Given a point and a line , draw the perpendicular through and call it . Let be any other line from which meets in . In a hyperbolic geometry, as moves off to infinity along , then the line approaches the limiting line , which is said to be parallel to at . The angle which makes with is then called the angle of parallelism for perpendicular distance , and is given bywhich is called Lobachevsky's formula.
A non-Euclidean geometry, also called Lobachevsky-Bolyai-Gauss geometry, having constant sectional curvature . This geometry satisfies all of Euclid's postulates except the parallel postulate, which is modified to read: For any infinite straight line and any point not on it, there are many other infinitely extending straight lines that pass through and which do not intersect .In hyperbolic geometry, the sum of angles of a triangle is less than , and triangles with the same angles have the same areas. Furthermore, not all triangles have the same angle sum (cf. the AAA theorem for triangles in Euclidean two-space). There are no similar triangles in hyperbolic geometry. The best-known example of a hyperbolic space are spheres in Lorentzian four-space. The Poincaré hyperbolic disk is a hyperbolic two-space. Hyperbolic geometry is well understood in two dimensions, but not in three dimensions.Geometric models of hyperbolic geometry..
In elliptic -space, the pole of an -flat is a point located at an arc length of radians away from each point of the -flat.
The triangle bounded by the polars of the vertices of a triangle with respect to a conic is called its polar triangle. The following table summarizes polar triangles of named triangle conics that correspond to named triangles.conictriangleBrocard inellipsesymmedial trianglecircumcircletangential triangleincirclecontact triangleKiepert parabolaSteiner triangleLemoine inellipseLemoine triangleMacBeath inconicMacBeath triangleMandart inellipseextouch triangleorthic inconicorthic trianglepolar circlereference triangleStammler hyperbolareference triangleSteiner circumellipseanticomplementary triangleSteiner inellipsemedial triangleYff parabolaYff contact triangleAnother usage of the term applies in the elliptic plane or on a sphere, where the pole of a line is a point that is at an arc length of radians from each point of the line, in the same way that the poles of the Earth are a quarter circle away from the..
In elliptic n-space, the flat pole of an -flat is a point located an arc length of radians distant from each point of the -flat. For an -dimensional spherical simplex, there are such poles, one for each of its facets. Passing an -flat through each subset of of these poles then divides the space into simplices. The polar simplex is the simplex having edges that are supplements of the dihedral angles of the original simplex.There are twice as many simplexes in spherical n-space, with diametrically opposite simplexes being congruent, so the chosen simplex is the one located in the same hemisphere as the original simplex.The polar simplex of a polar simplex is the original simplex. The principal circumcenter of a simplex is the incenter of its polar simplex, and the principal circumradius of a simplex is the complement of the inradius of its polar simplex. The altitudes of a simplex and its polar simplex lie on the lines connecting corresponding vertices...
Elliptic geometry is a non-Euclidean geometry with positive curvature which replaces the parallel postulate with the statement "through any point in the plane, there exist no lines parallel to a given line." In order to achieve a consistent system, however, the basic axioms of neutral geometry must be partially modified. Most notably, the axioms of betweenness are no longer sufficient (essentially because betweenness on a great circle makes no sense, namely if and are on a circle and is between them, then the relative position of is not uniquely specified), and so must be replaced with the axioms of subsets.Elliptic geometry is sometimes also called Riemannian geometry. It can be visualized as the surface of a sphere on which "lines" are taken as great circles. In elliptic geometry, the sum of angles of a triangle is ...
The Poincaré hyperbolic disk is a two-dimensional space having hyperbolic geometry defined as the disk , with hyperbolic metric(1)The Poincaré disk is a model for hyperbolic geometry in which a line is represented as an arc of a circle whose ends are perpendicular to the disk's boundary (and diameters are also permitted). Two arcs which do not meet correspond to parallel rays, arcs which meet orthogonally correspond to perpendicular lines, and arcs which meet on the boundary are a pair of limits rays. The illustration above shows a hyperbolic tessellation similar to M. C. Escher's Circle Limit IV (Heaven and Hell) (Trott 1999, pp. 10 and 83).The endpoints of any arc can be specified by two angles around the disk and . Define(2)(3)Then trigonometry shows that in the above diagram,(4)(5)so the radius of the circle forming the arc is and its center is located at , where(6)The half-angle subtended by the arc is then(7)so(8)The..
In the hyperbolic plane , a pair of lines can be parallel (diverging from one another in one direction and intersecting at an ideal point at infinity in the other), can intersect, or can be hyperparallel (diverge from each other in both directions).Taimina (2006) has crocheted numerous hyperbolic planes, originally as an instructive device.
Given a point and a line , draw the perpendicular through and call it . Let be any other line from which meets in . In a hyperbolic geometry, as moves off to infinity along , then the line approaches the limiting line , which is said to be parallel to at . The angle which makes with is then called the angle of parallelism for perpendicular distance , and is given byThis is known as Lobachevsky's formula.