Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
The hyperbolic plane is a plane where every point is a saddle point.
Hyperbolic plane geometry is also the geometry of pseudospherical surfaces, surfaces with a constant negative Gaussian curvature. Saddle surfaces have negative Gaussian curvature in at least some regions, where they locally resemble the hyperbolic plane.
The hyperboloid model of hyperbolic geometry provides a representation of events one temporal unit into the future in Minkowski space, the basis of special relativity. Each of these events corresponds to a rapidity in some direction.
When geometers first realised they were working with something other than the standard Euclidean geometry, they described their geometry under many different names; Felix Klein finally gave the subject the name hyperbolic geometry to include it in the now rarely used sequence elliptic geometry, parabolic geometry, and hyperbolic geometry.
In the former Soviet Union, it is commonly called Lobachevskian geometry, named after one of its discoverers, the Russian geometer Nikolai Lobachevsky.
Properties
Relation to Euclidean geometry
Hyperbolic geometry is more closely related to Euclidean geometry than it seems: the only axiomatic difference is the parallel postulate.When the parallel postulate is removed from Euclidean geometry the resulting geometry is absolute geometry.
There are two kinds of absolute geometry, Euclidean and hyperbolic.
All theorems of absolute geometry, including the first 28 propositions of book one of Euclid's Elements, are valid in Euclidean and hyperbolic geometry.
Propositions 27 and 28 of Book One of Euclid's Elements prove the existence of parallel/non-intersecting lines.
This difference also has many consequences: concepts that are equivalent in Euclidean geometry are not equivalent in hyperbolic geometry; new concepts need to be introduced.
Further, because of the angle of parallelism, hyperbolic geometry has an absolute scale, a relation between distance and angle measurements.
Lines
Single lines in hyperbolic geometry have exactly the same properties as single straight lines in Euclidean geometry. For example, two points uniquely define a line, and line segments can be infinitely extended.Two intersecting lines have the same properties as two intersecting lines in Euclidean geometry. For example, two distinct lines can intersect in no more than one point, intersecting lines form equal opposite angles, and adjacent angles of intersecting lines are supplementary.
When a third line is introduced, then there can be properties of intersecting lines that differ from intersecting lines in Euclidean geometry. For example, given two intersecting lines there are infinitely many lines that do not intersect either of the given lines.
These properties are all independent of the [|model] used, even if the lines may look radically different.
Non-intersecting / parallel lines
Non-intersecting lines in hyperbolic geometry also have properties that differ from non-intersecting lines in Euclidean geometry:This implies that there are through P an infinite number of coplanar lines that do not intersect R.
These non-intersecting lines are divided into two classes:
- Two of the lines are limiting parallels : there is one in the direction of each of the ideal points at the "ends" of R, asymptotically approaching R, always getting closer to R, but never meeting it.
- All other non-intersecting lines have a point of minimum distance and diverge from both sides of that point, and are called ultraparallel, diverging parallel or sometimes non-intersecting.
These limiting parallels make an angle θ with PB; this angle depends only on the Gaussian curvature of the plane and the distance PB and is called the angle of parallelism.
For ultraparallel lines, the ultraparallel theorem states that there is a unique line in the hyperbolic plane that is perpendicular to each pair of ultraparallel lines.
Circles and disks
In hyperbolic geometry, the circumference of a circle of radius r is greater than.Let, where is the Gaussian curvature of the plane. In hyperbolic geometry, is negative, so the square root is of a positive number.
Then the circumference of a circle of radius r is equal to:
And the area of the enclosed disk is:
Therefore, in hyperbolic geometry the ratio of a circle's circumference to its radius is always strictly greater than, though it can be made arbitrarily close by selecting a small enough circle.
If the Gaussian curvature of the plane is −1 then the geodesic curvature of a circle of radius r is:
Hypercycles and horocycles
In hyperbolic geometry, there is no line whose points are all equidistant from another line. Instead, the points that are all the same distance from a given line lie on a curve called a hypercycle.Another special curve is the horocycle, whose normal radii are all limiting parallel to each other.
Through every pair of points there are two horocycles. The centres of the horocycles are the ideal points of the perpendicular bisector of the line-segment between them.
Given any three distinct points, they all lie on either a line, hypercycle, horocycle, or circle.
The length of a line-segment is the shortest length between two points.
The arc-length of a hypercycle connecting two points is longer than that of the line segment and shorter than that of the arc horocycle, connecting the same two points.
The lengths of the arcs of both horocycles connecting two points are equal, and are longer than the arclength of any hypercycle connecting the points and shorter than the arc of any circle connecting the two points.
If the Gaussian curvature of the plane is −1, then the geodesic curvature of a horocycle is 1 and that of a hypercycle is between 0 and 1.
Triangles
Unlike Euclidean triangles, where the angles always add up to π radians, in hyperbolic space the sum of the angles of a triangle is always strictly less than π radians. The difference is called the defect. Generally, the defect of a convex hyperbolic polygon with sides is its angle sum subtracted from.The area of a hyperbolic triangle is given by its defect in radians multiplied by R, which is also true for all convex hyperbolic polygons. Therefore, all hyperbolic triangles have an area less than or equal to Rπ. The area of a hyperbolic ideal triangle in which all three angles are 0° is equal to this maximum.
As in Euclidean geometry, each hyperbolic triangle has an incircle. In hyperbolic space, if all three of its vertices lie on a horocycle or hypercycle, then the triangle has no circumscribed circle.
As in spherical and elliptical geometry, in hyperbolic geometry if two triangles are similar, they must be congruent.
Regular apeirogon and pseudogon
Special polygons in hyperbolic geometry are the regular apeirogon and pseudogon uniform polygons with an infinite number of sides.In Euclidean geometry, the only way to construct such a polygon is to make the side lengths tend to zero and the apeirogon is indistinguishable from a circle, or make the interior angles tend to 180° and the apeirogon approaches a straight line.
However, in hyperbolic geometry, a regular apeirogon or pseudogon has sides of any length.
The side and angle bisectors will, depending on the side length and the angle between the sides, be limiting or diverging parallel. If the bisectors are limiting parallel then it is an apeirogon and can be inscribed and circumscribed by concentric horocycles.
If the bisectors are diverging parallel then it is a pseudogon and can be inscribed and circumscribed by hypercycles.
Tessellations
Like the Euclidean plane it is also possible to tessellate the hyperbolic plane with regular polygons as faces.There are an infinite number of uniform tilings based on the Schwarz triangles where 1/p + 1/q + 1/r < 1, where p, q, r are each orders of reflection symmetry at three points of the fundamental domain triangle, the symmetry group is a hyperbolic triangle group. There are also infinitely many uniform tilings that cannot be generated from Schwarz triangles, some for example requiring quadrilaterals as fundamental domains.
Standardized Gaussian curvature
Though hyperbolic geometry applies for any surface with a constant negative Gaussian curvature, it is usual to assume a scale in which the curvature K is −1.This results in some formulas becoming simpler. Some examples are:
- The area of a triangle is equal to its angle defect in radians.
- The area of a horocyclic sector is equal to the length of its horocyclic arc.
- An arc of a horocycle so that a line that is tangent at one endpoint is limiting parallel to the radius through the other endpoint has a length of 1.
- The ratio of the arc lengths between two radii of two concentric horocycles where the horocycles are a distance 1 apart is e :1.
Cartesian-like coordinate systems
There are however different coordinate systems for hyperbolic plane geometry. All are based around choosing a point on a chosen directed line and after that many choices exist.
The Lobachevsky coordinates x and y are found by dropping a perpendicular onto the x-axis. x will be the label of the foot of the perpendicular. y will be the distance along the perpendicular of the given point from its foot.
Another coordinate system measures the distance from the point to the horocycle through the origin centered around and the length along this horocycle.
Other coordinate systems use the Klein model or the Poincaré disk model described below, and take the Euclidean coordinates as hyperbolic.