###### sphere

- In mathematics, the spherical mean of a function around a point is the average of all values of that function on a sphere of given radius centered at that point. (wikipedia.org)
- Spherical geometry is the geometry of the two-dimensional surface of a sphere. (wikipedia.org)
- Concepts of spherical geometry may also be applied to the oblong sphere, though minor modifications must be implemented on certain formulas. (wikipedia.org)
- Spherical trigonometry was studied by early Greek mathematicians such as Theodosius of Bithynia, a Greek astronomer and mathematician who wrote the Sphaerics, a book on the geometry of the sphere, and Menelaus of Alexandria, who wrote a book on spherical trigonometry called Sphaerica and developed Menelaus' theorem. (wikipedia.org)
- The book of unknown arcs of a sphere written by the Islamic mathematician Al-Jayyani is considered to be the first treatise on spherical trigonometry. (wikipedia.org)
- With points defined as the points on a sphere and lines as the great circles of that sphere, a spherical geometry has the following properties: Any two lines intersect in two diametrically opposite points, called antipodal points. (wikipedia.org)
- In spherical geometry, a spherical lune is an area on a sphere bounded by two half great circles which meet at antipodal points, and is an example of a digon, {2}θ, with dihedral angle θ. (wikipedia.org)
- A spherical wedge is the volume of space bounded by two planes passing through a sphere center and the surface of the sphere. (wikipedia.org)
- The surface area of a spherical lune is 2θ R2, where R is the radius of the sphere and θ is the dihedral angle in radians between the two half great circles. (wikipedia.org)
- When this angle equals 2π radians (360°) - i.e., when the second half great circle has moved a full circle, and the lune in between covers the sphere as a spherical monogon - the area formula for the spherical lune gives 4πR2, the surface area of the sphere. (wikipedia.org)
- Beman (2008) 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. (wikipedia.org)
- Spherical cap Ungula A. ^ A distinction is sometimes drawn between the terms "sphere" and "ball", where a sphere is regarded as being merely the outer surface of a solid ball. (wikipedia.org)
- In mathematics - specifically, in geometric measure theory - spherical measure σn is the "natural" Borel measure on the n-sphere Sn. (wikipedia.org)
- Spherical measure is often normalized so that it is a probability measure on the sphere, i.e. so that σn(Sn) = 1. (wikipedia.org)
- The relationship of spherical measure to Hausdorff measure on the sphere and Lebesgue measure on the ambient space has already been discussed. (wikipedia.org)
- An approximation for the volume of a thin spherical shell is the area of the outer sphere multiplied by the thickness t of the shell: V ≈ 4 π R 2 t , {\displaystyle V\approx 4\pi R^{2}t,} when t is very small compared to R ( t ≪ R {\displaystyle t\ll R} ). A Dyson sphere encloses a fictitious spherical shell around a star, as first described by author Olaf Stapledon. (wikipedia.org)
- A spherical design, part of combinatorial design theory in mathematics, is a finite set of N points on the d-dimensional unit d-sphere Sd 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 Sd divided by the area or measure of Sd). (wikipedia.org)
- 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 Earth. (wikipedia.org)
- A spherical lens has an aplanatic point (i.e., no spherical aberration) only at a radius that equals the radius of the sphere divided by the index of refraction of the lens material. (wikipedia.org)
- In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. (wikipedia.org)
- The spherical harmonics are a complete set of orthogonal functions on the sphere, and thus may be used to represent functions defined on the surface of a sphere, just as circular functions (sines and cosines) are used to represent functions on a circle via Fourier series. (wikipedia.org)

###### geometry

- In algebraic geometry, given a reductive algebraic group G and a Borel subgroup B, a spherical variety is a G-variety with an open dense B-orbit. (wikipedia.org)
- 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 unit vectors from the origin to any two points is less than or equal to t. (wikipedia.org)
- Two practical applications of the principles of spherical geometry are navigation and astronomy. (wikipedia.org)
- Spherical geometry is not elliptic geometry, but is rather a subset of elliptic geometry. (wikipedia.org)
- Locally, the projective plane has all the properties of spherical geometry, but it has different global properties. (wikipedia.org)
- In geometry, a spherical wedge or ungula is a portion of a ball bounded by two plane semidisks and a spherical lune (termed the wedge's base). (wikipedia.org)
- A spherical plain bearing is a bearing that permits angular rotation about a central point in two orthogonal directions (usually within a specified angular limit based on the bearing geometry). (wikipedia.org)
- In geometry, a spherical shell is a generalization of an annulus to three dimensions. (wikipedia.org)
- Spherical designs can be of value in approximation theory, in statistics for experimental design (being usable to construct rotatable designs), in combinatorics, and in geometry. (wikipedia.org)
- It relies on the mathematical methods of spherical geometry and the measurements of astrometry. (wikipedia.org)

###### Mathematics

- In category theory, a branch of mathematics, a spherical category is a pivotal category (a monoidal category with traces) in which left and right traces coincide. (wikipedia.org)
- Text with sinusoidal prose poems Asaro describes the mathematics of spherical harmonics in an essay at the end of the novel. (wikipedia.org)
- In pure and applied mathematics, particularly quantum mechanics and computer graphics and their applications, a spherical basis is the basis used to express spherical tensors. (wikipedia.org)

###### harmonics

- In mathematical terms, she has entered an alternate dimension defined by the functions known as spherical harmonics. (wikipedia.org)
- Some prose in the book is written in the shape of the sinusoidal functions found in the spherical harmonics. (wikipedia.org)
- The spherical harmonics are an orthonormal set of eigenfunctions used in many areas of math and physics, including quantum mechanics and electromagnetics. (wikipedia.org)
- Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. (wikipedia.org)
- Further, spherical harmonics are basis functions for SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3). (wikipedia.org)
- Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials of degree ℓ {\displaystyle \ell } in ( x , y , z ) {\displaystyle (x,y,z)} that obey Laplace's equation. (wikipedia.org)
- Functions that satisfy Laplace's equation are often said to be harmonic, hence the name spherical harmonics. (wikipedia.org)
- A specific set of spherical harmonics, denoted Y ℓ m ( θ , φ ) {\displaystyle Y_{\ell }^{m}(\theta ,\varphi )} or Y ℓ m ( r ) {\displaystyle Y_{\ell }^{m}({\mathbf {r} })} , are called Laplace's spherical harmonics, as they were first introduced by Pierre Simon de Laplace in 1782. (wikipedia.org)
- In 3D computer graphics, spherical harmonics play a role in a wide variety of topics including indirect lighting (ambient occlusion, global illumination, precomputed radiance transfer, etc.) and modelling of 3D shapes. (wikipedia.org)
- Spherical harmonics were first investigated in connection with the Newtonian potential of Newton's law of universal gravitation in three dimensions. (wikipedia.org)
- The functions Pi are the Legendre polynomials, and they are a special case of spherical harmonics. (wikipedia.org)
- In 1867, William Thomson (Lord Kelvin) and Peter Guthrie Tait introduced the solid spherical harmonics in their Treatise on Natural Philosophy, and also first introduced the name of "spherical harmonics" for these functions. (wikipedia.org)

###### optimal spherical codes

- These common structures are layers of optimal spherical codes in most cases, a surprising fact given the significant faceting of the icosahedron and dodecahedron. (pnas.org)

###### Weisstein

- Spherical pressure vessel Ball Solid torus Bubble W., Weisstein, Eric. (wikipedia.org)

###### bearings

- Self-aligning spherical bearings were first used by James Nasmyth around 1840 to support line shaft bearings in mills and machine shops. (wikipedia.org)
- Spherical bearings can be of a hydrostatic or mechanical construction. (wikipedia.org)
- The outer surface of the inner ring and the inner surface of the outer ring are spherical (or more correctly, toroidal) and are collectively considered the raceway and they slide against each other, either with a lubricant, a maintenance-free (typically polytetrafluoroethylene or PTFE) based liner, or they incorporate a rolling element such as a race of ball-bearings, allowing lower friction. (wikipedia.org)
- Spherical bearings are used in countless applications, wherever rotational motion must be allowed to change the alignment of its rotation axis. (wikipedia.org)
- While in practice, spherical bearings are not used here, it is a simple concept that illustrates a possible application of a spherical bearing. (wikipedia.org)
- In fact, spherical bearings are used in smaller sub-components of this type of suspension, for example certain types of constant-velocity joints. (wikipedia.org)
- Thrust spherical plain bearings can accommodate a radial load F r in addition to an axial load F a ( fig. 1 ). (skf.com)
- Spherical roller bearings are characterized by their self-aligning ability and high load-carrying capacity. (skf.com)
- Invented by SKF and refined over the years, SKF spherical roller bearings provide top-of-the-line performance. (skf.com)
- Most of SKF Explorer spherical roller bearings have been upgraded to the new SKF Explorer performance class specifications, featuring a combination of high-quality steel and an improved heat treatment process. (skf.com)
- Bearings Limited F Series Spherical Plain Thrust Bearings Support Radial Loads And Single Direction Axial Loads For Increased Versatility. (globalindustrial.com)
- Bearings Limited F Series Spherical Plain Thrust Bearings have a steel-on-steel sliding surface and are treated with MoS2(Molybdenum Disulfide) to reduce friction. (globalindustrial.com)
- Bearings Limited Spherical Plain Thrust Maintenance-Free Bearings require no additional lubrication and provide long-lasting durability. (globalindustrial.com)
- Bearings Limited GE Series Spherical Plain Bearings Are Made Of A Single Fracture Design For Increased Durability. (globalindustrial.com)
- Bearings Limited GE Series Spherical Plain Maintenance-Free Bearings require no additional lubrication and provide long-lasting durability. (globalindustrial.com)
- Bearings Limited GEM Spherical Plain Bearings Carry High Capacity Axial And Radial Loads For Industrial Applications. (globalindustrial.com)
- Bearings Limited GEM Spherical Plain Bearings are made of steel for long-lasting durability. (globalindustrial.com)
- Spherical plain bearings contain oil holes inside the bearing rings and annular grooves to provide lubrication. (globalindustrial.com)
- Bearings Limited GEG Spherical Plain Bearings have a steel-on-steel sliding surface and are treated with MoS2(Molybdenum Disulfide) to reduce friction. (globalindustrial.com)
- Bearings Limited COM Series Spherical Plain Bearings Are Made Of A Steel-On-Steel Sliding Surface And Chrome-Plated Ball To Prevent Corrosion. (globalindustrial.com)
- Bearings Limited COM Series Spherical Plain Bearings can be lubricated through annular grooves and oil holes inside inner ring. (globalindustrial.com)

###### displaystyle

- For d ≤ 2 {\displaystyle d\leq 2} the critical temperature occurs at absolute zero, resulting in no phase transition for the spherical model. (wikipedia.org)
- Then, for any x ∈ Sn−1 and any A ⊆ Sn−1, θ n ( { g ∈ O ( n ) ∣ g ( x ) ∈ A } ) = σ n − 1 ( A ) . {\displaystyle \theta ^{n}(\{g\in \mathrm {O} (n)\mid g(x)\in A\})=\sigma ^{n-1}(A).} In the case that Sn is a topological group (that is, when n is 0, 1 or 3), spherical measure σn coincides with (normalized) Haar measure on Sn. (wikipedia.org)
- the remaining factor can be regarded as a function of the spherical angular coordinates θ {\displaystyle \theta } and φ {\displaystyle \varphi } only, or equivalently of the orientational unit vector r {\displaystyle {\mathbf {r} }} specified by these angles. (wikipedia.org)

###### angular

- Spherical Harmonic involves an imagined universe based on the Hilbert space described by the angular wave functions that solve the Laplace Equation. (wikipedia.org)
- The spherical basis closely relates to the description of angular momentum in quantum mechanics and spherical harmonic functions. (wikipedia.org)

###### coordinates

- While spherical polar coordinates are one orthogonal coordinate system for expressing vectors and tensors using polar and azimuthal angles and radial distance, the spherical basis are constructed from the standard basis and use complex numbers. (wikipedia.org)
- Subsequently, in his 1782 memoire, Laplace investigated these coefficients using spherical coordinates to represent the angle γ between x1 and x. (wikipedia.org)

###### shapes

- This is an important effect, because spherical shapes are much easier to produce than aspherical ones. (wikipedia.org)
- Bacteria comes in three general shapes: spherical, rod-shaped and spiral. (brighthub.com)

###### trigonometry

- However, Gerolamo Cardano noted a century later that much of its material on spherical trigonometry was taken from the twelfth-century work of the Andalusi scholar Jabir ibn Aflah. (wikipedia.org)

###### platonic

- Here, we report densest found clusters of the Platonic solids in spherical confinement, for up to N = 60 constituent polyhedral particles. (pnas.org)

###### astronomy

- The earliest reliably documented mention of the spherical Earth concept dates from around the 6th century BC when it appeared in ancient Greek philosophy but remained a matter of speculation until the 3rd century BC, when Hellenistic astronomy established the spherical shape of the Earth as a physical given. (wikipedia.org)
- The primary elements of spherical astronomy are coordinate systems and time. (wikipedia.org)

###### approximation

- It is just too complicated to try to figure out the topography of a horse and how it affects the animal's coefficient of friction , when a spherical approximation will do just as well for learning the concepts. (everything2.com)

###### angle

- It can be bisected into two lunes of half the angle, or it can be bisected by an equatorial line into two right spherical triangles. (wikipedia.org)
- If AB is a semidisk that forms a ball when completely revolved about the z-axis, revolving AB only through a given α produces a spherical wedge of the same angle α. (wikipedia.org)

###### combinatorial

- He works out completely the case of groups of type A and conjectures that the combinatorial objects (homogeneous spherical data) he introduces indeed provide a combinatorial classification of spherical subgroups. (wikipedia.org)

###### friction

- a friction reducing member located between the spherical ball and the frame of the vehicle. (google.com)

###### Lune

- A spherical lune has two planes of symmetry. (wikipedia.org)
- The visibly lighted portion of the Moon visible from the Earth is a spherical lune. (wikipedia.org)
- The spherical lune is a lighted crescent shape seen from Earth. (wikipedia.org)

###### Wedge

- Harris, J. W. and Stocker, H. "Spherical Wedge. (wikipedia.org)
- A] A spherical wedge of α = π radians (180°) is called a hemisphere, while a spherical wedge of α = 2π radians (360°) constitutes a complete ball. (wikipedia.org)

###### projective

- A projective spherical variety is a Mori dream space. (wikipedia.org)

###### 1977

- The concept of a spherical design is due to Delsarte, Goethals, and Seidel (1977). (wikipedia.org)

###### triangles

- 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. (wikipedia.org)
- Each hosohedra can be divided by an equatorial bisector into two equal spherical triangles. (wikipedia.org)

###### orthogonal

- Spherical measure has a nice relationship to Haar measure on the orthogonal group. (wikipedia.org)

###### concentric

- After the 4th century BCE this was gradually replaced by a Greek scientific cosmology of a spherical earth surrounded by multiple concentric heavens. (wikipedia.org)

###### polyhedral

- In organic chemistry, spherical aromaticity is formally used to describe an unusually stable nature of some spherical compounds such as fullerenes, polyhedral boranes. (wikipedia.org)
- Our dense packings display a wide variety of symmetries and structures, and indicate that the presence of the spherical container suppresses packing effects due to polyhedral shape. (pnas.org)

###### earliest

- Though the earliest written mention of a spherical Earth comes from ancient Greek sources, there is no account of how the sphericity of the Earth was discovered. (wikipedia.org)
- Ball joint / Heim joint Race (bearing) Self-aligning ball bearing Shaker tilting chair one of the earliest patented applications of this technology Spherical roller bearing Rolt, L.T.C. (1986) . (wikipedia.org)

###### particles

- Spherical nucleic acids are structures that are made by taking a nanoparticle template and using chemistry to arrange short strands of DNA or RNA on the surface of those particles. (news-medical.net)

###### quantum mechanics

- 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 POVMs. (wikipedia.org)

###### outer

- A spherical bearing by itself consists 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. (wikipedia.org)

###### parameters

- 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. (wikipedia.org)
- For simple designs one can sometimes calculate parameters that minimize spherical aberration. (wikipedia.org)

###### given

- that is, given positive integers n and t, there is a number N(d,t) such that for every N ≥ N(d,t) there exists a spherical t-design of N points in dimension d. (wikipedia.org)

###### vector

- It was rigorously proved by Kac and C. J. Thompson that the spherical model is a limiting case of the N-vector model. (wikipedia.org)

###### finite

- For d greater than 2, the spherical model exhibits the typical ferromagnetic behaviour, with a finite Curie temperature where ferromagnetism ceases. (wikipedia.org)

###### shape

- It is often considered to be an imperfection of telescopes and other instruments which makes their focusing less than ideal due to the spherical shape of lenses and mirrors. (wikipedia.org)
- The reason that's important is that it forces the orientation and gives the architecture both its spherical shape and also the properties that I've been mentioning. (news-medical.net)

###### rotation

- For higher ranks, one may use either the commutator, or rotation definition of a spherical tensor. (wikipedia.org)

###### model

- Spherical fusion categories give rise to a family of three-dimensional topological state sum models (a particular formulation of a topological quantum field theory), the Turaev-Viro model, or rather Turaev-Viro-Barrett-Westbury model. (wikipedia.org)
- The spherical model in statistical mechanics is a model of ferromagnetism similar to the Ising model, which was solved in 1952 by T. H. Berlin and M. Kac. (wikipedia.org)
- The critical behaviour of the spherical model was derived in the completely general circumstances that the dimension d may be a real non-integer dimension. (wikipedia.org)
- M. Kac and C. J. Thompson, Spherical model and the infinite spin dimensionality limit, Physica Norvegica, 5(3-4):163-168, 1971. (wikipedia.org)
- To modern historians, these details confirm the truth of the Phoenicians' report and even open the possibility that the Phoenicians knew about the spherical model. (wikipedia.org)

###### often

- On another level, it is about the emotional toll that war exacts on those who survive it, a theme Asaro often explores in the Saga of the Skolian Empire, as exemplified by the subplot in Spherical Harmonic involving the attempts of Selei and her husband to come to terms with his shattering experiences during the war. (wikipedia.org)
- Such a set is often called a spherical t-design to indicate the value of t, which is a fundamental parameter. (wikipedia.org)

###### Greek

- Pythagoras Early Greek philosophers alluded to a spherical Earth, though with some ambiguity. (wikipedia.org)

###### commonly

- The spherical horse in simple harmonic motion is a canonical reference to a basic lack of real world practicality commonly encountered in physics , particularly as taught in the classroom . (everything2.com)
- A spherical bear in simple harmonic motion is another commonly encountered creature. (everything2.com)

###### existence

- The existence and structure of spherical designs with d = 1 (that is, in a circle) were studied in depth by Hong (1982). (wikipedia.org)

###### surface

- The propulsion mechanism has a spherical ball disposed within the socket and adapted to frictionally engage the ground surface. (google.com)

###### properties

- Even though the sequences can be identical, the properties of spherical nucleic acids are very different from linear nucleic acids. (news-medical.net)

###### Theory

- Paolo Bravi and Guido Pezzini, The spherical systems of the wonderful reductive subgroups, J. Lie Theory 25 (2015), 105-123. (wikipedia.org)
- Michel Brion, "Introduction to actions of algebraic groups" Knop, Friedrich (2014), "Localization of spherical varieties", Algebra & Number Theory, 8: 703-728, doi:10.2140/ant.2014.8.703 Losev, Ivan (2006). (wikipedia.org)

###### made

- Because of spherical aberration, telescopes shorter than f/10 are usually made with non-spherical mirrors or with correcting lenses. (wikipedia.org)

###### concept

- The concept of a spherical Earth displaced earlier beliefs in a flat Earth: In early Mesopotamian mythology, the world was portrayed as a flat disk floating in the ocean with a hemispherical sky-dome above, and this forms the premise for early world maps like those of Anaximander and Hecataeus of Miletus. (wikipedia.org)

###### science

- Spherical Harmonic is a science fiction novel from the Saga of the Skolian Empire by Catherine Asaro. (wikipedia.org)

###### means

- Spherical means for PDEs. (wikipedia.org)
- Positive" spherical aberration means peripheral rays are bent too much. (wikipedia.org)
- Negative" spherical aberration means peripheral rays are not bent enough. (wikipedia.org)

###### sometimes

- This article will focus on the spherical (sometimes oval) shaped bacteria. (brighthub.com)

###### area

- A typical value of refractive index for crown glass is 1.5 (see list), which indicates that only about 43% of the area (67% of diameter) of a spherical lens is useful. (wikipedia.org)

###### results

- the same spherical measure results from this choice of metric. (wikipedia.org)

###### measure

- There are several ways to define spherical measure. (wikipedia.org)