Image RechtwKugeldreieck.svg frame right Spherical triangle Sphericaltrigonometry is a branch of spherical ... and earth surface, orbital and space navigation . History See also History of trigonometrySpherical ... pioneer in sphericaltrigonometry and wrote a treatise on the subject. ref name MacTutor ... s, e.g., sin a b , and discovered the Law of sines sine formula for sphericaltrigonometry ref name ... what some consider the first treatise on sphericaltrigonometry, circa 1060, entitled The book of unknown arcs of a sphere , ref name MacTutor Al Jayyani in which sphericaltrigonometry was brought into its ... was the first to treat trigonometry as a mathematical discipline independent from astronomy, and he further developed sphericaltrigonometry, bringing it to its present form. ref name Britannica Cite ... triangle in sphericaltrigonometry. In his On the Sector Figure , he also stated the law of sines for plane ... navigation References Reflist Isaac Todhunter SphericalTrigonometry For the Use of Colleges and Schools ... SphericalTrigonometry a more thorough list of identities, with some derivation http mathworld.wolfram.com ...&seq 5&frames 0&view 50 SphericalTrigonometry &mdash for the use of colleges and schools by I. Todhunter ... Trigonometry Category Sphericaltrigonometry br Trigonometriezh sferek ca Trigonometria esf rica ... , who wrote a book on spherical triangles called Sphaerica and developed Menelaus theorem ... it was possible in ancient mathematics to compute the magnitudes of a spherical figure, in principle ... to spherical problems was very difficult in practice. ref citation first E. S. last Kennedy title The History of Trigonometry journal 31st Yearbook publisher National Council of Teachers of Mathematics ... of understanding spherical triangles is so divine and elevated that it is not appropriate ... , the general law of sines and the solution of a spherical triangle by means of the polar triangle ... s as numbers and method of solving a spherical triangle when all sides are unknown are likely to have ... more details
Merge from List of trigonometry topics date November 2011 The following outline is provided as an overview of and topical guide to trigonometryTrigonometry &ndash branch of mathematics that studies triangle s and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions , which describe those relationships and have applicability to cyclical phenomena, such as waves. Fields that use trigonometry main Uses of trigonometry Acoustics Architecture Astronomy Biology Cartography Chemistry Civil engineering Computer graphics Cryptography Crystallography Economics Electrical engineering Electronics Game development Geodesy Mechanical engineering Medical imaging Meteorology Music theory Number theory Oceanography Optics Pharmacy Phonetics ... History of trigonometry main Trigonometry Early history of trigonometry l1 History of trigonometry Law of cosines History History of the law of cosines History of trigonometryTrigonometry in calculus ... Trigonometric substitution Trigonometry scholars Pythagoras Lists main List of trigonometry topics ... symbols See also Algebra Geometry Outline of geometry Precalculus Spherical geometry Table of mathematical symbols External links Wikibooks Trigonometry http www.clarku.edu djoyce trig Dave s short trig course http www.ping.be ping1339 gonio.htm An introduction to trigonometry http catcode.com trig index.html Trigonometry FAQ http www.pupress.princeton.edu books maor Trigonometric Delights , by Eli ... baqaqi.chi.il.us buecher mathematics trigonometry index.html Trigonometry by Alfred Monroe Kenyon ... index.php?page trig Trigonometry on PlainMath.net Trigonometry Articles from PlainMath.Net http www.mathwords.com index trig.htm Trigonometry on Mathwords.com index of trigonometry entries on Mathwords.com ... Banneker s Trigonometry Puzzle at http mathdl.maa.org convergence 1 Convergence outline footer Category Outlines Trigonometry Category Trigonometry Category Mathematics related lists Trigonometry pt Anexo ... more details
solid angle s and polytopes such as tetrahedron s and n simplex n simplices . Trigonometry In sphericaltrigonometry , triangles on the surface of a sphere are studied. The spherical triangle identities ...Ordinary trigonometry studies triangle s in the Euclidean space euclidean plane R sup 2 sup . There are a number ... other than the real numbers of euclidean geometry. Generally, trigonometry can be the study of triples ... . Hyperbolic trigonometry 1 Study of hyperbolic triangle s in hyperbolic geometry with hyperbolic ... t , sinh t . 3 Gyrotrigonometry A form of trigonometry used in the gyrovector space ... Universal hyperbolic trigonometry ref citation arxiv 0909.1377 title Universal Hyperbolic Geometry I Trigonometry first N. J. last Wildberger year 2009 . ref an algebraic approach based on Rational trigonometry . Rational trigonometry a reformulation of trigonometry in terms of spread and quadrance rather than angle and length . Trigonometry in Galois fields Trigonometry for taxicab geometry ref citation url http www.physics.orst.edu tevian taxicab taxicab.pdf title Taxicab angles and trigonometry ... 0305 4470 33 24 309 issue 24 journal Journal of Physics A mr 1768742 pages 4525 4551 title Trigonometry of spacetimes a new self dual approach to a curvature signature in dependent trigonometry volume 33 year 2000 . ref Fuzzy qualitative trigonometry ref citation url http userweb.port.ac.uk liuh Papers LiuCoghill05c SMC.pdf contribution Fuzzy Qualitative Trigonometry first1 Honghai last1 Liu ... year 2005 volume 2 pages 1291 1296 . ref Operator trigonometry ref citation url http www.ict.nsc.ru jct getfile.php?id 159 title A computational trigonometry, and related contributions by Russians ... issue 3 pages 73 83 year 1999 . ref Lattice trigonometry ref citation last Karpenkov first Oleg arxiv ... of lattice trigonometry volume 102 year 2008 . ref Trigonometry on symmetric spaces ref citation last1 Aslaksen first1 Helmer last2 Huynh first2 Hsueh Ling contribution Laws of trigonometry in symmetric ... more details
the well known theorem of Menelaus as part of what is essentially sphericaltrigonometry in the typical ... role in sphericaltrigonometry and astronomy, but by far the most influential and significant ... sphericaltrigonometry , and discovered many trigonometric results. Bhaskara II was the first to discover ..., in the sense that only then did the object of study become the Sphericaltrigonometryspherical or plane ... Menelaus of Alexandria Book 3 deals with sphericaltrigonometry and includes Menelaus s theorem ... name musa He also discovered the law of sines for sphericaltrigonometry ref name Sesiano Jacques Sesiano ... , which is considered the first treatise on sphericaltrigonometry in its modern form. ref ... from astronomy, and he developed sphericaltrigonometry into its present form. ref name Britannica ... angled triangle in sphericaltrigonometry, and in his On the Sector Figure , he stated the law ... to express greater emphasis for the need of sphericaltrigonometry in calendrical science and astronomical ... that Shen s work in the lengths of arcs of circles provided the basis for sphericaltrigonometry developed ... used sphericaltrigonometry in his calculations to improve the Chinese calendar calendar system and Chinese ...Trigonometry History of science Trigonometry is a field of mathematics first compiled by 2nd century BCE by the Greek mathematician Hipparchus . The history of trigonometry and of trigonometric functions ... of trigonometry continued in Islamic mathematics , whence it was adopted as a separate subject in the Latin West beginning in the Renaissance with Regiomontanus . The development of modern trigonometry ... 1748 . Etymology The term trigonometry derives from the Greek language Greek trigonometria ... chord in trigonometry, and the Arabs had taken this over as jiba. In the Arabic language there is also .... The Hindus had given the name jiva to the half chord in trigonometry, and the Arabs had taken this over ... last Boyer authorlink Carl Benjamin Boyer title year 1991 chapter Greek Trigonometry and Mensuration ... more details
Unreferenced date December 2009 A spherical angle is a particular dihedral angle it is the angle between two intersecting arcs on a sphere , and is measured by the angle between the planes containing the arcs which would naturally also contain the centre of the sphere . See also Sphericaltrigonometry DEFAULTSORT Spherical Angle Category Sphericaltrigonometry pt ngulo esf rico ... more details
In mathematics, hyperbolic trigonometry can mean The use of the hyperbolic function s The use of gyrotrigonometry in hyperbolic geometry Mathdab ... more details
Infobox Album See Wikipedia WikiProject Albums Name Trigonometry Type album Artist Saafir Cover Mr. No No No.jpg Released January 20, 1998 Recorded 1997 Genre Hip Hop Length 50 16 Label Wrap Ichiban Records br small WRA 8163 small Producer Saafir br Shock G br JZ br J Groove Reviews Allmusic Rating 3 5 Allmusic class album id r332059 pure url yes link Last album Boxcar Sessions br 1994 This album Trigonometry br 1998 Next album The Hit List Saafir album The Hit List br 1999 Trigonometry is the second album by Saafir , under the alias Mr. No No. It was released on January 20, 1998, on Wrap Records and featured production from Saafir and Shock G . Track listing I m Saafir The Saucee Nomad 4 08 Major Knock 4 48 Goin Home 4 53 Stay Hi 4 08 J.Z. Theme 3 42 Broad Minded 4 56 Sendin U Signals 4 06 Street Scene 4 13 Birthday Suit 3 02 Just Ridin 3 48 In a Vest 4 15 Rock the Show 4 30 Samples Street Scene Memory Lane by Minnie Riperton 1990s hiphop album stub Category 1998 albums Category Saafir albums Category Ichiban Records albums ... more details
Onesource date January 2009 Rational trigonometry is a recently introduced approach to trigonometry that eschews ... geometric configurations, notably triangles. Rational trigonometry was introduced by Prof. Norman J. Wildberger in his 2005 book Divine Proportions Rational Trigonometry to Universal Geometry . The book ... is that the field of trigonometry is being taught and applied using choices in measurements and relations ... . Quadrance and spread Instead of distance and angle , rational trigonometry uses as its fundamental ... name horizons Cite journal first Norman J. last Wildberger title A Rational Approach to Trigonometry ..., in a typical trigonometry problem if rational numbers are assigned to all quadrances and spreads ..., Wildberger refers to the traditional trigonometry as classical trigonometry. It is otherwise ... of rational trigonometry is, applications aside, a special instance of the description of geometry ... who are first learning trigonometry are often not taught about the use of linear algebra ... comments . Trigonometry over arbitrary fields Rational trigonometry makes it possible to do trigonometry ... compared to angle In rational trigonometry, spread is a fundamental concept, somewhat but not precisely ... trigonometry Wildberger states that there are five basic laws in rational trigonometry. He also ... A sub 3 sub , respectively. As in classical trigonometry, if we know three of the six elements ... rational trigonometry or derived from Heron s formula, using the condition for collinearity that the triangle ... is framed in the terms of rational trigonometry. The spread of an angle is the square of its sine ... equation, so it is more inconvenient than the analogue in classical trigonometry. Calculating quadrance ... 1 2 a 1 2 b 1 2 a 2 2 b 2 2 . , math Ease of calculation Rational trigonometry makes some problems ... such as square roots, sine, and cosine compared with classical trigonometry . Such algorithms ... 2008 , http www.cs.utep.edu vladik 2008 olg08 01.pdf Rational trigonometry computational viewpoint ... more details
refimprove date March 2012 Amongst the lay public of non mathematicians and non scientists, trigonometry is known chiefly for its application to measurement problems, yet is also often used in ways that are far more subtle, such as its place in the music theory theory of music still other uses are more technical, such as in number theory . The mathematical topics of Fourier series and Fourier transforms rely heavily on knowledge of trigonometric functions and find application in a number of areas, including statistics . Thomas Paine s statement of the uses of trigonometry In Chapter XI of The Age of Reason , the American revolutionary and Age of Enlightenment Enlightenment thinker Thomas Paine ... to the motion of the heavenly bodies, are contained chiefly in that part of science that is called trigonometry ... speaks, and the extent of its uses are unknown. Some modern uses Scientific fields that make use of trigonometry ... involve trigonometry does not mean knowledge of trigonometry is needed in order to learn ... trigonometry. For example, a professor of music may perhaps know nothing of mathematics, but would .... In some of the fields of endeavor listed above it is easy to imagine how trigonometry could be used. For example, in navigation and land surveying, the occasions for the use of trigonometry are in at least some cases simple enough that they can be described in a beginning trigonometry textbook. In the case of music theory, the application of trigonometry is related to work begun by Pythagoras, who ... Many fields make use of trigonometry in more advanced ways than can be discussed in a single article ... is 1 2. Number theory There is a hint of a connection between trigonometry and number theory. Loosely ... dots, qquad frac 31 42 , qquad frac 37 42 , qquad frac 41 42 . math Then bring in trigonometry math ... Various types of equation s can be solved using trigonometry. For example, a Recurrence relation ... reflist Category Trigonometry Category Mathematical problem solving fr Applications de la ... more details
In trigonometry , it is common to use mnemonic s to help remember trigonometric identities and the relationships between the various trigonometric functions . For example, the sine , cosine , and tangent ratios in a right triangle can be remembered by representing them as strings of letters, for instance SOH CAH TOA in English S ine O pposite H ypotenuse C osine A djacent H ypotenuse T angent O pposite A djacent One way to remember the letters is to sound them out phonetically i.e. SOH CAH TO A . ref MathWorld title SOHCAHTOA urlname SOHCAHTOA ref Another method is to expand the letters into a sentence, such as S ome O ld H ippy C aught A nother H ippy T rippin O n A cid . ref A sentence that is more appropriate for high school is, Some old horse came a hopping through our alley. cite book title Memory A Very Short Introduction first Jonathan K. last Foster publisher Oxford year 2008 isbn 0192806750 page 128 ref Mnemonic chart Unreferenced section date June 2011 Another mnemonic permits all of the basic identities to be read off quickly. Although the word part of the mnemonic used to build the chart does not hold in English, the chart itself is fairly easy to reconstruct with a little thought. Functions appear on the left, co functions on the right, a 1 goes in the middle, triangles point down, and the entire drawing looks like a radiation symbol. File Trigonometric identity mnemonic.png thumb Trigonometric identities mnemonic Reading across the central 1 in any direction gives reciprocal identities math begin array ccc 1 over sin A csc A & text or & 1 over csc A sin A 1 over tan A cot A & text or & 1 over cot A tan A 1 over sec A cos A & text or & 1 over cos A sec A end array math Reading down any triangle gives the standard identities starting at the top and going clockwise math sin 2 A cos 2 A 1 math math 1 cot 2 A csc 2 A math math tan 2 A 1 sec 2 A math Reading a function ... References reflist Category Trigonometry Category Mnemonics ... more details
on sphericaltrigonometry. The book contains formulae for right handed triangles, the general law of sines, and the solution of a spherical triangle by means of the polar triangle. ref http www ... of the material there on sphericaltrigonometry was taken from the twelfth century work of the Spanish ... Sphericaltrigonometry Category Spherical astronomy ar cv cs Sf rick ...Distinguish2 the mathematical meaning of Non Euclidean geometry Image Triangles spherical geometry .jpg ... . Spherical geometry is the geometry of the two dimension al surface of a sphere . It is an example of a geometry which is not Euclidean. Two practical applications of the principles of spherical ... lines replaced by great circles. Thus, in spherical geometry angle s are defined between great circles, resulting in a sphericaltrigonometry that differs from ordinary trigonometry in many respects for example, the sum of the interior angles of a triangle exceeds 180 degrees. Spherical ... of spherical geometry, but it has different global properties. In particular, it is orientability non orientable , or one sided. Concepts of spherical geometry may also be applied to the oblong sphere, though minor modifications must be implemented on certain formulas. Higher dimensional spherical geometries exist see elliptic geometry . History Sphericaltrigonometry was studied by early Greek mathematics Greek mathematicians such as Menelaus of Alexandria , who wrote a book on sphericaltrigonometry called Sphaerica and developed Menelaus theorem . ref MacTutor id Menelaus title Menelaus ... de Vaux, were unquestionably the inventors of plane and spherical geometry, which did not, strictly ... J. Katz Princeton University Press ref Relation to Euclid s postulates Spherical geometry obeys two ... between any two points antipodal point s such as the north and south poles on a spherical globe are counterexamples ... add up to 180 . Since spherical geometry violates the parallel postulate, there exists no such triangle ... more details
geometry hemi cube is the spherical cube. See also Spherical geometry Sphericaltrigonometry Polyhedron ...Image Trunc icosa.jpg 250px thumb right The most familiar spherical polyhedron is the soccer ball , thought of as a spherical Truncated icosahedron Applications truncated icosahedron . In mathematics , a spherical polyhedron is a tessellation tiling of the sphere in which the surface is divided or partitioned by great arc s into bounded regions called spherical polygons. Much of the theory of symmetrical polyhedron polyhedra is most conveniently derived in this way. The most familiar spherical polyhedron is the soccer ball , thought of as a spherical Truncated icosahedron Applications truncated icosahedron . Some polyhedra, such as the hosohedron hosohedra and their dual polyhedron duals the dihedron dihedra , exist as spherical polyhedra but have no flat faced analogue. In the examples below, 2, 6 is a hosohedron and 6, 2 is the dual dihedron. History The first known man made polyhedra are spherical ... B zj n Abu l Wafa wrote the first serious study of spherical polyhedra. Two hundred years ago, at the start of the 19th Century, Louis Poinsot Poinsot used spherical polyhedra to discover the four ... 64px BR Truncated octahedron 4.6.6 Image Spherical snub tetrahedron.png 64px BR Icosahedron 3.2.3.3.3.3 ... Uniform tiling 432 t012.png 64px BR Great rhombicuboctahedron 4.6.8 Image Spherical snub cube.png ... 4.6.10 File Spherical snub dodecahedron.png 64px BR Snub dodecahedron 3.2.3.3.3.5 Dihedral ... 64px BR Dihedron 2.12.12 Image hexagonal dihedron.png 64px BR Dihedron 2.6.2.6 File Spherical ... 2,6 File Spherical truncated trigonal prism.png 64px BR Hexagonal prism 2.4.6.4 File Spherical truncated hexagonal prism.png 64px BR Dodecagonal prism 4.4.12 File Spherical hexagonal antiprism.png 64px BR Hexagonal antiprism 3.2.3.2.3.6 Relation to tilings of the projective plane Spherical polyhedra ... plane, projective polyhedra correspond under 2 fold cover to spherical polyhedra that are symmetric ... more details
angle maximization problem Sphericaltrigonometry main Sphericaltrigonometry Half side formula ...Merge to Outline of trigonometry date November 2011 This is a list of trigonometry topics , by Wikipedia page. Some topics appear in more than one section below. Geometric foundations of trigonometry Altern base Angle Angular distance Angle excess Chord geometry Circle also see List of circle topics Unit circle Degree angle Grad angle Hypotenuse Opposites theorem pi pi pi Ptolemy s theorem Pythagorean theorem Radian Right angle Thales theorem Trigonometric function Triangle also see List of triangle ... Conway triangle notation Exact trigonometric constants Generalized trigonometry Gudermannian function Lissajous curve Polar sine Rational trigonometry Spread polynomials Numerical mathematics Abbe error ... polynomial Trigonometric series Trigonometry in Galois fields Trigonometry in science and engineering main Uses of trigonometry Physics Abbe sine condition Greninger chart Phasor Snell s law ... History of trigonometry Aristarchus inequality ryabha a s sine table Bhaskara I s sine approximation ... in trigonometry All Students Take Calculus See also List of exponential topics Hyperbolic function External links Wikibooks Trigonometry http www.clarku.edu djoyce trig Dave s short trig course http www.ping.be ping1339 gonio.htm An introduction to trigonometry http catcode.com trig index.html Trigonometry FAQ http www.pupress.princeton.edu books maor Trigonometric Delights , by Eli Maor , Princeton ... buecher mathematics trigonometry index.html Trigonometry by Alfred Monroe Kenyon and Louis Ingold ... trig Trigonometry on PlainMath.net Trigonometry Articles from PlainMath.Net http www.mathwords.com index trig.htm Trigonometry on Mathwords.com index of trigonometry entries on Mathwords.com http mathdl.maa.org convergence 1 ?pa content&sa viewDocument&nodeId 212&bodyId 81 Benjamin Banneker s Trigonometry ... Trigonometry Category Trigonometry ar ... more details
Spherical function can refer to Spherical harmonics Zonal spherical function mathdab Long comment to avoid being listed on short pages ... more details
File LaoHaiKugelschicht1.png thumb A spherical segment. In geometry , a spherical segment is the solid defined by cutting a sphere with a pair of Parallel geometry parallel planes . It can be thought of as a spherical cap with the top truncated, and so it corresponds to a spherical frustum . The surface of the spherical segment excluding the bases is called spherical zone . If the radius of the sphere is called R , the radius of the spherical segment bases r sub 1 sub and r sub 2 sub , and the height of the segment the distance from one parallel plane to the other called h , the volume of the spherical segment is then math V frac pi h 6 3r 1 2 3r 2 2 h 2 , math The area of the spherical zone which excludes the top and bottom bases is given by math A 2 pi R h , math See also Spherical cap Spherical wedge Spherical sector External links Commonscat Spherical segment MathWorld id SphericalSegment title Spherical segment MathWorld id Zone title Spherical zone http mathforum.org dr.math faq formulas faq.sphere.html spheresegment Summary of spherical formulas Category Spherical geometry geometry stub de Kugelschicht nn Kulebelte pl Warstwa kulista ru ... more details
In geometry , a spherical shell is a generalization of an Annulus mathematics annulus to three dimensions. A spherical shell is the region between two concentric sphere s of differing radii. See also Ball mathematics Ball Chord geometry Chord External links MathWorld id SphericalShell title Spherical shell DEFAULTSORT Spherical Shell Category Elementary geometry Category Geometric shapes ... more details
In geometry , a spherical cone is the surface of revolution obtained by rotating a sector of a circle about one of its straight edges. External links http mathworld.wolfram.com SphericalCone.html Spherical cone at MathWorld Category Geometric shapes ... more details
cleanup date March 2008 Unreferenced date January 2008 Image Unimate.jpg thumb 240px The original Unimate A spherical robot is a robot with two rotary joint s and one prismatic joint in other words, two rotary axes and one linear axis. Spherical robots have an arm which forms a spherical coordinate system . Portal Robotics DEFAULTSORT Spherical Robot Category Spherical orb robots robo stub et Sf riline robot ... more details
File Ausschnitt.PNG thumb A spherical sector. In geometry , a spherical sector is a portion of a sphere defined by a conical boundary with apex at the center of the sphere. It can be described as the union of a spherical cap and the cone formed by the center of the sphere and the base of the cap. If the radius of the sphere is denoted by r and the height of the cap by h , the volume of the spherical sector is math V frac 2 pi r 2 h 3 . math This may also be written as math V frac 2 pi r 3 3 1 cos theta , math where math &theta is half the cone geometry cone angle, i.e., the angle between the rim of the cap and the direction to the middle of the cap as seen from the sphere center. If the radius of the base of the cap is denoted by a , the surface area of the spherical sector is math A pi r 2h a . math See also Circular sector the analogous 2D object. Spherical cap Spherical segment Spherical wedge External links Commonscat Spherical sector MathWorld id SphericalSector title Spherical sector MathWorld id SphericalCone title Spherical cone http mathforum.org dr.math faq formulas faq.sphere.html spherecap Summary of spherical formulas Category Spherical geometry geometry stub de Kugelausschnitt nn Kjeglesektor pl Wycinek kuli ru ... more details
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 product of any two points is less than or equal to t . The kissing number problem may be stated as the problem of finding the maximal N for a given n for which a spherical code with parameters n , N ,1 2 exists. The Tammes problem may be stated as the problem of finding a spherical code with minimal t for given n and N . External links Mathworld urlname SphericalCode title Spherical code http www.research.att.com njas packings A library of putatively optimal spherical codes Category Coding theory Geometry stub ... more details
Spherical packing may refer to Sphere packing Spherical code disambig Short pages monitor This long comment was added to the page to prevent it being listed on Special Shortpages. It and the accompanying monitoring template were generated via Template Longcomment. Please do not remove the monitor template without removing the comment as well. ... more details
File Spherical Wedge.svg thumb right 250px A spherical wedge with radius r and angle of the wedge In geometry , a spherical wedge or ungula is a portion of a ball mathematics ball bounded by two plane semidisk s and a Lune mathematics Spherical geometry spherical lune termed the wedge s base . The angle between the radii lying within the bounding semidisks is the Dihedral angle dihedral angle of the wedge . If AB is a semidisk that forms a ball when completely revolved about the z axis, revolving AB only through a given produces a spherical wedge of the same angle . ref cite book title Geometry, plane, solid, and spherical, in six books author P. Morton publisher Baldwin and Cradock year 1830 page 180 ref Beman 2008 ref cite book title New Plane and Solid Geometry page 338 author D. W. Beman publisher BiblioBazaar, LLC year 2008 isbn 0554447010 ref remarks that a spherical wedge is to the sphere of which it is a part as the angle of the wedge is to a perigon. ref a A A spherical wedge of radian s 180 is called a Sphere hemisphere , while a spherical wedge of 2 radians 360 constitutes a complete ball. The volume of a spherical wedge can be intuitively related to the AB definition in that while the volume of a ball of radius r is given by math tfrac 4 3 pi r 3 math , the volume a spherical wedge of the same radius r is given by ref name Hart math V frac alpha 2 pi cdot frac 4 3 pi r 3 frac 2 3 alpha r 3. math Extrapolating the same principle and considering that the surface ... BiblioBazaar, LLC year 2009 isbn 1103118048 ref states that the volume of a spherical wedge ... . ref a A Hence, and through derivation of the spherical wedge volume formula, it can be concluded that, if math V s math is the volume of the sphere and math V w math is the volume of a given spherical ... S s frac alpha 2 pi math See also Spherical cap Notes A. note a A distinction is sometimes drawn between ... Beman 2008 and Hart 2008 do. References reflist Category Spherical geometry ... more details
appears to be warped onto a sphere. image Spherical aberration 2.svg thumb right Spherical aberration ... lens with spherical surfaces bottom, positive spherical aberration suffers from spherical aberration ... to the axis. It therefore does not produce a perfect focal point. Drawing is exaggerated. image spherical ... , zero centre , and positive bottom spherical aberration. Images to the left are defocused toward the inside, images on the right toward the outside. image spherical aberration slice.jpg thumb right Longitudinal sections through a focused beam with top negative, centre zero, and bottom positive spherical aberration. The lens is to the left. Spherical aberration is an optical effect observed in an optical ... has an Aplanat aplanatic point i.e., no spherical aberration only at a radius that equals the radius ... 43 of the area 67 of diameter of a spherical lens is useful. It is often considered to be an imperfection ... due to the sphere spherical shape of lenses and mirrors. This is an important effect, because spherical ... spherical shapes. So great is the difference in cost of manufacture that for most mass produced lens systems, the cost of design and multiple spherical elements is often less than simpler designs that include aspheric lenses. Positive spherical aberration means peripheral rays are bent too much. Negative spherical aberration means peripheral rays are not bent enough. The effect is proportional to the fourth ... spherical mirrors with focal ratio s shorter than f 10, light from a distant point source such as a star ... as sharply as if the aberration were not present. Because of spherical aberration, telescopes shorter than f 10 are usually made with non spherical mirrors or with correcting lenses. In lens .... For simple designs one can sometimes calculate parameters that minimize spherical aberration. For example, in a design consisting of a single lens with spherical surfaces and a given object distance o, image distance i, and refractive index n, one can minimize spherical aberration by adjusting the radii ... more details
, n 92, p. 789. ref leading Islamic mathematics Muslim mathematicians to develop sphericaltrigonometry ...Image John Gower world Vox Clamantis detail.jpg thumb right Medieval artistic representation of a spherical ... spherical Earth dates back to ancient Greek philosophy from around the 6th century BC, ref ... established the spherical shape of the earth as a physical given. The Hellenistic paradigm was gradually .... Arthur A. Clark. http www.archive.org details primerviajeentor00piga ref The concept of a spherical ... was portrayed as a flat disk floating in the ocean and surrounded by a spherical sky, ref Cite document ... evidence of a spherical Earth comes from ancient Greek sources, there is no account of how the sphericity ... Early Greek philosophers alluded to a spherical Earth, though with some ambiguity. ref name dicks1 ... arguments supporting the idea of a spherical Earth Every portion of the Earth tends ... , who lived in the Seleucia region of Mesopotamia , stated that the Earth is spherical and actually ... the distance of the sun in earth radii. Roman Empire From its Greek origins, the idea of a spherical ... AD , who suggested that the spherical shape of the Earth was probably known to seafarers around the Mediterranean ... or 8th century BC. Strabo cited various phenomena observed at sea as suggesting that the Earth was spherical ... Knowledge of the spherical shape of the Earth was received in scholarship of Late Antiquity as a matter ... of a spherical earth surrounded by the spheres of planets, vehemently supported by astronomers ... sphere , state that the Earth is spherical and that its circumference is 4,967 yojana s, which ... of the spherical shape of the Earth in Scandinavia is a 12th century Old Icelandic translation ... Ages who were aware that the earth was spherical, has been compiled by Reinhard Kr ger, professor .....jpg thumb Spherical earth with the four seasons. Illustration in 12th century book Liber Divinorum ... have thought he referred to a spherical Earth ref Isidore, Etymologiae, XIV.ii.1 3 Wesley M. Stevens ... more details
A spherical design , part of combinatorial design theory in mathematics , is a finite set of N points on the d dimensional unit hypersphere S sup d sup such that the average value of any polynomial f of degree t or less on the set equals the average value of f on the whole sphere that is, the integral of f over S sup d sup divided by the area or measure of S sup d sup . Such a set is often called a spherical t design to indicate the value of t , which is a fundamental parameter. Spherical t designs for different values of N and t can be found precomputed at http www.research.att.com njas sphdesigns. Spherical designs can be of value in approximation theory , in statistics for experimental design being usable to construct rotatable design s , in combinatorics , and in geometry . The main problem is to find examples, given d and t , that are not too large. However, such examples may be hard to come by. Spherical t designs have also recently been appropriated in quantum mechanics in the form of quantum t designs with various applications to quantum information theory , quantum computing and POVM s. The concept of a spherical design is due to Delsarte, Goethals, and Seidel 1977 . The existence and structure of spherical designs with d 1 that is, in a circle was studied in depth by Hong ... a spherical t design of N points in dimension d . However, their proof gave no idea of how big N d ... small spherical designs many of them are related to finite group action s on the sphere and are of great interest in themselves. One application of spherical designs is for whole sphere data collection. Spherical t designs meet the accurately approximate integrals by sums criteria for good pixelizations ...., Goethals, J.M., and Seidel, J.J. 1977 , Spherical codes and designs. Geometriae Dedicata vol. 6, pp. 363 388. Hong, Yiming 1982 , On spherical t designs in R sup 2 sup . European Journal of Combinatorics ... and spherical designs. Advances in Mathematics , vol. 52, pp. 213 240. Much more general than spheres ... more details
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