Geometry/Hyperbolic and Elliptic Geometry
There are precisely three different classes of three-dimensional constant-curvature geometry: Euclidean, hyperbolic and elliptic geometry. The three geometries are all built on the same first four axioms, but each has a unique version of the fifth axiom, also known as the parallel postulate. The 1868 Essay on an Interpretation of Non-Euclidean Geometry by Eugenio Beltrami (1835 - 1900) proved the logical consistency of the two Non-Euclidean geometries, hyperbolic and elliptic.
The Parallel Postulate
The parallel postulate is as follows for the corresponding geometries.
Euclidean geometry: Playfair's version: "Given a line l and a point P not on l, there exists a unique line m through P that is parallel to l." Euclid's version: "Suppose that a line l meets two other lines m and n so that the sum of the interior angles on one side of l is less than 180°. Then m and n intersect in a point on that side of l." These two versions are equivalent; though Playfair's may be easier to conceive, Euclid's is often useful for proofs.
Hyperbolic geometry: Given an arbitrary infinite line l and any point P not on l, there exist two or more distinct lines which pass through P and are parallel to l.
Elliptic geometry: Given an arbitrary infinite line l and any point P not on l, there does not exist a line which passes through P and is parallel to l.
Hyperbolic geometry is also known as saddle geometry or Lobachevskian geometry. It differs in many ways to Euclidean geometry, often leading to quite counter-intuitive results. Some of these remarkable consequences of this geometry's unique fifth postulate include:
1. The sum of the three interior angles in a triangle is strictly less than 180°. Moreover, the angle sums of two distinct triangles are not necessarily the same.
2. Two triangles with the same interior angles have the same area.
Models of Hyperbolic Space
The following are four of the most common models used to describe hyperbolic space.
1. The Poincaré Disc Model. Also known as the conformal disc model. In it, the hyperbolic plane is represented by the interior of a circle, and lines are represented by arcs of circles that are orthogonal to the boundary circle and by diameters of the boundary circle. Preserves hyperbolic angles.
2. The Klein Model. Also known as the Beltrami-Klein model or projective disc model. In it, the hyperbolic plane is represented by the interior of a circle, and lines are represented by chords of the circle. This model gives a misleading visual representation of the magnitude of angles.
3. The Poincaré Half-Plane Model. The hyperbolic plane is represented by one-half of the Euclidean plane, as defined by a given Euclidean line l, where l is not considered part of the hyperbolic space. Lines are represented by half-circles orthogonal to l or rays perpendicular to l. Preserves hyperbolic angles.
4. The Lorentz Model. Spheres in Lorentzian four-space. The hyperbolic plane is represented by a two-dimensional hyperboloid of revolution embedded in three-dimensional Minkowski space.
Based on this geometry's definition of the fifth axiom, what does parallel mean? The following definitions are made for this geometry. If a line l and a line m do not intersect in the hyperbolic plane, but intersect at the plane's boundary of infinity, then l and m are said to be parallel. If a line p and a line q neither intersect in the hyperbolic plane nor at the boundary at infinity, then p and q are said to be ultraparallel.
The Ultraparallel Theorem
For any two lines m and n in the hyperbolic plane such that m and n are ultraparallel, there exists a unique line l that is perpendicular to both m and n.
Elliptic geometry differs in many ways to Euclidean geometry, often leading to quite counter-intuitive results. For example, directly from this geometry's fifth axiom we have that there exist no parallel lines. Some of the other remarkable consequences of the parallel postulate include: The sum of the three interior angles in a triangle is strictly greater than 180°.
Models of Elliptic Space
Spherical geometry gives us perhaps the simplest model of elliptic geometry. Points are represented by points on the sphere. Lines are represented by circles through the points.