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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
developed spherical trigonometry into its present form. He listed the six distinct cases of a right-angled triangle in spherical trigonometry, and in his
History_of_trigonometry
Area of geometry, about angles and lengths
tables of sine values, and used them to solve problems in trigonometry and spherical trigonometry. In the 2nd century AD, the Greco-Egyptian astronomer Ptolemy
Trigonometry
Geometry of the surface of a sphere
geodesy, spherical geometry and the metrical tools of spherical trigonometry are in many respects analogous to Euclidean plane geometry and trigonometry, but
Spherical_geometry
Shortest distance between two points on the surface of a sphere
Isoazimuthal Loxodromic navigation Meridian arc Rhumb line Spherical geometry Spherical trigonometry Versor Admiralty Manual of Navigation, Volume 1, The Stationery
Great-circle_distance
Triangle in hyperbolic geometry
the angles and sides are analogous to those of spherical trigonometry; the length scale for both spherical geometry and hyperbolic geometry can for example
Hyperbolic_triangle
Direction that Muslims face while praying salah
and allows the exact calculation (hisab) of the qibla using a spherical trigonometric formula that takes the coordinates of a location and of the Kaaba
Qibla
Persian astronomer (1201–1274)
spherical trigonometry. This followed earlier work by Greek mathematicians such as Menelaus of Alexandria, who wrote a book on spherical trigonometry
Nasir_al-Din_al-Tusi
Set of points equidistant from a center
postulate. In spherical trigonometry, angles are defined between great circles. Spherical trigonometry differs from ordinary trigonometry in many respects
Sphere
Formula for the great-circle distance between two points on a sphere
of a more general formula in spherical trigonometry, the law of haversines, that relates the sides and angles of spherical triangles. The first table of
Haversine_formula
development of trigonometry. He "innovated new trigonometric functions, created a table of cotangents, and made some formulas in spherical trigonometry." These
Mathematics in the medieval Islamic world
Mathematics_in_the_medieval_Islamic_world
19th-century survey to measure the Indian subcontinent
applied to all distances calculated from simple trigonometry: Curvature of the Earth The non-spherical nature of the curvature of the Earth Gravitational
Great_Trigonometrical_Survey
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
Scottish mathematician (1550–1617)
listing the natural logarithms of trigonometric functions. The book also has a discussion of theorems in spherical trigonometry, usually known as Napier's Rules
John_Napier
Shortest paths on a bounded deformed sphere-like quadric surface
circles (all of which are closed) and the problems reduce to ones in spherical trigonometry. However, Newton (1687) showed that the effect of the rotation of
Geodesics_on_an_ellipsoid
Relation between sides of a right triangle
the radius R approaches infinity. For practical computation in spherical trigonometry with small right triangles, cosines can be replaced with sines using
Pythagorean_theorem
Partition of a sphere's surface into polygons
p≥1 Wikimedia Commons has media related to Spherical polyhedra. Spherical geometry Spherical trigonometry Polyhedron Projective polyhedron Toroidal polyhedron
Spherical_polyhedron
Greek astronomer, geographer and mathematician (c. 190 – c. 120 BCE)
others. He developed trigonometry and constructed trigonometric tables, and he solved several problems of spherical trigonometry. His other reputed achievements
Hipparchus
Andalusian philosopher and mathematician
sphere, which is considered "the first treatise on spherical trigonometry", although spherical trigonometry in its ancient Hellenistic form was dealt with
Ibn_Mu'adh_al-Jayyani
In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for
List of trigonometric identities
List_of_trigonometric_identities
Measurement of electromagnetic radiation
cycles. Distribution is based on a fundamental identity from spherical trigonometry, the spherical law of cosines: cos ( c ) = cos ( a ) cos ( b ) +
Solar_irradiance
Indian mathematician and astronomer (1114–1185)
Siddhanta-Śiromaṇi, Bhaskara developed spherical trigonometry along with a number of other trigonometric results. (See Trigonometry section below.) Bhaskara's arithmetic
Bhāskara_II
Problem of finding unknown lengths and angles of a triangle
Spherical trigonometry on Math World. Intro to Spherical Trig. Includes discussion of The Napier circle and Napier's rules Spherical Trigonometry —
Solution_of_triangles
Overview of and topical guide to trigonometry
to trigonometry: Trigonometry – branch of mathematics that studies the relationships between the sides and the angles in triangles. Trigonometry defines
Outline_of_trigonometry
1 minus the cosine of an angle
distances between two points on the surface of the Earth using spherical trigonometry for applications in navigation. Inman also used the terms nat. versine
Versine
Angle between the zenith and the centre of the Sun's disc
derived by applying the cosine law to the zenith-pole-Sun spherical triangle, the spherical trigonometry is a relatively esoteric subject. By introducing the
Solar_zenith_angle
scientific fields, notably in mathematics and astronomy (algebra, spherical trigonometry), and in chemistry, etc. which were later also transmitted to the
Islamic world contributions to Medieval Europe
Islamic_world_contributions_to_Medieval_Europe
Generalization of Pythagorean theorem
In trigonometry, the law of cosines (also known as the cosine formula or cosine rule or Al-Kashi’s theorem) relates the lengths of the sides of a triangle
Law_of_cosines
Origin and evolution of the symbols used to write equations and formulas
The state of trigonometry advanced during the Song dynasty (960–1279), when Chinese mathematicians had greater need of spherical trigonometry in calendrical
History of mathematical notation
History_of_mathematical_notation
Fundamental result in geometry
adjacent angles (see spherical trigonometry). In the limit where the three side lengths tend to 0 {\displaystyle 0} , the spherical excess also tends to
Sum_of_angles_of_a_triangle
i}} . The usual laws for planar trigonometry of a triangle hold for this triangle. Consider the projective (spherical) triangle at the point P i {\displaystyle
Trigonometry_of_a_tetrahedron
Quaternion of norm 1 (unit quaternion)
"Representation of versors by spherical arcs". Elements of Quaternions. pp. 71–72. Hardy, A.S. (1887). "Applications to spherical trigonometry". Elements of Quaternions
Versor
60–120—Greece, Nicomachus 70–140—Greece, Menelaus of Alexandria, Spherical trigonometry 78–139—China, Zhang Heng c. 2nd century—In Greece, Ptolemy of Alexandria
Timeline_of_mathematics
Study of triangles in other spaces than the Euclidean plane
tetrahedrons and n-simplices. In spherical trigonometry, triangles on the surface of a sphere are studied. The spherical triangle identities are written
Generalized_trigonometry
Process of monitoring and controlling the movement of a craft or vehicle
These include initially meridional parts, then developments in spherical trigonometry and logarithms enabled navigators from the 1700s onwards to navigate
Navigation
Characterizes spherical triangles with fixed base and area
"Trigonométrie. Recherches de trigonométrie sphérique" [Trigonometry. Research on spherical trigonometry], Annales de Mathématiques Pures et Appliquées, 15:
Lexell's_theorem
Theorem in geometry
geometry, Legendre's theorem on spherical triangles, named after Adrien-Marie Legendre, is stated as follows: Let ABC be a spherical triangle on the unit sphere
Legendre's theorem on spherical triangles
Legendre's_theorem_on_spherical_triangles
Spherical geometry analog of a straight line
a 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.
Great_circle
or space. This includes spherical trigonometry, hyperbolic trigonometry, gyrotrigonometry, and universal hyperbolic trigonometry. Geometric algebra an alternative
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Trigonometric result for hyperbolic triangles
to the planar law of cosines from plane trigonometry, or the spherical law of cosines in spherical trigonometry. It can also be related to the relativistic
Hyperbolic_law_of_cosines
Indian mathematician-astronomer (476–550)
part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry, and spherical trigonometry. It also contains continued fractions, quadratic equations
Aryabhata
Mathematics used in Ancient China
need of spherical trigonometry in calendar science and astronomical calculations. The polymath and official Shen Kuo (1031–1095) used trigonometric functions
Chinese_mathematics
Alexandria (c. 100 AD) pioneered spherical trigonometry through Menelaus' theorem. The most complete and influential trigonometric work of antiquity is the Almagest
History_of_mathematics
Development of linear transformations forming the Lorentz group
imaginary trigonometric functions, Frank (1909) and Varićak (1910) used hyperbolic functions, Bateman and Cunningham (1909–1910) used spherical wave transformations
History of Lorentz transformations
History_of_Lorentz_transformations
Persian mathematician and astronomer (940–998)
astronomer who worked in Baghdad. He made important innovations in spherical trigonometry, and his work on arithmetic for businessmen contains the first instance
Abu_al-Wafa'_al-Buzjani
Celestial coordinate system in spherical coordinates, with the Sun as its center
this definition, the galactic poles and equator can be found from spherical trigonometry and can be precessed to other epochs; see the table. The IAU recommended
Galactic_coordinate_system
Property of all triangles on a Euclidean plane
Law of sines In trigonometry, the law of sines (sometimes called the sine formula or sine rule) is a mathematical equation relating the lengths of the
Law_of_sines
First publication of complete tables of logarithms, 1614
of tables of trigonometric functions and their Napierian logarithms. These tables greatly simplified calculations in spherical trigonometry, which are central
Mirifici Logarithmorum Canonis Descriptio
Mirifici_Logarithmorum_Canonis_Descriptio
Finnish-Swedish mathematician and astronomer (1740–1784)
was the prominent mathematician of his time who contributed to spherical trigonometry with new and interesting solutions, which he took as a basis for
Anders_Johan_Lexell
Field of knowledge
algebra. Other achievements of the Islamic period include advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system
Mathematics
Chinese astronomer and mathematician (1231–1316)
Works Bureau. Throughout his life he also did extensive work with spherical trigonometry. After Kublai Khan's death, Guo continued to be an advisor to Kublai's
Guo_Shoujing
Islamic mathematician (c. 780 – c. 850)
as-Sindhind contained tables for the trigonometric functions of sines and cosine. A related treatise on spherical trigonometry is attributed to him. Al-Khwārizmī
Al-Khwarizmi
Change of variable for integrals involving trigonometric functions
{2\,dt}{1+t^{2}}}.} The tangent of half an angle is important in spherical trigonometry and was sometimes known in the 17th century as the half tangent
Tangent half-angle substitution
Tangent_half-angle_substitution
Shape with ten sides
University Press, p. 9, ISBN 9780521098595. The elements of plane and spherical trigonometry, Society for Promoting Christian Knowledge, 1850, p. 59. Note that
Decagon
Period of cultural flourishing from 786 to 1258
al-Jayyānī is one of the several scholars to whom the invention of the spherical law of sines is attributed; he wrote "The Book of Unknown Arcs of a Sphere"
Islamic_Golden_Age
Mechanical linkage consisting of four links connected by joints in a loop
Linkage (mechanical) Pumpjack Six-bar linkage Slider-crank linkage Spherical trigonometry Straight line mechanism (Approximate straight lines are primarily
Four-bar_linkage
1791–1853 geodetic survey of Britain
data was used to calculate the sides of the triangles by using spherical trigonometry. The final results were inconclusive, for triangulation was inferior
Principal Triangulation of Great Britain
Principal_Triangulation_of_Great_Britain
Using measures of converging rays to improve fixed points for mapping
surface. If the curvature of the Earth must be allowed for, then spherical trigonometry must be used. With ℓ {\displaystyle \ell } being the distance between
Triangulation_(surveying)
Mathematical expression of circle like slices of sphere
followers on spherical geometry", Gaṇita Bhārati, 36: 53–108, arXiv:1409.4736 Todhunter, Isaac; Leathem, John Gaston (1901), Spherical Trigonometry (Revised ed
Spherical_circle
Transparent dry-erase sphere used to teach spherical geometry
interactive geometry software is typically limited to the flat plane. Spherical trigonometry is fundamental to ancient astronomy and astrology, celestial navigation
Lénárt_sphere
recognized Earth's sphericity, leading Muslim mathematicians to develop spherical trigonometry in order to further mensuration and to calculate the distance and
History_of_geodesy
5-sided star shaped polygon
Logarithms) along with rules that link the values of trigonometric functions of five parts of a right spherical triangle (two angles and three sides). The properties
Pentagramma_mirificum
14th-century astronomical instrument
astrolabes, Richard developed the rectangulus as an instrument for spherical trigonometry and to measure the angles between planets and other astronomical
Rectangulus
Flight or sailing route along the shortest path between two points on a globe's surface
the globe. The great circle path may be found using spherical trigonometry; this is the spherical version of the inverse geodetic problem. If a navigator
Great-circle_navigation
Sphere with radius one, usually centered on the origin of the space
used as a model for spherical geometry because it has constant sectional curvature of 1, which simplifies calculations. In trigonometry, circular arc length
Unit_sphere
Astronomical instrument
in Yantrarāja: Kim Plofker (February 2000). "The astrolabe and spherical trigonometry in medieval India". Journal for the History of Astronomy: 37–54
Astrolabe
Branch of astronomy about the celestial sphere
and location on Earth. It relies on the mathematical methods of spherical trigonometry and the measurements of astrometry. This is the oldest branch of
Spherical_astronomy
Emperor of Yuan China from 1271 to 1294
that period. Muslim mathematicians introduced Euclidean geometry, spherical trigonometry, and Arabic numerals in China. Kublai brought siege engineers Ismail
Kublai_Khan
Canadian historian of mathematics
book, Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry, concerns spherical trigonometry. In 2016 he received a Deborah and Franklin Haimo
Glen_Van_Brummelen
American mathematician
textbook on spherical projections expanding the Appendix in Simpson's Trigonometry book. A Treatise on Plane and Spherical Trigonometry: Including the
Enoch_Lewis_(mathematician)
Shape with three sides
sides. Relations between angles and side lengths are a major focus of trigonometry. In particular, the sine, cosine, and tangent functions relate side lengths
Triangle
Idempotent linear transformation from a vector space to itself
Banerjee (2004) for application of sums of projectors in basic spherical trigonometry. The term oblique projections is sometimes used to refer to non-orthogonal
Projection_(linear_algebra)
India in place of the chords of arc used in Greek trigonometry. Ptolemy's Almagest (a geocentric spherical Earth cosmic model) was translated at least five
Astronomy in the medieval Islamic world
Astronomy_in_the_medieval_Islamic_world
Polyhedron with four faces
Maurer. pp. 105–132. Retrieved 7 August 2018. Todhunter, I. (1886), Spherical Trigonometry: For the Use of Colleges and Schools, p. 129 ( Art. 163 ) Lévy,
Tetrahedron
Using distance measures along a shape's edges to determine position in space
Issue 5; May 1986. Spherical Trigonometry, Isaac Todhunter, MacMillan; 5th edition, 1886. A treatise on spherical trigonometry, and its application
True-range_multilateration
Family of solutions to related differential equations
of half-integer order in terms of the standard trigonometric functions, and therefore for the spherical Bessel functions. In particular, for non-negative
Bessel_function
Distance measured along the surface of the Earth
willing to accept a possible error of 0.5%, one can use formulas of spherical trigonometry on the sphere that best approximates the surface of the Earth. The
Geographical_distance
Common point(s) shared by two lines in Euclidean geometry
Academic Press. pp. 140–145. ISBN 9780120887354. Todhunter, I. (1893). Spherical Trigonometry: For the Use of Colleges and Schools (3rd ed.). Macmillan. pp. 16–25
Line–line_intersection
Coordinates comprising a distance and an angle
location on the Earth. From the 9th century onward they were using spherical trigonometry and map projection methods to determine these quantities accurately
Polar_coordinate_system
Function used in computer graphics
geometry, spherical linear interpolation, commonly abbreviated slerp, is a function which interpolates between two points on a sphere, such that spherical distance
Spherical linear interpolation
Spherical_linear_interpolation
Development of mathematics in South Asia
Calculated the solar year to 9 decimal places. Trigonometry: Developments of spherical trigonometry The trigonometric formulas: sin ( a + b ) = sin ( a
Indian_mathematics
Adaptation of the standard Mercator projection
The angles of the two graticules are related by using spherical trigonometry on the spherical triangle NM′P defined by the true meridian through the
Transverse Mercator projection
Transverse_Mercator_projection
Theorem in differential geometry
subsumed as special cases of Gauss–Bonnet. In spherical trigonometry and hyperbolic trigonometry, the area of a triangle is proportional to the amount by
Gauss–Bonnet_theorem
Indian mathematician (1813–1870)
mathematician who had specialised in spherical trigonometry, so that they could be a part of the Great Trigonometric Survey. In 1832, under the leadership
Radhanath_Sikdar
Lunisolar calendar
dynasty Shòushí calendar (授時曆; 授时历; 'season granting calendar') used spherical trigonometry to find the length of the tropical year. The calendar had a 365
Chinese_calendar
exposition of the whole system of plane and spherical trigonometry. This work is really the first in history on trigonometry as an independent branch of pure mathematics
List of Islamic scholars described as father or founder of a field
List_of_Islamic_scholars_described_as_father_or_founder_of_a_field
Relation between the side lengths and angles of a spherical triangle
In spherical trigonometry, the half-side formula relates the angles and lengths of the sides of spherical triangles, which are triangles drawn on the surface
Half-side_formula
Development of the mathematical function
of tables of trigonometric functions and their natural logarithms. These tables greatly simplified calculations in spherical trigonometry, which are central
History_of_logarithms
Azimuth angle of the Sun's position
vector pointing toward the Sun, through vector analysis rather than spherical trigonometry, as follows: ϕ s = δ , λ s = − 15 ( T G M T − 12 + E m i n / 60
Solar_azimuth_angle
Shape with eleven sides
2307/2299029, JSTOR 2299029. Loomis, Elias (1859), Elements of Plane and Spherical Trigonometry: With Their Applications to Mensuration, Surveying, and Navigation
Hendecagon
Topics referred to by the same term
episode "The Gang, a Guy and a Bakery" of USA High Angle excess, in spherical trigonometry Excess insurance, a type of liability insurance Excess, in chemistry
Excess
Special mathematical functions defined on the surface of a sphere
generalized by taking expansions in spherical harmonics rather than trigonometric functions. Moreover, analogous to how trigonometric functions can equivalently
Spherical_harmonics
Group realized geometrically by reflections across the sides of a triangle
in the center. The resulting tesselation has 4 × 6=24 spherical triangles (it is the spherical disdyakis cube). These groups are finite, which corresponds
Triangle_group
Means of calculating position
positioning system Robotic mapping Simultaneous localization and mapping Spherical trigonometry Voyage of the James Caird Waypoint Transport portal Adams, Cecil
Dead_reckoning
Menelaus" as part of what is essentially spherical trigonometry in the typical Greek form – a geometry or trigonometry of chords in a circle. In the circle
Timeline of scientific discoveries
Timeline_of_scientific_discoveries
Unit of length
and read off the distance in nautical miles. The Earth is not perfectly spherical but an oblate spheroid, so the length of a minute of latitude increases
Mile
German mathematician and astronomer (1436–1476)
motion of the earth in a letter to a friend. Much of the material on spherical trigonometry in Regiomontanus' On Triangles was taken directly[dubious – discuss]
Regiomontanus
Cartesian geographic coordinate system
numerous calculations on them, making Cartesian geometry preferable to spherical trigonometry when computing power was at a premium. In recent years, the rise
Projected_coordinate_system
Topics referred to by the same term
Solid trigonometry may refer to: solid geometry, geometry of three-dimensional Euclidean space spherical trigonometry, deals with the trigonometric functions
Solid_trigonometry
Irish–American computer programmer (1921–2006)
her studies, she took every mathematics course offered, including spherical trigonometry, differential calculus, projective geometry, partial differential
Kathleen_Antonelli
SPHERICAL TRIGONOMETRY
SPHERICAL TRIGONOMETRY
SPHERICAL TRIGONOMETRY
Female
Hungarian
Hungarian pet form of German Irma, IRMUSKA means "entire, whole."
Girl/Female
American, Anglo, Australian, British, English, German
Prosperous Friend; Rich in Friendship; Female Version of Edwin
Boy/Male
Hindu, Indian
My Eyes
Female
Norse
Old Norse name derived from the word sága, SÃGA means "the seeing one." In mythology, this is the name of a goddess of the Aesir, and possibly another name for Frigg.
Girl/Female
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Tamil, Telugu
Helping Nature; Somebody who Gives Shelter; Goddess Parvati
Girl/Female
Norse
A Valkyrie.
Girl/Female
Indian
Green
Male
French
 Old French form of German Reinhold, REINALD means "wise ruler."
Boy/Male
Australian, French, Greek, Latin
Masculine
Surname or Lastname
English (Nottinghamshire)
English (Nottinghamshire) : nickname for a thin person, from Middle English spray ‘slender branch’ (of uncertain origin).
SPHERICAL TRIGONOMETRY
SPHERICAL TRIGONOMETRY
SPHERICAL TRIGONOMETRY
SPHERICAL TRIGONOMETRY
SPHERICAL TRIGONOMETRY
a.
Having the form of a sphere; like a sphere; globular; orbicular; as, a spherical body.
a.
Alt. of Schetical
n.
Freedom from spherical aberration.
adv.
Spherically.
a.
Made convex; protuberant in a spherical form.
a.
Spherical.
a.
Exactly spherical; globular.
a.
Spherical; orbicular; orblike; circular.
a.
Having the form of a globe; spherical.
a.
Round; circular; spherical.
a.
Alt. of Spheric
a.
Round; spherical; starlike.
n.
The eye, as luminous and spherical.
a.
See Spheroidal.
a.
Having the form of a bunch of grapes; like a cluster of grapes, as a mineral presenting an aggregation of small spherical or spheroidal prominences.
v. t.
To form into roundness; to make spherical, or spheral; to perfect.
n.
A rudimentary form of crystallite, spherical in shape.
n.
The doctrine of the sphere; the science of the properties and relations of the circles, figures, and other magnitudes of a sphere, produced by planes intersecting it; spherical geometry and trigonometry.
n.
A portion of a spherical or other convex surface.
a.
Globular; spherical; orbicular.