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Geometry of the surface of a sphere
Spherical geometry or spherics (from Ancient Greek σφαιρικά) is the geometry of the two-dimensional surface of a sphere or the n-dimensional surface of
Spherical_geometry
Set of points equidistant from a center
on the sphere. Spherical geometry is a form of elliptic geometry, which together with hyperbolic geometry makes up non-Euclidean geometry. The sphere is
Sphere
Non-Euclidean geometry
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
Elliptic_geometry
Polygon with one edge and one vertex
segment. Most definitions of a polygon in Euclidean geometry do not admit the monogon. In spherical geometry, a monogon can be constructed as a vertex on a
Monogon
Type of non-Euclidean geometry
rarely used sequence elliptic geometry (spherical geometry), parabolic geometry (Euclidean geometry), and hyperbolic geometry. In the former Soviet Union
Hyperbolic_geometry
Geometry of figures on the surface of a sphere
Spherical trigonometry is the branch of spherical geometry and trigonometry that deals with the metrical relationships between the sides and angles of
Spherical_trigonometry
Coordinates comprising a distance and two angles
In mathematics, a spherical coordinate system specifies a given point in three-dimensional space by using a distance and two angles as its three coordinates
Spherical_coordinate_system
Section of a sphere
In geometry, a spherical cap or spherical dome is a portion of a sphere or of a ball cut off by a plane. It is also a spherical segment of one base, i
Spherical_cap
Exterior angle of a triangle is greater than either of the remote interior angles
geometry is used (see Foundations of geometry) this assertion of Euclid can be proved. The exterior angle theorem is not valid in spherical geometry nor
Exterior_angle_theorem
Overview of and topical guide to geometry
Ruppeiner geometry Solid geometry Spherical geometry Symplectic geometry Synthetic geometry Systolic geometry Taxicab geometry Toric geometry Transformation
Outline_of_geometry
Shape with three sides
tetrahedron. In non-Euclidean geometries, three "straight" segments also determine a "triangle", for instance, a spherical triangle or hyperbolic triangle
Triangle
Region between parallel planes intersecting a sphere
geometry, a spherical segment is the solid defined by cutting a sphere or a ball with a pair of parallel planes. It can be thought of as a spherical cap
Spherical_segment
Three dimensional analogue of uniformization conjecture
one of three geometries (Euclidean, spherical, or hyperbolic). In three dimensions, it is not always possible to assign a single geometry to a whole topological
Geometrization_conjecture
Geometric shape; radial slice of a sphere
In geometry, a spherical wedge or ungula is a portion of a ball bounded by two plane semidisks and a spherical lune (termed the wedge's base). The angle
Spherical_wedge
Area on a sphere bounded by two semicircles joined at antipodal points
In spherical geometry, a spherical lune (or biangle) is an area on a sphere bounded by two half great circles which meet at antipodal points. It is an
Spherical_lune
Spherical triangle with three right angles
In geometry, an octant of a sphere is a spherical triangle with three right angles and three right sides. It is sometimes called a trirectangular (spherical)
Octant_of_a_sphere
Special mathematical functions defined on the surface of a sphere
scientific fields. The table of spherical harmonics contains a list of common spherical harmonics. Since the spherical harmonics form a complete set of
Spherical_harmonics
Branch of mathematics
of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by
Differential_geometry
Fundamental result in geometry
hypotenuse equals the sum of the squares of the other two sides. Spherical geometry does not satisfy several of Euclid's axioms, including the parallel
Sum_of_angles_of_a_triangle
Partition of a sphere's surface into polygons
In geometry, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded
Spherical_polyhedron
Intersection of a sphere and cone emanating from its center
In geometry, a spherical sector, also known as a spherical cone, is a portion of a ball that is bounded by a spherical cap and the cone that connects
Spherical_sector
Curve on the sphere analogous to an ellipse or hyperbola
Publication No. 112. Higgs, Peter W. (1979). "Dynamical symmetries in a spherical geometry I". Journal of Physics A: Mathematical and General. 12 (3): 309–323
Spherical_conic
Mathematical expression of circle like slices of sphere
spherical geometry, a spherical circle (often shortened to circle) is the locus of points on a sphere at constant spherical distance (the spherical radius)
Spherical_circle
Shortest distance between two points on the surface of a sphere
distance Isoazimuthal Loxodromic navigation Meridian arc Rhumb line Spherical geometry Spherical trigonometry Versor Admiralty Manual of Navigation, Volume 1
Great-circle_distance
Multiple proofs regarding Earth's approximately spherical shape
The roughly spherical shape of Earth can be empirically evidenced by many different types of observation, ranging from ground level, flight, or orbit
Empirical evidence for the spherical shape of Earth
Empirical_evidence_for_the_spherical_shape_of_Earth
Two geometries based on axioms closely related to those specifying Euclidean geometry
non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the
Non-Euclidean_geometry
Relationship between two figures of the same shape and size, or mirroring each other
similarity and not congruence in Euclidean space. However, in spherical geometry and hyperbolic geometry (where the sum of the angles of a triangle varies with
Congruence_(geometry)
Relation between sides of a right triangle
> c2. Here two cases of non-Euclidean geometry are considered—spherical geometry and hyperbolic plane geometry; in each case, as in the Euclidean case
Pythagorean_theorem
Transparent dry-erase sphere used to teach spherical geometry
writing surface for exploring spherical geometry, invented by Hungarian István Lénárt as a modern replacement for a spherical blackboard. It can be used
Lénárt_sphere
Mathematics of smooth surfaces
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most
Differential geometry of surfaces
Differential_geometry_of_surfaces
Common point(s) shared by two lines in Euclidean geometry
[further explanation needed] In spherical and elliptic geometries, every pair of lines intersects, while in hyperbolic geometry there exist infinitely many
Line–line_intersection
Ancient Greek mathematician (fl. 300 BC)
set of axioms. He also wrote works on perspective, conic sections, spherical geometry, number theory, and mathematical rigour. In addition to the Elements
Euclid
Branch of mathematics
Geometry is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is
Geometry
Relation used in geometry
affine geometries and Euclidean geometry is a special instance of this type of geometry. In some other geometries, such as hyperbolic geometry, lines
Parallel_(geometry)
Plane of reference that divides the sphere into two hemispheres
The fundamental plane in a spherical coordinate system is a plane of reference that divides the sphere into two hemispheres. The geocentric latitude of
Fundamental plane (spherical coordinates)
Fundamental_plane_(spherical_coordinates)
Basis used to express spherical tensors
and their applications, a spherical basis is the basis used to express spherical tensors.[definition needed] The spherical basis closely relates to the
Spherical_basis
Spherical tank
[citation needed] Today, spherical tanks are designed to codes such as ASME VIII, PD 5500, or EN 13445. The spherical geometry minimizes both the mechanical
Horton_sphere
Three-dimensional geometric shape
In geometry, a spherical shell (a ball shell) is a generalization of an annulus to three dimensions. It is the region of a ball between two concentric
Spherical_shell
Geometry without the parallel postulate
with elliptic geometry or spherical geometry: the notion of ordering or betweenness of points on lines, used to axiomatize absolute geometry, is inconsistent
Absolute_geometry
Geometric object
In four-dimensional geometry, the spherinder, or spherical cylinder or spherical prism, is a geometric object, defined as the Cartesian product of a 3-ball
Spherinder
Function used in computer graphics
In geometry, spherical linear interpolation, commonly abbreviated slerp, is a function which interpolates between two points on a sphere, such that spherical
Spherical linear interpolation
Spherical_linear_interpolation
Topics referred to by the same term
refer to: Vertex (graph theory), a vertex in a mathematical graph Vertex (geometry), a point where two or more curves, lines, or edges meet. Node (autonomous
Node
oriented arrangement of linear nucleic acids in a three-dimensional, spherical geometry. This novel three-dimensional architecture is responsible for many
Spherical_nucleic_acid
Property of geometry, also used to generalize the notion of "distance" in metric spaces
so the arrangement is forbidden by the tetrahedral inequality. In spherical geometry, a triangle is formed from three points and three great circle arcs
Triangle_inequality
Shape with four equal sides and angles
balls for taxicab geometry and Chebyshev distance, two forms of non-Euclidean geometry. Although spherical geometry and hyperbolic geometry both lack polygons
Square
Spherics (sometimes spelled sphaerics or sphaerica) is a term used in the history of mathematics for historical works on spherical geometry, exemplified
Spherics
Circle-packing on the surface of a sphere
Interesting Geometry. New York: Penguin Books. pp. 31. ISBN 0-14-011813-6. Bagchi, Bhaskar (1997). "How to Stay Away from Each Other in a Spherical Universe"
Tammes_problem
Spherical geometry analog of a straight line
great circle is a geodesic of the sphere, so that great circles in spherical geometry are the natural analog of straight lines in Euclidean space. For any
Great_circle
290 BC) – astronomy, spherical geometry Euclid (fl. 300 BC) – Elements, Euclidean geometry (sometimes called the "father of geometry") Apollonius of Perga
List_of_geometers
Property of segments that have the same length and the same direction
In Euclidean geometry, equipollence is a homogeneous relation between directed line segments. Two segments are said to be equipollent when they have the
Equipollence_(geometry)
Natural number
degenerate, collapsing to a line segment between the two vertices. In spherical geometry, however, non-degenerate digons can exist. Two distinct points in
2
Quadrilateral with four right angles
and not all equal, though opposite angles are equal. Other geometries, such as spherical, elliptic, and hyperbolic, have so-called rectangles with opposite
Rectangle
Pair of diametrically opposite points on a circle, sphere, or hypersphere
results in spherical geometry depend on choosing non-antipodal points, and degenerate if antipodal points are allowed; for example, a spherical triangle
Antipodal_point
Complement of latitude; polar angle
In a spherical coordinate system, a colatitude is the complementary angle of a given latitude, i.e. the difference between a right angle and the latitude
Colatitude
Subclass of manifold
the following sections. The spherical manifolds are exactly the manifolds with spherical geometry, one of the eight geometries of Thurston's geometrization
Spherical_3-manifold
4th-century BC Ancient Greek astromer, mathematician and geographer
works On the Moving Sphere and On Risings and Settings, both about spherical geometry. Autolycus was born in Pitane, a town of Aeolis within Ionia, Asia
Autolycus_of_Pitane
Model of hyperbolic geometry
analogous to the gnomonic projection of spherical geometry, in that geodesics (great circles in spherical geometry) are mapped to straight lines. This model
Beltrami–Klein_model
On inflection points of spherical curves
MR 0243466 Weiner, Joel L. (1977), "Global properties of spherical curves", Journal of Differential Geometry, 12 (3): 425–434, doi:10.4310/jdg/1214434093, MR 0514446
Tennis_ball_theorem
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
Descartes's_theorem
Triangle in hyperbolic geometry
some properties that are analogous to those of triangles in spherical or elliptic geometry: Two triangles with the same angle sum are equal in area. There
Hyperbolic_triangle
Spherical polyhedron composed of lunes
In spherical geometry, an n-gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite
Hosohedron
In the theory of analytic geometry for real three-dimensional space, the curve formed from the intersection between a sphere and a cylinder can be a circle
Sphere–cylinder_intersection
Ancient Greek astronomer and mathematician
astronomer and mathematician from Bithynia who wrote the Spherics, a treatise about spherical geometry, as well as several other books on mathematics and astronomy
Theodosius_of_Bithynia
Study of geometries as axiomatic systems
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
Foundations_of_geometry
Type of geometry
In mathematics, a Klein geometry is a type of geometry motivated by Felix Klein in his influential Erlangen program. More specifically, it is a homogeneous
Klein_geometry
Spherical triangle used in astronavigation
The navigational triangle or PZX triangle is a spherical triangle used in astronavigation to determine the observer's position on the globe. It is composed
Navigational_triangle
Topics referred to by the same term
hyperbolic planes, elliptic planes two-dimensional spherical geometry. Plane curve Inversive geometry Geometrography This disambiguation page lists mathematics
Plane geometry (disambiguation)
Plane_geometry_(disambiguation)
Fundamental object of geometry
In geometry, a point is an abstract idealization of an exact position, without size, in physical space, or its generalization to other kinds of mathematical
Point_(geometry)
Branch of differential geometry and differential topology
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds
Symplectic_geometry
Great circle on the celestial sphere that is perpendicular to the horizon
In spherical geometry, a vertical circle is a great circle on the celestial sphere that is perpendicular to the horizon. Therefore, it contains the vertical
Vertical_circle
Quadrilateral symmetric across a diagonal
quadrilateral. The fourth angle is acute in hyperbolic geometry and obtuse in spherical geometry. Every kite is an orthodiagonal quadrilateral, meaning
Kite_(geometry)
Lunisolar Chinese calendar
volumes of books, it offered an encyclopedic account of Euclidean geometry, spherical geometry and trigonometry, with extensive translations and references
Chongzhen_calendar
In differential geometry, the tangent indicatrix of a closed space curve is a curve on the unit sphere intimately related to the curvature of the original
Tangent_indicatrix
Portuguese designer and author (born 1978)
his third point, Lima hypothesizes on how the circular framing and spherical geometry of our visual field, which cause a distortion similar to a "fish-eye
Manuel_Lima
algebraic geometry. Elliptic geometry a type of non-Euclidean geometry (it violates Euclid's parallel postulate) and is based on spherical geometry. It is
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Hyperbolic geometry is a non-Euclidean geometry where the first four axioms of Euclidean geometry are kept but the fifth axiom, the parallel postulate
Constructions in hyperbolic geometry
Constructions_in_hyperbolic_geometry
Sphere with radius one, usually centered on the origin of the space
of the unit sphere. The unit sphere is often used as a model for spherical geometry because it has constant sectional curvature of 1, which simplifies
Unit_sphere
Ancient Greek spherical geometry treatise
The Spherics (Greek: τὰ σφαιρικά, tà sphairiká) is a three-volume treatise on spherical geometry written by the Hellenistic mathematician Theodosius of
Theodosius'_Spherics
Straight figure with zero width and depth
Euclidean geometry. In elliptic geometry we see a typical example of this. In the spherical representation of elliptic geometry, lines are represented by great
Line_(geometry)
Chess variants played on spherical boards
The other pieces have the moves from standard chess adapted to the spherical geometry. For example, the rook can move any number of fields orthogonally
Spherical_chess
American mathematician (1810–1853)
appointed to succeed him in 1855. He published an Elementary Treatise of Spherical Geometry and Trigonometry (New Haven, 1848), and Tables of Logarithms of Numbers
Anthony_Dumond_Stanley
Geometrical problem
In analytic geometry, a line and a sphere can intersect in three ways: no intersection at all, intersection at exactly one point, and intersection at two
Line–sphere_intersection
differential geometry Metric tensor Riemannian manifold Pseudo-Riemannian manifold Levi-Civita connection Non-Euclidean geometry Elliptic geometry Spherical geometry
List of differential geometry topics
List_of_differential_geometry_topics
Type of geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that
Projective_geometry
Measure of distance in physical space
geometry used in general relativity is an example of such a geometry. In spherical geometry, length is measured along the great circles on the sphere and
Length
Device for testing fissionable materials
characteristics of uranium-233, uranium-235, and plutonium-239 in spherical geometries surrounded by a relatively thick natural uranium neutron reflector
Flattop_(critical_assembly)
Tensor operator generalizes the notion of operators which are scalars and vectors
class of these are spherical tensor operators which apply the notion of the spherical basis and spherical harmonics. The spherical basis closely relates
Tensor_operator
Framework of distances and directions
framework. In the 19th and 20th centuries mathematicians began to examine geometries that are non-Euclidean, in which space is conceived as curved, rather
Space
Mathematical relation in spherical triangles
In spherical trigonometry, the law of cosines (or, more specifically, the law of cosines for sides) is a theorem relating the three sides and one of the
Spherical_law_of_cosines
Study of geometry using a coordinate system
In mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts
Analytic_geometry
German mathematician (1794–1874)
examined the model of geometry on a "sphere" of imaginary radius, which he called "logarithmic-spherical" (now called hyperbolic geometry). He published his
Franz_Taurinus
In geometry and coding theory, a spherical code with parameters (n,N,t) is a set of N points on the unit hypersphere in n dimensions for which the dot
Spherical_code
Metastable excited state of a nuclide
This geometry can result in quantum-mechanical states where the distribution of protons and neutrons is so much further from spherical geometry that de-excitation
Nuclear_isomer
Geometric axiom
postulate. An elliptic geometry is geometry that does not satisfy the converse of the postulate. In particular, in spherical geometry, two lines meet in exactly
Parallel_postulate
French astronomer (born 1943)
aims to develop an extremely large astronomical interferometer with spherical geometry that might theoretically show features on Earth-like worlds around
Antoine_Émile_Henry_Labeyrie
Shear layer connecting differentially rotating regions
axially flow from the fast rotating sphere to slower one. In cylindrical geometries, the thickness of both the two outer layers is E k 1 / 4 {\displaystyle
Stewartson_layer
Timurid sultan, astronomer and mathematician (1394–1449)
his work in astronomy-related mathematics, such as trigonometry and spherical geometry, as well as his general interests in the arts and intellectual activities
Ulugh_Beg
Generalized sphere of dimension n (mathematics)
n} -sphere is the setting for n {\displaystyle n} -dimensional spherical geometry. Considered extrinsically, as a hypersurface embedded in ( n + 1
N-sphere
Geometry without using coordinates
Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is geometry without the use of coordinates. It relies on the axiomatic
Synthetic_geometry
Dutch graphic artist (1898–1972)
The Hague, pointed out that Parmigianino (1503–1540) had explored spherical geometry and reflection in his 1524 Self-portrait in a Convex Mirror, depicting
M._C._Escher
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