Image John Gower world Vox Clamantis detail.jpg thumb right Medieval artistic representation of a sphericalEarth with compartments representing earth , air , and water c. 1400 . The concept of a Sphere sphericalEarth dates back to ancient Greek philosophy from around the 6th century BC, ref ... the spherical shape of the earth as a physical given. The Hellenistic paradigm was gradually ... of a sphericalEarth comes from ancient Greek sources, there is no account of how the sphericity ... philosophers alluded to a sphericalEarth, though with some ambiguity. ref name dicks1 cite book ... supporting the idea of a sphericalEarth Every portion of the Earth tends toward the center until by compression ... region of Mesopotamia , stated that the Earth is spherical and actually orbits the Sun , influenced ... in earth radii. Roman Empire From its Greek origins, the idea of a sphericalearth, along with much ... cited various phenomena observed at sea as suggesting that the Earth was spherical. He observed that elevated ... centuries AD. ref name Pingree 1978 , 533, 554f. The Greek concept of a sphericalearth surrounded ... . Christian world Image Hildegard von Bingen Werk Gottes , 12. Jh..jpg thumb Sphericalearth with the four ... to a sphericalEarth ref Isidore, Etymologiae, XIV.ii.1 3 Wesley M. Stevens, The Figure of the Earth ... accepted the terminology of the sphericalearth from Hyginus without taking the time to understand ... 1179 , depicts the sphericalearth several times in her work Liber Divinorum Operum . http brunelleschi.imss.fi.it ... de Sphaera , based on Ptolemy, in which he considers the Earth to be spherical. ref Olaf Pedersen ... Earth. Likewise, the fact that Bertold von Regensburg mid 13th century used the sphericalEarth as a sermon ... world Astronomy in medieval Islam Islamic astronomy inherited the idea of a sphericalearth ... discovery p.90 ref A circumnavigation alone does not prove that the earth is spherical ..., No. 1 1976 , pp. 106 127 107 109 ref Summary of evidence for a sphericalearth These are given in an order ... more details
Distinguish2 the mathematical meaning of Non Euclidean geometry Image Triangles spherical geometry .jpg ... triangle on the face of the earth, the sum of the angles is very nearly 180. The surface of a sphere ... . 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 spherical trigonometry 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 Spherical trigonometry was studied by early Greek mathematics Greek mathematicians such as Menelaus of Alexandria , who wrote a book on spherical ... de Vaux, were unquestionably the inventors of plane and spherical geometry, which did not, strictly ... on spherical trigonometry. 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 groups.dcs.st ... of the material there on spherical trigonometry was taken from the twelfth century work of the Spanish .... Katz Princeton University Press ref See also SIGI Great circle distance Spherical distance Spherical polyhedron Half side formula References Reflist External links commonscat Spherical geometry http www.math.uncc.edu droyster math3181 notes hyprgeom node5.html Spherical Geometry University of North ... mathworld urlname SphericalGeometry title Spherical Geometry http www.navigation spreadsheets.com ... more details
Spherical function can refer to Spherical harmonics Zonal spherical function mathdab Long comment to avoid being listed on short pages ... more details
Image RechtwKugeldreieck.svg frame right Spherical triangle Spherical trigonometry is a branch of spherical ... between the sides and the angle s. This is of great importance for calculations in astronomy and earth surface, orbital and space navigation . History See also History of trigonometry Spherical triangles ... wrote a book on spherical triangles called Sphaerica and developed Menelaus theorem . ref MacTutor ... in ancient mathematics to compute the magnitudes of a spherical figure, in principle, by use of the table of Chord geometry chords and Menelaus theorem, the application of the theorem to spherical ... spherical triangles is so divine and elevated that it is not appropriate to extend its ... in spherical trigonometry and wrote a treatise on the subject. ref name MacTutor Khwarizmi MacTutor ... a b , and discovered the Law of sines sine formula for spherical trigonometry ref name Sesiano 157 Jacques ... the first treatise on spherical trigonometry, circa 1060, entitled The book of unknown arcs of a sphere , ref name MacTutor Al Jayyani in which spherical trigonometry was brought into its modern ... law of sines and the solution of a spherical triangle by means of the polar triangle . This treatise ... and method of solving a spherical triangle when all sides are unknown are likely to have influenced ... spherical trigonometry, bringing it to its present form. ref name Britannica Cite web title trigonometry ... accessdate 2008 07 21 ref He listed the six distinct cases of a right angled triangle in spherical trigonometry. In his On the Sector Figure , he also stated the law of sines for plane and spherical triangles, and discovered the law of tangents for spherical triangles. ref name Berggren 518 ... coincide with the center of the sphere. For example, simplifying the shape of the Earth the geoid to a sphere ... of a geodesic . An area on the sphere bounded by arcs of great circle s is called a spherical polygon . Note that, unlike the case on a plane, spherical digon biangles two sided analogs ... 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 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 ... 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 pl Warstwa kulista ru ... 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
For the mathematical functions Spherical harmonics Infobox book name Spherical Harmonic title orig translator image include the file and the image size image caption author Catherine Asaro illustrator cover artist country language series Saga of the Skolian Empire subject genre Science Fiction publisher Tor Books pub date December 2001 english pub date media type Hardcover pages 512 isbn ISBN 978 0 312 89063 6 br ISBN 0 312 89063 X oclc dewey congress preceded by The Quantum Rose followed by The Moon s Shadow Spherical Harmonic is a science fiction novel from the Saga of the Skolian Empire by Catherine Asaro . It tells the story of Dyhianna Selei Dehya , the Ruby Pharaoh of the Skolian Empire ... title Spherical Harmonic publisher Powells.com accessdate 15 December 2009 ref Plot summary Spherical ... ruler of modern Skolia. Spherical Harmonic takes place following the Radiance War, a conflict fought .... Just prior to the opening scene of Spherical Harmonic , Dyhianna Selei escapes a Eubian ... by the functions known as spherical harmonics. As the book opens, she is coalescing on a moon ... prose in the book is written in the shape of the sinusoidal functions found in the spherical harmonics ... ref Asaro describes the mathematics of spherical harmonics in an essay at the end of the novel. ref ... husband Eldrinson Althor Valdoria Eldrinson . Her plan to go to Earth and free members of the Ruby ... of the war. Context In one sense, Spherical Harmonic is a space adventure about the recovery ... , as exemplified by the subplot in Spherical Harmonic involving the attempts of Selei and her husband ... ref Library Journal 2001 ref ref http www.amazon.com Spherical Harmonic Saga Skolian Empire dp 0812568826 ... known for the use of Catherine Asaro Mathematical fiction mathematics in her novels. Spherical Harmonic ... that solve the Laplace Equation . The spherical harmonics are an orthonormal set of eigenfunctions ... there are fictional extrapolations of the mathematical theory. The novel Spherical Harmonic overlaps ... more details
Spherical astronomy or positional astronomy is the branch of astronomy that is used to determine the location of objects on the celestial sphere , as seen at a particular date, time, and location on the Earth . It relies on the mathematical methods of spherical geometry and the measurements of astrometry . This is the oldest branch of astronomy and dates back to ancient history antiquity . Observations of celestial objects have been, and continue to be, important for religious and astrology astrological purposes, as well as for timekeeping and navigation . The science of actually measuring positions of celestial objects in the sky is known as astrometry. The primary elements of spherical astronomy are coordinate systems and time. The coordinates of objects on the sky are listed using the equatorial coordinate system , which is based on the projection of the Earth s equator onto the celestial sphere. The position of an object in this system is given in terms of right ascension and declination . The latitude and local time can then be used to derive the position of the object in the horizontal coordinate system , consisting of the altitude and azimuth . The coordinates of celestial objects such as stars and galaxies are tabulated in a star catalog , which gives the position for a particular ... to change slightly over time. The effects of these changes in the movement of the Earth are compensated ... contains the orbit of a planet, usually in reference to Earth Elongation astronomy Elongation refers ... point inferior and superior planets Superior planets have a larger orbit than the earth while the inferior and superior planets inferior planets Mercury and Venus orbit the sun inside the Earth s orbit ... and C, from the U.S. Naval Observatory. References Robin M. Green, Spherical Astronomy , 1985 ... M. Green, Textbook on Spherical Astronomy , 1977 in literature 1977 , Cambridge University Press ... Sidereal time Solstice Category Spherical astronomy Category Astrometry Category Astrological aspects ... 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
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 Spherical trigonometry DEFAULTSORT Spherical Angle Category Spherical trigonometry pt ngulo esf rico ... 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 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
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
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 ..., P.D., and Zaslavsky, Thomas 1984 , Averaging sets A generalization of mean values and spherical ... 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 is to the volume ... . 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
A spherical bearing is a Bearing mechanical bearing that permits angular rotation about a central point in two orthogonal directions usually within a specified angular limit based on the bearing geometry . Typically these bearings support a rotating shaft in the bore of the inner ring that must move not only rotationally, but also at an angle. Construction Construction of spherical bearings can be hydrostatic or strictly mechanical. A spherical bearing by itself can consist of an outer ring and an inner ring and a locking feature that makes the inner ring captive within the outer ring in the axial direction only. The outer surface of the inner ring and the inner surface of the outer ring are collectively considered the raceway and they slide against each other, either with a lubricant or a maintenance free polytetrafluoroethylene PTFE based liner. Some spherical bearings incorporate a rolling element such as a race of ball bearings, allowing lower friction. History The Swede Sven Wingquist 1876 1953 invented the spherical bearing in 1907. He founded a global company, SKF AB Svenska Kullagerfabriken , still the world s leading producer of industrial bearings. Application Spherical bearings are used in countless applications, wherever rotational motion must be allowed to change the alignment of its rotation axis. A prime example is a tie rod on a vehicle suspension. The mechanics of the suspension allow the axle to move up and down, but the linkages are designed to control that motion in one direction only and they must allow motion in the other directions. Spherical bearings have been used in car suspension s, driveshafts, heavy machinery, sewing machines, and many other applications. See also Ball joint Heim joint Race bearing Self aligning ball bearing External links http www.bearings.machinedesign.com guiEdits Content BDE 6 4 bdemech6 31.aspx Spherical bearing 101 Category Bearings tech stub pl o ysko bary kowe ... more details
In differential geometry , the spherical image of a Curve Lengths of curves unit speed curve is given by taking the curve s tangent vector s as points, all of which must lie on the unit sphere . The movement of the spherical image describes the changes in the original curve s direction ref O Neill, B. Elementary Differential Geometry , 1961, pg 71. ref If math alpha math is a unit speed curve, that is math alpha prime 1 math , and math T math is the Frenet frame unit tangent vector field along math alpha math , then the curve math sigma T math is the spherical image of math alpha math . All points of math sigma math must lie on the unit sphere because math sigma T 1 math . References references Category Differential geometry differential geometry stub ... more details
File SphericalCow2.gif thumb 200px right A spherical cow Hey Diddle Diddle jumps over the moon . Spherical cow is a metaphor for highly simplified scientific model abstract models of complex real life phenomena. ref name Robin Shelton and J. Allie Cliffe http lheawww.gsfc.nasa.gov docs xray research snrs spherical cow.html Spherical Cows Bot generated title ref Jokes The phrase comes from a joke about Theoretical physics theoretical physicists Milk production at a dairy farm was low, so the farmer wrote to the local university, asking for help from academia. A multidisciplinary team of professors was assembled, headed by a theoretical physicist, and two weeks of intensive on site investigation took place. The scholars then returned to the university, notebooks crammed with data, where the task of writing the report was left to the team leader. Shortly thereafter the physicist returned to the farm, saying to the farmer I have the solution, but it only works in the case of spherical cows in a vacuum. It is told in many variants. ref cite web url http www.physics.csbsju.edu stats WAPP2 cow.html title Spherical Cow A Simple Model accessdate 2007 02 19 author Kirkman, T. W. year 1996 work http www.physics.csbsju.edu stats Statistics to Use ref In Russian, it is called a spherical horse in a vacuum, ref cite web url http ru.wiktionary.org wiki title accessdate 2010 07 15 year 2010 ref ref cite web url http lurkmore.ru title accessdate 2010 10 15 year 2010 ref from a joke about a physicist who said he could predict the winner of any horse race to multiple decimal points provided it was a spherical horse moving through a vacuum. ref name HefleyHefley2008 cite book author1 ... In the TV show The Big Bang Theory a variation of the joke is told with spherical chickens. The point ... snrs spherical cow.html NASA Exploration of the Universe Division Supernova models as spherical ... more details
Infobox VG title image File SphericalWorldsAmigaBoxFront.jpg 250px Spherical Worlds Box caption developer 4Matted Future Creations publisher distributor designer series engine version released 1996 genre Shoot em up modes Single player ratings platforms Amiga media requirements input Spherical Worlds is a shoot em up computer game for the Amiga , released by the Austrians Austrian company Rockstar Vienna neo Software Produktions GmbH in 1996 ref cite web url http hol.abime.net 2058 title Spherical Worlds on Abime.net ref ref cite web url http www.lemonamiga.com ?mainurl http 3A www.lemonamiga.com games details.php 3Fid 3D987 title Spherical Worlds on Lemon Amiga ref . The game is essentially an top view shoot em up where you control a spherical battle droid shooting your way through the enemy battleship. You can equip your droid with weapon and tool enhancements, such as additional weapons, power up s, extra lives, Guided missile guided missiles through a special weapon shop shown between game levels. Occasionally between the levels you can also play a bonus 3D pre rendered tunnel level where you can collect bonus items that bring you additional credits that are converted into money for power ups. On Advanced Graphics Architecture AGA Amigas the game had some enhancements such as more enemies or bullets on the screen. References reflist External links http www.infobyte.hr sphericalworlds Spherical Worlds web site Category 1996 video games Category Amiga games Category CD32 games Category Multidirectional shooters scroll shooter videogame stub ... more details
Image Spherical mean.png right thumb The spherical mean of a function math u math shown in red is the average of the values math u y math top, in blue with math y math on a sphere of given radius around a given point bottom, in blue . In mathematics , the spherical mean of a function mathematics function around a point is the average of all values of that function on a sphere of given radius centered at that point. Definition Consider an open set U in the Euclidean space R sup n sup and a continuous function u defined on U with real number real or complex number complex values. Let x be a point in U and r > 0 be such that the closed set closed ball mathematics ball B x , r of center x and radius r is contained in U . The spherical mean over the sphere of radius r centered at x is defined as math frac 1 omega n 1 r int limits partial B x, r u y , mathrm d S y math where &part B x , r is the n sphere n &minus 1 sphere forming the boundary topology boundary of B x , r , d S denotes integration with respect to spherical measure and &omega sub n &minus 1 sub r is the surface area of this n &minus 1 sphere. Equivalently, the spherical mean is given by math frac 1 omega n 1 int limits y 1 u x ry , mathrm d S y math where &omega sub n &minus 1 sub is the area of the n &minus 1 sphere of radius 1. The spherical mean is often denoted as math int limits partial B x, r , u y , mathrm d S y . math Properties and uses From the continuity of math u math it follows that the function math r to int limits partial B x, r , u y ,dS y math is continuous, and its Limit of a function limit as math r to 0 math is math u x . math Spherical means are used in finding the solution of the wave equation math u tt c 2 Delta u math for math t 0 math with prescribed boundary ... Shalimova, I. A. title Spherical means for PDEs publisher VSP date 1997 pages isbn 9067642118 External links planetmath reference id 5568 title Spherical mean Category Partial differential equations ... more details
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 ... polyhedra correspond under 2 fold cover to spherical polyhedra that are symmetric under reflection ... is the spherical cube. See also Spherical geometry Spherical trigonometry Polyhedron Projective polyhedron ... more details
Merge from Dome mathematics date October 2011 File Spherical Cap.svg thumb 200px The spherical cap is the purple section. In geometry , a spherical cap is a portion of a sphere cut off by a Plane mathematics plane . If the plane passes through the center of the sphere, so that the height of the cap is equal to the radius of the sphere, the spherical cap is called a Sphere Hemisphere hemisphere . If the radius of the sphere is math r math , the radius of the base of the cap is math a math , and the height of the cap is math h math , then the volume of the spherical cap is math V frac pi h 6 3a 2 h 2 , math and the curved surface area of the spherical cap is math A 2 pi r h. math The parameters math a math , math h math and math r math are not independent math r 2 r h 2 a 2 r 2 h 2 2rh a 2, math math r frac a 2 h 2 2h math . Substituting this into the area formula gives math A 2 pi frac a 2 h 2 2h h pi a 2 h 2 . math Note also that in the upper hemisphere of the diagram, math scriptstyle h r sqrt r 2 a 2 math , and in the lower hemisphere math scriptstyle h r sqrt r 2 a 2 math hence in either hemisphere math scriptstyle a sqrt h 2r h math and so an alternative expression for the volume is math V frac pi h 2 3 3r h math . Hyperspherical cap Generally, the math n math dimensional volume of a hyperspherical cap of height math h math and radius math r math in math n math dimensional Euclidean space is given by math V frac pi frac n 1 2 , r n , Gamma left frac n 1 2 right int limits ... mathematics Solid angle contains formula for n sphere caps Spherical segment Spherical sector Spherical ... abstract ?doi ajms.2011.66.70 . External links Commonscat Spherical caps MathWorld id SphericalCap title Spherical cap , derivation and some additional formulas http formularium.org ?go 81 Online calculator for spherical cap volume and area http mathforum.org dr.math faq formulas faq.sphere.html spherecap Summary of spherical formulas Category Spheres Elementary geometry stub ca Casquet esf ric ... more details
Image Harmoniki.png right thumb 300px Visual representations of the first few spherical harmonics. Red ... the function is negative. In mathematics , spherical harmonics are the angular portion of a set of solutions to Laplace s equation . Represented in a system of Spherical coordinate system spherical coordinates , Laplace s spherical harmonics math Y ell m math are a specific set of spherical harmonics ... account of various approaches to spherical harmonics in three dimensions can be found in Chapter IV of harvnb MacRobert 1967 . The term Laplace spherical harmonics is in common use see harvnb Courant Hilbert 1962 and harvnb Meijer Bauer 2004 . ref Spherical harmonics are important in many theoretical ... computer graphics , spherical harmonics play a special role in a wide variety of topics including ... radiance transfer , etc. and recognition of 3D shapes. History Spherical harmonics were first investigated ... are the Legendre polynomials , and they are a special case of spherical harmonics. Subsequently, in his 1782 memoire, Laplace investigated these coefficients using spherical coordinates to represent ... spherical harmonics in their Treatise on Natural Philosophy , and also first introduced the name of spherical harmonics for these functions. The solid harmonics were homogeneous function homogeneous ... frac partial 2 u partial z 2 0. math By examining Laplace s equation in spherical coordinates, Thomson and Tait recovered Laplace s spherical harmonics. The term Laplace s coefficients was employed by William ... others reserved this designation for the zonal spherical harmonics that had properly been introduced ... of vibration in a vibrating string string , the spherical harmonics represent the fundamental modes ... be generalized by taking expansions in spherical harmonics rather than trigonometric functions. This was a boon for problems possessing spherical symmetry , such as those of celestial mechanics originally studied by Laplace and Legendre. The prevalence of spherical harmonics already in physics set ... more details
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