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Triangle in hyperbolic geometry
In hyperbolic geometry, a hyperbolic triangle is a triangle in the hyperbolic plane. It consists of three line segments called sides or edges and three
Hyperbolic_triangle
Region of the Cartesian plane bounded by a hyperbola and two radii
functions. When in standard position, a hyperbolic sector determines a hyperbolic triangle, the right triangle with one vertex at the origin, base on the
Hyperbolic_sector
Symmetric subdivision in hyperbolic geometry
In hyperbolic geometry, a uniform hyperbolic tiling (or regular, quasiregular or semiregular hyperbolic tiling) is an edge-to-edge filling of the hyperbolic
Uniform tilings in hyperbolic plane
Uniform_tilings_in_hyperbolic_plane
Type of non-Euclidean geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate
Hyperbolic_geometry
Shape with three sides
"straight" segments also determine a "triangle", for instance, a spherical triangle or hyperbolic triangle. A geodesic triangle is a region of a general two-dimensional
Triangle
Type of hyperbolic triangle
In hyperbolic geometry an ideal triangle is a hyperbolic triangle whose three vertices all are ideal points. Ideal triangles are also sometimes called
Ideal_triangle
Group realized geometrically by reflections across the sides of a triangle
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
Triangle_group
Spherical triangle that can be used to tile a sphere
Euclidean plane, or the hyperbolic plane. Each Schwarz triangle on a sphere defines a finite group, while on the Euclidean or hyperbolic plane they define an
Schwarz_triangle
Fundamental result in geometry
fails. Contrarily to the spherical case, the sum of the angles of a hyperbolic triangle is less than 180°, and can be arbitrarily close to 0°. Thus one has
Sum_of_angles_of_a_triangle
Argument of the hyperbolic functions
on hyperbolic analogies to the corresponding circular (trigonometric) functions by regarding a hyperbolic angle as defining a hyperbolic triangle. The
Hyperbolic_angle
as Euclidean polygons. In particular, the sum of the angles of a hyperbolic triangle is less than 180 degrees. Coxeter decompositions are named after
Coxeter decompositions of hyperbolic polygons
Coxeter_decompositions_of_hyperbolic_polygons
Triangle containing a 90-degree angle
A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular
Right_triangle
Triangle with circular arc edges
circular triangles Hart circle, a circle associated with certain circular triangles Hyperbolic triangle, a triangle that has straight sides in hyperbolic geometry
Circular_triangle
Relation between sides of a right triangle
cosh is the hyperbolic cosine. This formula is a special form of the hyperbolic law of cosines that applies to all hyperbolic triangles: cosh c R =
Pythagorean_theorem
Point in a triangle that can be seen as its middle under some criteria
In geometry, a triangle center or triangle centre is a point in the triangle's plane that is in some sense in the middle of the triangle. For example,
Triangle_center
In the theory of Riemann surfaces and hyperbolic geometry, the triangle group (2,3,7) is particularly important for its connection to Hurwitz surfaces
(2,3,7)_triangle_group
Topics referred to by the same term
mathematics, hyperbolic trigonometry can mean: The study of hyperbolic triangles in hyperbolic geometry (traditional trigonometry is the study of triangles in plane
Hyperbolic_trigonometry
Trigonometric result for hyperbolic triangles
In hyperbolic geometry, the "law of cosines" is a pair of theorems relating the sides and angles of triangles on a hyperbolic plane, analogous to the planar
Hyperbolic_law_of_cosines
Hyperbolic analogues of trigonometric functions
1. The hyperbolic functions may be defined in terms of the legs of a right triangle covering this sector. In complex analysis, the hyperbolic functions
Hyperbolic_functions
Geometry of figures on the surface of a sphere
trigonometry Great-circle distance or spherical distance Hyperbolic triangle Lenart sphere Schwarz triangle Spherical geometry Spherical polyhedron Triangulation
Spherical_trigonometry
Theorem in differential geometry
cases of Gauss–Bonnet. In spherical trigonometry and hyperbolic trigonometry, the area of a triangle is proportional to the amount by which its interior
Gauss–Bonnet_theorem
Orientation-preserving mapping class group of the torus
the hyperbolic plane by congruent hyperbolic triangles known as the V6.6.∞ Infinite-order triangular tiling is created. Note that each such triangle has
Modular_group
Geometrical term
being coterminal. If, in a hyperbolic triangle, the pairs of sides are limiting parallel, then the triangle is an ideal triangle. A ray A a {\displaystyle
Limiting_parallel
Schwarz triangles, the Schwarz triangles are ordered by their densities. The analogous cases of Euclidean tilings are also listed, and those of hyperbolic tilings
List of uniform polyhedra by Schwarz triangle
List_of_uniform_polyhedra_by_Schwarz_triangle
Characterizes spherical triangles with fixed base and area
parallel to the base. An analogous theorem can also be proven for hyperbolic triangles, for which the apex lies on a hypercycle. Given a fixed base A B
Lexell's_theorem
Concept in mathematics
hyperbolic spaces as they are 0-hyperbolic (i.e. all triangles are tripods). The 1-skeleton of the triangulation by Euclidean equilateral triangles is
Hyperbolic_metric_space
Category of coordinate systems
In the hyperbolic plane, as in the Euclidean plane, each point can be uniquely identified by two real numbers. Several qualitatively different ways of
Coordinate systems for the hyperbolic plane
Coordinate_systems_for_the_hyperbolic_plane
hypercubic honeycomb Triangle Automedian triangle Delaunay triangulation Equilateral triangle Golden triangle Hyperbolic triangle (non-Euclidean geometry)
List_of_mathematical_shapes
Angle in certain right triangles in the hyperbolic plane
hyperbolic geometry, angle of parallelism Π ( a ) {\displaystyle \Pi (a)} is the angle at the non-right angle vertex of a right hyperbolic triangle having
Angle_of_parallelism
Mathematics of smooth surfaces
is complete. A hyperbolic triangle is a geodesic triangle for this metric: any three points in D are vertices of a hyperbolic triangle. If the sides have
Differential geometry of surfaces
Differential_geometry_of_surfaces
Mathematical concept
precisely in geometric group theory, a hyperbolic group, also known as a word hyperbolic group or Gromov hyperbolic group, is a finitely generated group
Hyperbolic_group
tilings on sphere, euclidean plane, and hyperbolic plane can be made by Wythoff construction within a fundamental triangle, (p q r), defined by internal angles
Lists of uniform tilings on the sphere, plane, and hyperbolic plane
Lists_of_uniform_tilings_on_the_sphere,_plane,_and_hyperbolic_plane
Topics referred to by the same term
may refer to: Angle of parallelism, in hyperbolic geometry, the angle at one vertex of a right hyperbolic triangle that has two hyperparallel sides Axial
Parallelism
Generalized manifold
constructs Fuchsian groups as hyperbolic reflection groups generated by reflections in the edges of a geodesic triangle in the hyperbolic plane for the Poincaré
Orbifold
Doughnut-shaped surface of revolution
angles of a hyperbolic triangle T determine T up to congruence.) As a result, the Gauss–Bonnet theorem shows that the area of each triangle can be calculated
Torus
Mathematical space used to study hyperbolic geometry
space is a mathematical concept proposed by Abraham A. Ungar for studying hyperbolic geometry in analogy to the way vector spaces are used in Euclidean geometry
Gyrovector_space
Constant a such that af(x) is a probability measure
used to establish the hyperbolic functions cosh and sinh from the lengths of the adjacent and opposite sides of a hyperbolic triangle. Normalization (statistics)
Normalizing_constant
In mathematics, a Riemann surface
subgroup of the group generated by reflections in the sides of a hyperbolic triangle with angles π 2 , π 3 , π 8 {\displaystyle {\tfrac {\pi }{2}},{\tfrac
Bolza_surface
Covering by shapes without overlaps or gaps
Schwarz triangle is a spherical triangle that can be used to tile a sphere. It is possible to tessellate in non-Euclidean geometries such as hyperbolic geometry
Tessellation
Pictorial representation of symmetry
subdivided, e.g. into hyperbolic and other Coxeter groups. However, there are multiple non-equivalent definitions for hyperbolic Coxeter groups. We use
Coxeter–Dynkin_diagram
inequality Heilbronn triangle problem Heptagonal triangle Heronian triangle Heron's formula Hofstadter points Hyperbolic triangle (non-Euclidean geometry)
List_of_triangle_topics
Two geometries based on axioms closely related to those specifying Euclidean geometry
defect of triangles in hyperbolic geometry is positive, the defect of triangles in Euclidean geometry is zero, and the defect of triangles in elliptic
Non-Euclidean_geometry
Tiling of the hyperbolic plane
3,7) triangle group, and a fundamental domain for this action is the (2,3,7) Schwarz triangle. This is the smallest hyperbolic Schwarz triangle, and thus
Heptagonal_tiling
Point at infinity in hyperbolic geometry
theorem still hold for an omega triangle, defined by two points in hyperbolic space and an omega point. The hyperbolic distance between an ideal point
Ideal_point
Form of differential geometry
defined by a tower of principal congruence subgroups of the (2,3,7) hyperbolic triangle group satisfy the bound s y s π 1 ( Σ g ) ≥ 4 3 log g , {\displaystyle
Systolic_geometry
Property of geometry, also used to generalize the notion of "distance" in metric spaces
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length
Triangle_inequality
Group that admits a formal description in terms of reflections
Coxeter groups include the triangle groups corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane, and the Weyl groups
Coxeter_group
Swiss polymath (1728–1777)
As the triangle gets larger or smaller, the angles change in a way that forbids the existence of similar hyperbolic triangles, as only triangles that have
Johann_Heinrich_Lambert
Concept in geometry
regular tiling of the hyperbolic plane with a Schläfli symbol of {3,7}. The symmetry group of the tiling is the (2,3,7) triangle group, and a fundamental
Order-7_triangular_tiling
Study of triangles in other spaces than the Euclidean plane
plane triangle identities. Hyperbolic trigonometry: Study of hyperbolic triangles in hyperbolic geometry with hyperbolic functions. Hyperbolic functions
Generalized_trigonometry
Basic framework of mathematics
(1728–1777) started to build hyperbolic geometry and introduced the hyperbolic functions and computed the area of a hyperbolic triangle (where the sum of angles
Foundations_of_mathematics
Isometric automorphisms of a hyperbolic space
In geometry, hyperbolic motions are isometric automorphisms of a hyperbolic space. Under composition of mappings, the hyperbolic motions form a continuous
Hyperbolic_motion
Concept in geometry
. The hyperbolic case is similar, with the area of a disk of intrinsic radius R in the (constant curvature − 1 {\displaystyle -1} ) hyperbolic plane given
Area_of_a_circle
Conformal mappings in complex analysis
spherical triangle if α + β + γ > 1, or a hyperbolic triangle if α + β + γ < 1. When α + β + γ = 1, then the triangle is a Euclidean triangle with straight
Schwarz_triangle_function
Property of all triangles on a Euclidean plane
rule) is a mathematical equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, a sin α = b sin β
Law_of_sines
Semiregular tiling of the hyperbolic plane
semiregular dual tiling of the hyperbolic plane. It is constructed by congruent right triangles with 4, 6, and 14 triangles meeting at each vertex. The image
3-7_kisrhombille
Interpolation method in computer graphics
used to achieve continuous lighting on triangle meshes by computing the lighting at the corners of each triangle and linearly interpolating the resulting
Gouraud_shading
geometry, triangle inequalities are inequalities involving the parameters of triangles, that hold for every triangle, or for every triangle meeting certain
List_of_triangle_inequalities
Compact Riemann surface of genus 3
algebraic integers. The group Γ(I) is a subgroup of the (2,3,7) hyperbolic triangle group. Namely, Γ(I) is a subgroup of the group of elements of unit
Klein_quartic
Right triangle with a feature making calculations on the triangle easier
geometry and hyperbolic geometry, there are infinitely many different shapes of right isosceles triangles. Another type of special right triangle is the 30°-60°-90°
Special_right_triangle
defined by a tower of principal congruence subgroups of the (2,3,7) hyperbolic triangle group satisfy the bound s y s ( Σ g ) ≥ 4 3 log g , {\displaystyle
Systoles_of_surfaces
looking at hyperbolic space. hyperbolic trigonometry the study of hyperbolic triangles in hyperbolic geometry, or hyperbolic functions in Euclidean geometry
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Meanings of mass in special relativity
December 12, 2017 – via HUIT Sites Hosting. Ungar, Abraham A. (2010). Hyperbolic Triangle Centers: The Special Relativistic Approach. Dordrecht: Springer.
Mass_in_special_relativity
1959 woodcut by M. C. Escher
tessellation of the hyperbolic plane by right triangles with angles of 30°, 45°, and 90°; triangles with these angles are possible in hyperbolic geometry but
Circle_Limit_III
This article lists the regular polytopes in Euclidean, spherical and hyperbolic spaces. This table shows a summary of regular polytope counts by rank.
List_of_regular_polytopes
Shape with four equal sides and angles
two forms of non-Euclidean geometry. Although spherical geometry and hyperbolic geometry both lack polygons with four equal sides and right angles, they
Square
In addition, the angles in a hyperbolic triangle add up to less than 180° (a defect), while those on a spherical triangle add up to more than 180° (an
Angular_defect
index 2 Fuchsian subgroup of orientation-preserving elements. The hyperbolic triangle groups are notable NEC groups. Others are listed in Orbifold notation
Non-Euclidean crystallographic group
Non-Euclidean_crystallographic_group
Periodic tiling of the hyperbolic disk
In geometry, the order-3 apeirogonal tiling is a regular tiling of the hyperbolic plane. It is represented by the Schläfli symbol {∞,3}, having three regular
Order-3_apeirogonal_tiling
Space where every point locally resembles a hyperbolic space
In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in
Hyperbolic_manifold
Linear map that preserves areas
argument adding and subtracting triangles of area 1⁄2, one triangle being {(0,0), (0,1), (1,1)}, shows the hyperbolic sector area is equal to the area
Squeeze_mapping
Topics referred to by the same term
with every triangle Nine-point hyperbola, hyperbola constructed from a given triangle and point Unit hyperbola, geometric figure Hyperbolic (disambiguation)
Hyperbola_(disambiguation)
Equation for radii of tangent circles
definition of curvature, the theorem also applies in spherical geometry and hyperbolic geometry. In higher dimensions, an analogous quadratic equation applies
Descartes'_theorem
Russian-American mathematician (1963–2026)
research dealt with low-dimensional geometry and topology, Kleinian groups, hyperbolic geometry, geometric group theory, geometric representation theory in Lie
Michael_Kapovich
Notation for tesselations
tilings in Euclidean or hyperbolic space. The Wythoff construction begins by choosing a generator point on a fundamental triangle. This point must be chosen
Wythoff_symbol
Russian mathematician (1792–1856)
distance the point is off the given line. In hyperbolic geometry the sum of angles in a hyperbolic triangle must be less than 180 degrees. Non-Euclidean
Nikolai_Lobachevsky
Theorem about triangles
sines and hyperbolic sines, respectively. Projective geometry Median (geometry) – an application Circumcevian triangle Menelaus's theorem Triangle Stewart's
Ceva's_theorem
Spatial geometry with curvature
to be open or hyperbolic. Triangles which lie on the surface of an open space will have a sum of angles which is less than 180°. Triangles which lie on
Curved_space
Another relevant list is that of K. Takeuchi, who classified the (hyperbolic) triangle groups that are arithmetic groups (85 examples). Émile Picard sought
Schwarz's_list
Relationship between two figures of the same shape and size, or mirroring each other
Euclidean space. However, in spherical geometry and hyperbolic geometry (where the sum of the angles of a triangle varies with size) AAA is sufficient for congruence
Congruence_(geometry)
Theorem in algebraic geometry
If the fundamental domain is a triangle with the vertex angles π/p, π/q and π/r, defining a tiling of the hyperbolic plane, then p, q, and r are integers
Hurwitz's automorphisms theorem
Hurwitz's_automorphisms_theorem
Quadrilateral with four right angles
angles are equal. Other geometries, such as spherical, elliptic, and hyperbolic, have so-called rectangles with opposite sides equal in length and equal
Rectangle
Generalization of Pythagorean theorem
theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides a {\displaystyle a} , b {\displaystyle
Law_of_cosines
On rays from a point to a line, with equal inscribed circles between adjacent rays
the sides of the triangles r N h − r N = tanh N b 2 . {\displaystyle {\frac {r_{N}}{h-r_{N}}}=\tanh {\frac {Nb}{2}}.} Hyperbolic function Japanese
Equal_incircles_theorem
Three linked but pairwise separated rings
n-colorings. As links, they are Brunnian, alternating, algebraic, and hyperbolic. In arithmetic topology, certain triples of prime numbers have analogous
Borromean_rings
German polymath and scholar (1777–1855)
was a drawing of a tessellation of the unit disk by "equilateral" hyperbolic triangles with all angles equal to π / 4 {\displaystyle \pi /4} . An example
Carl_Friedrich_Gauss
Geometric mean and hyperbolic angle as coordinates in quadrant I
diagonals divide the rhombus into four congruent right triangles. The angle MOA is the hyperbolic angle parameter u of cosh and sinh, and tanh u = sinh
Hyperbolic_coordinates
Regular tiling of the plane
parallelogons. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees. The triangular tiling
Triangular_tiling
mathematician Mikhail Gromov. The Gromov product can also be used to define δ-hyperbolic metric spaces in the sense of Gromov. Let (X, d) be a metric space and
Gromov_product
Geometry of the surface of a sphere
elliptic geometry, to which spherical geometry is closely related, and hyperbolic geometry; each of these new geometries makes a different change to the
Spherical_geometry
Mathematical function relating circular and hyperbolic functions
In mathematics, the Gudermannian function relates a hyperbolic angle measure ψ {\textstyle \psi } to a circular angle measure ϕ {\textstyle \phi } called
Gudermannian_function
Euclidean geometry without distance and angles
Minkowski geometry, lines that are hyperbolic-orthogonal remain in that relation when the plane is subjected to hyperbolic rotation. An axiomatic treatment
Affine_geometry
3 intersections of any triangle's adjacent angle trisectors form an equilateral triangle
in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, called the first Morley triangle or simply
Morley's_trisector_theorem
Schwarz triangle (2,3,7) or a realization as a hyperbolic reflection group), but rather to the ordinary triangle group (the von Dyck group) D(2,3,7) of
Hurwitz_surface
Quaternion of norm 1 (unit quaternion)
saw the modelling power of hyperbolic versors operating on the split-complex number plane, and in 1891 he introduced hyperbolic quaternions to extend the
Versor
Subdivision of the plane by lines
set of points. Arrangements of lines have also been considered in the hyperbolic plane, and generalized to pseudolines, curves that have similar topological
Arrangement_of_lines
Geometry without the parallel postulate
systems, giving rise to Euclidean or hyperbolic geometry. Thus every theorem of absolute geometry is a theorem of hyperbolic geometry and Euclidean geometry
Absolute_geometry
Fundamental trigonometric functions
sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side
Sine_and_cosine
Theoretical physics phenomenon
This does not hold for relativistic velocity addition; instead a hyperbolic triangle arises whose edges are related to the rapidities of the boosts. Changing
Wigner_rotation
Reals with an extra square root of +1 adjoined
algebra, a split-complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit j satisfying j 2 = 1 {\displaystyle
Split-complex_number
HYPERBOLIC TRIANGLE
HYPERBOLIC TRIANGLE
HYPERBOLIC TRIANGLE
Girl/Female
Hindu, Indian
Falgun Month
Girl/Female
Hindu, Indian, Marathi
Crest Jewel; Disciplined; Cultured
Surname or Lastname
English
English : variant spelling of Council.
Boy/Male
Hindu
Supreme spirit, Lord of the lords, A name of Lord Rama
Boy/Male
Hindu
Home, Banner, Golden
Girl/Female
Indian
Intelligence, Mind
Boy/Male
Indian, Punjabi, Sikh
God's Warrior
Boy/Male
Muslim
The avenger
Female
Spanish
Portuguese and Spanish form of Hebrew Debowrah, DÉBORA means "bee."
Boy/Male
Tamil
Silver or courage
HYPERBOLIC TRIANGLE
HYPERBOLIC TRIANGLE
HYPERBOLIC TRIANGLE
HYPERBOLIC TRIANGLE
HYPERBOLIC TRIANGLE
n.
The use of hyperbole.
n.
One who uses hyperboles.
a.
Having the form, or nearly the form, of an hyperbola.
v. i.
To speak or write with exaggeration.
a.
Having some property that belongs to an hyperboloid or hyperbola.
n.
The act of exaggerating; the act of doing or representing in an excessive manner; a going beyond the bounds of truth reason, or justice; a hyperbolical representation; hyperbole; overstatement.
a.
Of or pertaining to an hyperbaton; transposed; inverted.
a.
Alt. of Hyperbolical
n.
A figure by which a grave and magnificent word is put for the proper word; amplification; hyperbole.
a.
Belonging to the hyperbola; having the nature of the hyperbola.
n.
A figure of speech in which the expression is an evident exaggeration of the meaning intended to be conveyed, or by which things are represented as much greater or less, better or worse, than they really are; a statement exaggerated fancifully, through excitement, or for effect.
v. t.
To state or represent hyperbolically.
imp. & p. p.
of Hyperbolize
n.
A curve formed by a section of a cone, when the cutting plane makes a greater angle with the base than the side of the cone makes. It is a plane curve such that the difference of the distances from any point of it to two fixed points, called foci, is equal to a given distance. See Focus. If the cutting plane be produced so as to cut the opposite cone, another curve will be formed, which is also an hyperbola. Both curves are regarded as branches of the same hyperbola. See Illust. of Conic section, and Focus.
p. pr. & vb. n.
of Hyperbolize
adv.
In the form of an hyperbola.
n.
A surface of the second order, which is cut by certain planes in hyperbolas; also, the solid, bounded in part by such a surface.
n.
Diminution; a species of hyperbole, representing a thing as being less than it really is.
a.
Relating to, containing, or of the nature of, hyperbole; exaggerating or diminishing beyond the fact; exceeding the truth; as, an hyperbolical expression.
a.
Exaggerated; excessive; hyperbolical.