
	Where results make sense

Topic: Unit sphere

Related Topics

celestial sphere

Riemann sphere

Sphere Builders

Dyson sphere

armillary sphere

3 sphere

sphere of influence

Public sphere

Greater East Asia Co Prosperity Sphere

Blue Spheres

private sphere

Alexander horned sphere

In the News (Thu 24 Dec 09)

Northern Trust

Marvin Harrison

Peter Chernin

Gary Locke

Match Play

Penelope Cruz

Mickey Rourke

AR Rahman

Danny Boyle

Hugh Jackman

	Sphere - Wikipedia, the free encyclopedia
		The sphere has the smallest surface area among all surfaces enclosing a given volume and it encloses the largest volume among all closed surfaces with a given surface area.
		For this reason, the sphere appears in nature: for instance bubbles and small water drops are roughly spherical, because the surface tension locally minimizes surface area.
		The circumscribed cylinder for a given sphere has a volume which is 3/2 times the volume of the sphere, and also a surface area which is 3/2 times the surface area of the sphere.
	en.wikipedia.org /wiki/Sphere (1116 words)

	3-sphere - Wikipedia, the free encyclopedia
		In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere.
		Roughly speaking, a glome is to a sphere as a sphere is to a circle.
		Just as the set of all unit complex numbers is important in complex geometry, the set of all unit quaternions is important to the geometry of the quaternions.
	en.wikipedia.org /wiki/3-sphere (2134 words)

	PlanetMath: sphere
		A sphere is defined as the locus of the points in three dimensions that are equidistant from a particular point called the center.
		A unit sphere is a sphere with radius 1.
		This is version 19 of sphere, born on 2001-10-15, modified 2006-08-08.
	planetmath.org /encyclopedia/Sphere.html (400 words)

	Extremal Values
		To summarize, the minimum and maximum of q on the unit sphere are eigen values of the underlying matrix m, and the vectors that point to these extremal values are eigen vectors of m.
		Furthermore, q(x), on the unit sphere, is trapped between the smallest and largest entry on the main diagonal, i.e.
		As a corollary, q(x) is everywhere positive on the unit sphere iff all eigen values are positive, and the same for q(x) negative.
	www.mathreference.com /la-qf,extreme.html (652 words)

	Math Forum - Ask Dr. Math
		Date: 01/21/2002 at 16:43:23 From: Jordan Kratzer Subject: The "Unit sphere" I know that there is a unit circle and that it has to do with sine and cosine and the placement of certain points on that circle.
		Yes, the unit sphere in three-dimensional space consists of all points (x,y,z) such that the distance from the origin r satisfies the equation r^2 = x^2 + y^2 + z^2 = 1 Because the unit sphere is a surface, and not a circle, two angles are required to specify a given point on the sphere.
		For example, a common set of such angles are the "azimuthal angle" (you can think of lines of longitude on a globe) and the "polar angle," the angle between a vertical line drawn through the line x = y = 0 and the line that connects the point of interest to the origin.
	mathforum.org /library/drmath/view/54194.html (288 words)

	Coloring the rational quantum sphere and the Kochen-Specker theorem
		We review and extend recent findings of Godsil and Zaks [1], who published a constructive coloring of the rational unit sphere with the property that for any orthogonal tripod formed by rays extending from the origin of the points of the sphere, exactly one ray is red, white and fl.
		As a corollary, the rational unit sphere can be colored by two colors such that, for any arbitrary orthogonal tripod spanned by rays through its origin, one vector is colored by color #1 and the other rays are colored by color #2.
		The rational unit sphere is not closed under certain geometrical operations such as taking an orthogonal ray of the subspace spanned by two non-collinear rays (the cross product of the associated vectors).
	tph.tuwien.ac.at /~svozil/publ/2000-co.htm (2481 words)

	On coloring the rational quantum sphere
		We discuss types of colorings of the rational quantum sphere similar to the one suggested recently by Meyer [1], in particular the consequences for the Kochen-Specker theorem and for the correlation functions of entangled subsystems.
		Recently, Godsil and Zaks [2] published a constructive coloring of the rational unit sphere with the property that for any orthogonal tripod formed by rays extending from the origin of the points of the sphere, exactly one ray is red, white and fl.
		As a corollary, the rational unit sphere can be colored by two colors such that, for any arbitrary orthogonal tripod spanned by rays through its origin, one vector is colored by color #1 and the other two rays are colored by color #2.
	tph.tuwien.ac.at /~svozil/publ/coloring.htm (2007 words)

	Weighted Approximation of Functions on the Unit Sphere (ResearchIndex)
		0.1: Polynomial Interpolation On The Unit Sphere - Xu
		2 Harmonic analysis and approximation on the unit sphere Scien..
		Polynomial Interpolation on the Unit Sphere and on the Unit Ball - Xu (2002)
	citeseer.ist.psu.edu /609113.html (571 words)

	Polynomial Interpolation On The Unit Sphere - Xu (ResearchIndex)
		Polynomial Interpolation On The Unit Sphere - Xu (ResearchIndex)
		Yuan Xu, Polynomial Interpolation on the unit sphere, SIAM J. Numer Anal.
		(context) - Bojanov, Xu Polynomial interpolation on the unit ball and on the unit sp..
	citeseer.ist.psu.edu /601074.html (560 words)

	Pacific Energy Center Tour: Three-Dimensional Unit Sphere
		The unit sphere, located on the front wall of the lighting classroom, is a sphere with a one-foot radius, which defines the relationship between lumens, candela, and footcandles.
		That point source, of uniform intensity equal to one candela, produces an amount of light flowing through one square foot of the sphere's area that is defined as one lumen.
		The amount of light falling on that one square foot of the sphere is equal to one footcandle.
	www.pge.com /003_save_energy/003c_edu_train/pec/tour/light1.shtml (204 words)

	The Volume of the Hypersphere
		The sphere in n dimensions is the set of points that are 1 unit away from the origin.
		The sphere touches the cube along the axes, but watch what happens as we follow the diagonal line from the origin to the far corner of the cube.
		The surface area of the unit sphere is 3 times 4π/3, or 4π.
	www.mathreference.com /ca-int,hsp.html (492 words)

	Distributing Points on a Sphere (Site not responding. Last check: 2007-08-24)
		The first approach is to randomly distribute the required number of points on a sphere of the desired radius.
		A simple way to randomly (uniform) distribute points on sphere is called the "hypercube rejection method".
		To apply this to a unit cube at the origin, choose coordinates (x,y,z) each uniformly distributed on the interval [-1,1].
	local.wasp.uwa.edu.au /~pbourke/geometry/spherepoints (319 words)

	STROUD - Numerical Integration in M Dimensions
		SPHERE_UNIT_04_ND approximates an integral on the surface of the unit sphere in ND.
		SPHERE_UNIT_05_ND approximates an integral on the surface of the unit sphere in ND.
		SPHERE_UNIT_07_1_ND approximates an integral on the surface of the unit sphere in ND.
	www.csit.fsu.edu /~burkardt/f_src/stroud/stroud.html (2051 words)

	Distances In Bounded Regions
		Thus the average squared distance between two point in a unit disk is 1, and the average cubed distance is 2048/(525p).
		Now, what about the unit cube, or the unit 4D hyper-cube? The nice thing about the parametric distance density equations is that they immediately generalize to higher dimensions. The parametric equations for the distance density of a d-dimensional unit cube are
		The proposition that the density of v is proportional to the intersection of B with B[®v] is very general, and applies to non-convex (and even non-connected regions), not just simple regions like spheres and cubes. For example, it can be used to determine the distance distribution for points of a torus.
	www.mathpages.com /home/kmath324/kmath324.htm (974 words)

	University of New South Wales - UNSW - Distributing points on the sphere (Site not responding. Last check: 2007-08-24)
		Given a set of points on the unit sphere one quantity of interest is the volume of the convex hull of the set of points.
		The minimum energy problem is to find a set of m points on the unit sphere which minimize their Riesz energy.
		on the sphere forms a fundamental system if a polynomial of degree at most n which is zero at the points must be zero at every point on the sphere.
	www.maths.unsw.edu.au /school/articles/me100.html (1864 words)

	[No title] (Site not responding. Last check: 2007-08-24)
		Dear Dr. Rusin: The most popular (based on the sci.math newsgroup) approach for covering a unit sphere by N equal circles (caps) is based on definition of circles centers such that N electrons located in their centers will have minimum potential energy on the unit sphere.
		Consequently if that occurs, then probably we receive equal distribution of N points on the unit sphere, but we don't receive centers for the best covering circles.
		You are quite right; using an inverse-square-law force will leave N points well distributed around a sphere, but there is no guarantee that this distribution is optimal in any of several other senses, including the placement of non-overlapping disks.
	www.math.niu.edu /Papers/Rusin/known-math/96/other.force (377 words)

	spectral (Site not responding. Last check: 2007-08-24)
		It is compact: it's closed since it's defined by an equality of a continuous function, and it's bounded since we know it's the unit sphere.
		restricted to the unit sphere is the smallest eigenvalue
		Solution: This is the same function as in Example 1, and the first constraint is just the unit sphere constraint.
	www.math.columbia.edu /~loftin/ao/spectral/spectral.html (704 words)

	[No title]
		and its validity implies at once that the maximum area tetrahedron inscribed in the unit sphere is the regular one.
		Finally, I would be hard-pressed to disbelieve your original conjecture in sci.math, that the regular tetrahedron maximizes the surface area of all tetrahedra in the sphere, but frankly I would am loathe to begin considering this because the most elementary technique, while sure to succeed eventually, would invoke at least half a dozen variables.
		By the way, this example shows that as the size of the sphere increases, the location of the optimal vertex changes in a fundamental way: to embed that same triangle in a horizontal circle within a larger sphere, the circle will have to drop down from the equator towards the South Pole.
	www.math.niu.edu /~rusin/known-math/98/tetrahed_area (2365 words)

	STROUD - N-Dimensional Quadrature
		sphere_unit_3_nd.m, approximates an integral on the surface of the unit sphere in ND;
		sphere_unit_4_nd.m, approximates an integral on the surface of the unit sphere in ND;
		sphere_unit_5_nd.m, approximates an integral on the surface of the unit sphere in ND;
	www.csit.fsu.edu /~burkardt/m_src/stroud/stroud.html (1904 words)

	Electronic Journal of Differential Equations
		A method for compressing functions on the unit sphere is presented.
		This method is based on a Ramanujan set of rotations, and generates an equidistributed system of points.
		This method is flexible and easy to implement as it needs only few transformations to cover the whole unit sphere with spherical caps.
	ejde.math.txstate.edu /conf-proc/07/a2/abstr.html (81 words)

	Proving a relation is a partial order - Page 2
		is a continuous operator on the compact unit sphere.
		In this case, the compact domain happens to be the unit sphere.
		I mean, linear functions don't normally do things like map a sphere or a circle into something that spirals towards the origin, but the issue is proving it.
	www.physicsforums.com /showthread.php?p=782913 (2605 words)

	Rational approximation on the unit sphere in , John T. Anderson, Alexander J. Izzo, John Wermer
		Rational approximation on the unit sphere in C
		------, The Henkin transform and approximation on the unit sphere in $C^2,$ unpublished manuscript.
		W. Rudin, Function theory in the unit ball of $\bold C^n,$ Springer-Verlag, Berlin, 1980.
	projecteuclid.org /getRecord?id=euclid.mmj/1080837737 (367 words)

	Robotics Institute: Vanishing Point Calculation as a Statistical Inference on the Unit Sphere (Site not responding. Last check: 2007-08-24)
		For each group, the vanishing point location is estimated as the polar axis of an equatorial distribution on the unit sphere, and the statistical error of the estimate is determined.
		The sensitivity of the estimates to the number of lines in a convergent cluster is studied.
		Collins and R. Weiss, "Vanishing Point Calculation as a Statistical Inference on the Unit Sphere," International Conference on Computer Vision, December, 1990, pp.
	www.ri.cmu.edu /pubs/pub_1935.html (238 words)

	EleNTorus Software
		The parametric representation of the unit 1-sphere (circle) is
		If this circle is rotated about an axis passing through the center, a sphere results.
		This representation is constructed from the previous by an addition formula and suffix factors.
	pw1.netcom.com /~elements/ntorus.html (442 words)

	Algebras of invariant differential operators
		Let G be the identity component of the isometry group for an arbitrary curved two-point homogeneous space M.
		We consider algebras of G-invariant differential operators on bundles of unit spheres over M.
		The generators of this algebra and the corresponding relations for them are found.
	stacks.iop.org /0305-4470/36/7361 (225 words)

	Rigidity of compact submanifolds in a unit sphere, Guang Hua Chen, Xiao Rong Zou
		Rigidity of compact submanifolds in a unit sphere, Guang Hua Chen, Xiao Rong Zou
		Rigidity of compact submanifolds in a unit sphere
		[1] S. KOBAYASHI, Minimal submanifolds of a sphere with second fundamental form of constant length in Shiing-Shen Chern Selected Papers, 393-409, Springer Verlag, 1978.
	projecteuclid.org /getRecord?id=euclid.kmj/1138043355 (94 words)

	Generating unit sphere? - GameDev.Net Discussion Forums
		Home » Community » Forums » General Programming » Generating unit sphere?
		Say I want the sphere to contain 10 vectors, how do I put the vectors in an average fashion so that there won't be too remarkable weights in some places?
		Posted - 4/19/2002 7:03:29 PM Read the How to uniformly distribute points on a sphere faq.
	www.gamedev.net /community/forums/topic.asp?topic_id=91027 (115 words)

	metric tensor on unit sphere
		Mathematician's and physicist's notations of spherical coordinates seem to be opposite of each other.
		Exactly what coordinate system are you using on the surface of the sphere?
		A standard choice (I would have thought) would be spherical coordinates, with
	www.physicsforums.com /showthread.php?p=1041708#post1041708 (432 words)

	Mailgate: sci.stat.math: Re: joint normal distribution integral over the unit sphere
		Mailgate: sci.stat.math: Re: joint normal distribution integral over the unit sphere
		Subject: Re: joint normal distribution integral over the unit sphere
		Re: joint normal distribution integral o, Ray Koopman
	mailgate.supereva.it /sci/sci.stat.math/msg10376.html (185 words)

	Energy Citations Database (ECD) - Energy and Energy-Related Bibliographic Citations
		Energy Citations Database (ECD) Document #4083770 - TRANSPORT THEORY: DISCRETE ORDINATES QUADRATURE OVER THE UNIT SPHERE.
		Availability information may be found in the Availability, Publisher, Research Organization, Resource Relation and/or Author (affiliation information) fields and/or via the "Full-text Availability" link.
		TRANSPORT THEORY: DISCRETE ORDINATES QUADRATURE OVER THE UNIT SPHERE.
	www.osti.gov /energycitations/product.biblio.jsp?osti_id=4083770 (98 words)

Try your search on: Qwika (all wikis)