it does define a distance on the points of projective space.

A notable property of the projective elliptic geometry is that for even dimensions, such as the plane, the geometry is non-orientable. It erases the distinction between clockwise and counterclockwise rotation by identifying them.

Stereographic model

A model representing the same space as the hyperspherical model can be obtained by means of stereographic projection. Let Eⁿ represent Rⁿ ∪ {∞}, that is, $n$ -dimensional real space extended by a single point at infinity. We may define a metric, the chordal metric, on Eⁿ by

\delta (u,v)={\frac {2\|u-v\|}{\sqrt {(1+\|u\|^{2})(1+\|v\|^{2})}}}

where $u$ and $v$ are any two vectors in Rⁿ and $\|\cdot \|$ is the usual Euclidean norm. We also define

\delta (u,\infty )=\delta (\infty ,u)={\frac {2}{\sqrt {1+\|u\|^{2}}}}.

The result is a metric space on Eⁿ, which represents the distance along a chord of the corresponding points on the hyperspherical model, to which it maps bijectively by stereographic projection. We obtain a model of spherical geometry if we use the metric

d(u,v)=2\arcsin \left({\frac {\delta (u,v)}{2}}\right).

Elliptic geometry is obtained from this by identifying the antipodal points $u$ and $- u / ‖ u ‖ 2$ , and taking the distance from $v$ to this pair to be the minimum of the distances from $v$ to each of these two points.

Self-consistency

Because spherical elliptic geometry can be modeled as, for example, a spherical subspace of a Euclidean space, it follows that if Euclidean geometry is self-consistent, so is spherical elliptic geometry. Therefore it is not possible to prove the parallel postulate based on the other four postulates of Euclidean geometry.

Tarski proved that elementary Euclidean geometry is complete: there is an algorithm which, for every proposition, can show it to be either true or false.^[8] (This does not violate Gödel's theorem, because Euclidean geometry cannot describe a sufficient amount of arithmetic for the theorem to apply.^[9]) It therefore follows that elementary elliptic geometry is also self-consistent and complete.

Notes

1 2 Duncan Sommerville (1914) The Elements of Non-Euclidean Geometry, chapter 3 Elliptic geometry, pp 88 to 122, George Bell & Sons
↑ Coxeter 1969 94
↑ H. S. M. Coxeter (1965) Introduction to Geometry, page 92
↑ Cayley, Arthur (1859), "A sixth memoir upon quantics", Philosophical Transactions of the Royal Society of London , 149: 61–90, doi: 10.1098/rstl.1859.0004 , ISSN 0080-4614, JSTOR 108690
↑ Rafael Artzy (1965) Linear Geometry, Chapter 3–8 Quaternions and Elliptic Three-space, pp. 186–94,Addison-Wesley
↑ W.R. Hamilton(1844 to 1850) On quaternions or a new system of imaginaries in algebra, Philosophical Magazine, link to David R. Wilkins collection at Trinity College, Dublin
↑ Lemaître, Georges (1948), "Quaternions et espace elliptique", Pontificia Academia Scientiarum, Acta , 12: 57–78, ISSN 0370-2138
↑ Tarski (1951)
↑ Franzén 2005, pp. 25–26.

Related Research Articles

Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry, Elements. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated earlier, Euclid was the first to organize these propositions into a logical system in which each result is proved from axioms and previously proved theorems.

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" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics.

A sphere is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. Formally, a sphere is the set of points that are all at the same distance $r$ from a given point in three-dimensional space. That given point is the center of the sphere, and $r$ is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians.

In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or relaxing the metric requirement. In the former case, one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras, which give rise to kinematic geometries that have also been called non-Euclidean geometry.

Spherical geometry or spherics is the geometry of the two-dimensional surface of a sphere or the $n$ -dimensional surface of higher dimensional spheres.

<span class="mw-page-title-main">Hyperbolic geometry</span> Non-Euclidean geometry

In mathematics, hyperbolic geometry is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:

In mathematics, two points of a sphere are called antipodal or diametrically opposite if they are the endpoints of a diameter, a straight line segment between two points on a sphere and passing through its center.

<span class="mw-page-title-main">Real projective plane</span> Compact non-orientable two-dimensional manifold

In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has basic applications to geometry, since the common construction of the real projective plane is as the space of lines in R³ passing through the origin. The real projective plane is then an extension of the (ordinary) plane — to every point $(v 1, v 2)$ of the ordinary plane, the line spanned by $(v 1, v 2,1)$ is associated (i.e., the real projective plane is the projective completion of the ordinary plane, cf. also the homogeneous coordinates below) while there are also some “points in the infinity”.

In projective geometry, duality or plane duality is a formalization of the striking symmetry of the roles played by points and lines in the definitions and theorems of projective planes. There are two approaches to the subject of duality, one through language and the other a more functional approach through special mappings. These are completely equivalent and either treatment has as its starting point the axiomatic version of the geometries under consideration. In the functional approach there is a map between related geometries that is called a duality. Such a map can be constructed in many ways. The concept of plane duality readily extends to space duality and beyond that to duality in any finite-dimensional projective geometry.

In geometry, parallel lines are coplanar infinite straight lines that do not intersect at any point. Parallel planes are planes in the same three-dimensional space that never meet. Parallel curves are curves that do not touch each other or intersect and keep a fixed minimum distance. In three-dimensional Euclidean space, a line and a plane that do not share a point are also said to be parallel. However, two noncoplanar lines are called skew lines. Line segments and Euclidean vectors are parallel if they have the same direction.

<span class="mw-page-title-main">Line (geometry)</span> Straight figure with zero width and depth

In geometry, a straight line, usually abbreviated line, is an infinitely long object with no width, depth, or curvature, an idealization of such physical objects as a straightedge, a taut string, or a ray of light. Lines are spaces of dimension one, which may be embedded in spaces of dimension two, three, or higher. The word line may also refer, in everyday life, to a line segment, which is a part of a line delimited by two points.

In mathematics, Hilbert's fourth problem in the 1900 list of Hilbert's problems is a foundational question in geometry. In one statement derived from the original, it was to find — up to an isomorphism — all geometries that have an axiomatic system of the classical geometry, with those axioms of congruence that involve the concept of the angle dropped, and `triangle inequality', regarded as an axiom, added.

In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle. The triangle can be an ordinary Euclidean triangle, a triangle on the sphere, or a hyperbolic triangle. Each triangle group is the symmetry group of a tiling of the Euclidean plane, the sphere, or the hyperbolic plane by congruent triangles called Möbius triangles, each one a fundamental domain for the action.

In mathematics, a versor is a quaternion of norm one. Each versor has the form

In geometry, a three-dimensional space is a mathematical space in which three values (coordinates) are required to determine the position of a point. Most commonly, it is the three-dimensional Euclidean space, that is, the Euclidean space of dimension three, which models physical space. More general three-dimensional spaces are called 3-manifolds. The term may also refer colloquially to a subset of space, a three-dimensional region, a solid figure.

In geometry, the Beltrami–Klein model, also called the projective model, Klein disk model, and the Cayley–Klein model, is a model of hyperbolic geometry in which points are represented by the points in the interior of the unit disk and lines are represented by the chords, straight line segments with ideal endpoints on the boundary sphere.

In a Euclidean space, the sum of angles of a triangle equals the straight angle . A triangle has three angles, one at each vertex, bounded by a pair of adjacent sides.

In mathematics, a plane is a two-dimensional space or flat surface that extends indefinitely. A plane is the two-dimensional analogue of a point, a line and three-dimensional space.

Foundations of geometry is the study of geometries as axiomatic systems. There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean geometries. These are fundamental to the study and of historical importance, but there are a great many modern geometries that are not Euclidean which can be studied from this viewpoint. The term axiomatic geometry can be applied to any geometry that is developed from an axiom system, but is often used to mean Euclidean geometry studied from this point of view. The completeness and independence of general axiomatic systems are important mathematical considerations, but there are also issues to do with the teaching of geometry which come into play.

In spherical geometry, Lexell's theorem holds that every spherical triangle with the same surface area on a fixed base has its apex on a small circle, called Lexell's circle or Lexell's locus, passing through each of the two points antipodal to the two base vertices.

References

Alan F. Beardon, The Geometry of Discrete Groups , Springer-Verlag, 1983
H. S. M. Coxeter (1942) Non-Euclidean Geometry, chapters 5, 6, & 7: Elliptic geometry in 1, 2, & 3 dimensions, University of Toronto Press, reissued 1998 by Mathematical Association of America, ISBN 0-88385-522-4.
H.S.M. Coxeter (1969) Introduction to Geometry, §6.9 The Elliptic Plane, pp. 92–95. John Wiley & Sons.
"Elliptic geometry", Encyclopedia of Mathematics , EMS Press, 2001 [1994]
Felix Klein (1871) "On the so-called noneuclidean geometry" Mathematische Annalen 4:573–625, translated and introduced in John Stillwell (1996) Sources of Hyperbolic Geometry, American Mathematical Society ISBN 0-8218-0529-0.
Boris Odehnal "On isotropic congruences of lines in elliptic three-space"
Eduard Study (1913) D.H. Delphenich translator, "Foundations and goals of analytical kinematics", page 20.
Alfred Tarski (1951) A Decision Method for Elementary Algebra and Geometry. Univ. of California Press.
Franzén, Torkel (2005). Gödel's Theorem: An Incomplete Guide to its Use and Abuse. AK Peters. ISBN 1-56881-238-8.
Alfred North Whitehead (1898) Universal Algebra Archived 2014-09-03 at the Wayback Machine , Book VI Chapter 2: Elliptic Geometry, pp 371–98.

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[DS-1] 1 2 Duncan Sommerville (1914) The Elements of Non-Euclidean Geometry, chapter 3 Elliptic geometry, pp 88 to 122, George Bell & Sons

[2] Coxeter 1969 94

[3] H. S. M. Coxeter (1965) Introduction to Geometry, page 92

[4] Cayley, Arthur (1859), "A sixth memoir upon quantics", Philosophical Transactions of the Royal Society of London , 149: 61–90, doi: 10.1098/rstl.1859.0004 , ISSN 0080-4614, JSTOR 108690

[5] Rafael Artzy (1965) Linear Geometry, Chapter 3–8 Quaternions and Elliptic Three-space, pp. 186–94,Addison-Wesley

[6] W.R. Hamilton(1844 to 1850) On quaternions or a new system of imaginaries in algebra, Philosophical Magazine, link to David R. Wilkins collection at Trinity College, Dublin

[7] Lemaître, Georges (1948), "Quaternions et espace elliptique", Pontificia Academia Scientiarum, Acta , 12: 57–78, ISSN 0370-2138

[8] Tarski (1951)

[9] Franzén 2005, pp. 25–26.

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