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Topic: Spherical geometry


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  Spherical geometry - Wikipedia, the free encyclopedia
Spherical geometry is the geometry of the two-dimensional surface of a sphere.
Thus, in spherical geometry angles 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 geometry is the simplest model of elliptic geometry, in which a line has no parallels through a given point.
en.wikipedia.org /wiki/Spherical_geometry   (268 words)

  
 Spherical geometry - Wikipedia, the free encyclopedia   (Site not responding. Last check: 2007-11-07)
Thus, in spherical geometry angles 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 can exceed 180 degrees).
Spherical geometry has important practical uses in celestial navigation and astronomy.
An important related geometry related to that modeled by the sphere is called the projective plane; it is obtained by identifying antipodes (pairs of opposite points) on the sphere.
www.encyclopedia-online.info /Spherical_geometry   (255 words)

  
 Spherical geometry   (Site not responding. Last check: 2007-11-07)
Spherical geometry is the geometry of the two- dimensional surface of a sphere.
Spherical geometry is the simplest model of elliptic geometry,in which a line has no parallels through a given point.
An important related geometry related to that modeled by the sphere is called the projective plane ; it is obtained by identifying antipodes (pairs of opposite points) on the sphere.Locally, the projective plane has all the properties of spherical geometry, but it has different global properties.
www.therfcc.org /spherical-geometry-33432.html   (233 words)

  
 Spherical geometry   (Site not responding. Last check: 2007-11-07)
Spherical geometry is the geometry of the two- dimension al surface of a sphere.
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 can exceed 180 degrees).
An important related geometry related to that modeled by the sphere is called the projective plane ; it is obtained by identifying antipodes (pairs of opposite points) on the sphere.
www.serebella.com /encyclopedia/article-Spherical_geometry.html   (444 words)

  
 Spherical/Elliptical Geometry   (Site not responding. Last check: 2007-11-07)
Spherical geometry replaces the standard flat plan with the plane being the surface of a sphere.
The end result is that in spherical geometry, lines always intersect in exactly two points, whereas in elliptical geometry, lines always intersect in one point.
This type of geometry is especially useful in describing the Earth's surface.
members.tripod.com /noneuclidean/ellipse.html   (237 words)

  
 Riemannian geometry - Wikipedia, the free encyclopedia
In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i.e.
Riemannian geometry was first put forward in generality by Bernhard Riemann in the nineteenth century.
It deals with a broad range of geometries whose metric properties vary from point to point, as well as two standard types of Non-Euclidean geometry, spherical geometry and hyperbolic geometry, as well as Euclidean geometry itself.
en.wikipedia.org /wiki/Riemannian_geometry   (831 words)

  
 Spherical Geometry and Trigonometry
Spherical geometry and trigonometry used to be important topics in a technical education because they were essential for navigation.
Thus the area of a spherical triangle on a unit sphere is equal to the spherical exess, which is the sum of vertex angles in excess of π radians.
A side of a spherical triangle is the intersection of a plane passing through the center of a sphere with the surface of the sphere.
www2.sjsu.edu /faculty/watkins/sphere.htm   (1395 words)

  
 [No title]   (Site not responding. Last check: 2007-11-07)
Geometers also study non-planar triangles in noneuclidean geometries, such as spherical triangles in spherical geometry and hyperbolic triangles in hyperbolic geometry.
For spherical and hyperbolic spatial geometries, the probability of detection of the topology by direct observation depends on the spatial curvature.
Because spherical geometry is rather different from ordinary Euclidean geometry, the equations for distance take on a different form.
www.worldhistory.com /wiki/S/Spherical-geometry.htm   (592 words)

  
 Mathematical mysteries: Strange Geometries
Geometry in which the fifth postulate is assumed is known as "Euclidean" or "flat" geometry.
It is the geometry of the surface of a sphere, known as "spherical geometry".
In spherical geometry, the curvature is positive, in hyperbolic geometry, it is negative.
plus.maths.org /issue18/xfile/index.html   (1318 words)

  
 Sphere - Wikipedia, the free encyclopedia
In three-dimensional Euclidean geometry, a sphere is the set of points in R
For this reason, the sphere appears in nature: for instance bubbles and small water drops are roughly spherical, because the surface tension minimizes surface area.
This terminology is also used for astronomical bodies such as the planet Earth, even though it is neither spherical nor even spheroidal (see geoid).
en.wikipedia.org /wiki/Sphere   (1116 words)

  
 Spherical trigonometry: Facts and details from Encyclopedia Topic   (Site not responding. Last check: 2007-11-07)
Spherical trigonometry is a part of spherical geometry spherical geometry quick summary:
A triangle is one of the basic shapes of geometry: a two-dimensional figure with three vertices and three sides which are straight line segments....
The great-circle distance is the shortest distance between any two points on the surface of a sphere measured along a path on the surface of the sphere...
www.absoluteastronomy.com /encyclopedia/s/sp/spherical_trigonometry.htm   (1395 words)

  
 elliptical geometry
The original form of elliptical geometry, known as spherical geometry or Riemannian geometry, was pioneered by Bernard Riemann and Ludwig Schläfli and treats lines as great circles on the surface of a sphere.
Another odd property of spherical geometry is that the sum of the angles of a triangle is greater then 180°.
It was Felix Klein who first saw clearly how to rid spherical geometry of its one blemish: the fact that two lines have not one but two common points.
www.daviddarling.info /encyclopedia/E/elliptical_geometry.html   (362 words)

  
 Amazonite Info   (Site not responding. Last check: 2007-11-07)
The shortest distance between two points on a spherical surface, measured on the surface, is the distance along the great circle through those points.
The n sphere ofunit radius centred at the origin is denoted S n and is often referred to as the n sphere.
Spherical geometry and spherical trigonometry are methods of determining magnitudes and figures on a spherical surface.
www.amazonite.jewelry-boxes-etc.com   (1254 words)

  
 A Unified Algebraic Framework for Classical Geometry
Spherical trigonometry was thoroughly developed in modern form by Euler in his 1782 paper [E1782].
Spherical geometry in n-dimensions was first studied by Schlafli in his 1852 treatise, which was published posthumously in [S1901].
Because the three geometries are obtained by interpreting null vectors of the same Minkowski space differently, natural correspondences exist among geometric entities and constraints of these geometries.
modelingnts.la.asu.edu /html/UAFCG.html   (2056 words)

  
 Spherical Trigonometry   (Site not responding. Last check: 2007-11-07)
One of the primary concerns in astronomy throughout history was the positioning of the heavenly bodies, for which spherical trigonometry was required.
The spherical triangle in the diagram above is the same as the trihedral, except that the sides a, b and c are great circle paths on the surface of the sphere centred at V rather than straight lines (as in the trihedral).
To solve a spherical triangle problem, known values are substituted into the equations, which are then solved for the remaining unknown values.
www.hps.cam.ac.uk /starry/sphertrig.html   (567 words)

  
 NonEuclid: Non-Euclidean Geometery   (Site not responding. Last check: 2007-11-07)
Euclidean Geometry was of great practical value to the ancient Greeks as they used it (and we still use it today) to design buildings and survey land.
One of the most useful non-Euclidean geometries is Spherical Geometry which describes the surface of a sphere.
Spherical Geometry is used by pilots and ship captains as they navigate around the world.
www.cs.unm.edu /~joel/NonEuclid/noneuclidean.html   (333 words)

  
 Spherical geometry | TutorGig.co.uk Encyclopedia   (Site not responding. Last check: 2007-11-07)
DVD See all 6 results in Spherical geometry..
Elements of descriptive geometry; with its applications to spherical p..
Elements of descriptive geometry, with their application to spherical..
www.tutorgig.co.uk /encyclopedia/getdefn.jsp?keywords=Spherical_geometry   (431 words)

  
 [No title]   (Site not responding. Last check: 2007-11-07)
Spherical geometry (which is clearly not Euclidean) was in existence and studied by at least the ancient Babylonians, Indians, and Greeks more than 2,000 years ago.
Spherical geometry was of importance for astronomical observations and astrological calculations.
In this chapter we will introduce the geometry of the hyperbolic plane as the intrinsic geometry of a particular surface in 3-space, in much the same way that we introduced spherical geometry by looking at the intrinsic geometry of the sphere in 3-space.
www.mathsci.appstate.edu /~sjg/class/3610/hen.html   (623 words)

  
 Hugues Hoppe's home page
The geometry clipmap introduced in Losasso and Hoppe 2004 is a new level-of-detail structure for rendering terrains.
Geometry images have the potential to simplify the rendering pipeline, since they eliminate the "gather" operations associated with vertex indices and texture coordinates.
Neither the topology, the presence of boundaries, nor the geometry of M are assumed to be known in advance — all are inferred automatically from the data.
research.microsoft.com /~hoppe   (8986 words)

  
 Trigonometry
Since the proof of the spherical Law of Tangents is identical to that of the correspoinding plane Law of Tangents, we state the spherical Law of Tangents without proof.
An interesting problem of spherical trigonometry is that of finding the area of a spherical cap of either a cone or pyramid, with its apex at the center of the sphere.
For each law, we give the spherical (a, b, c are the sides; A, B, C are the angles), its dual for the polar triangle (A. C are the angles; a, b, c are the sides), and the plane (a, b, c are the norms of the sides; A, B, C are the angles) version.
www.rism.com /Trig/Trig02.htm   (8729 words)

  
 Electric Field, Spherical Geometry
The electric flux is then just the electric field times the area of the spherical surface.
The electric field is seen to be identical to that of a point charge Q at the center of the sphere.
The electric field inside a sphere of uniform charge is radially outward (by symmetry), but a spherical Gaussian surface would enclose less than the total charge Q. The charge inside a radius r is given by the ratio of the volumes:
hyperphysics.phy-astr.gsu.edu /hbase/electric/elesph.html   (349 words)

  
 Spherical Geometry   (Site not responding. Last check: 2007-11-07)
50) this.border=1; this.alt='Thumbnails by Thumbshots.org';">The Geometry of the Sphere
Spherical Geometry, which demonstrates the intrinsic curvature of a spherical surface..
The geometry on a sphere is an example of a spherical or elliptic geometry.
www.sphericalgeometry.info   (1005 words)

  
 Module 6
In spherical geometry, the “great circles” are not limited like the latitude lines, any antipodal points can be joined by a great circle.
We will define the measure of the angle between two spherical lines to be the same measure as the corresponding angle in the tangent plane.
It was remarkable in that he actually worked with the quadrilaterals of spherical geometry without really recognizing what he’d done.
www.uh.edu /~hollyer/Module6/m6ppt   (695 words)

  
 Spherical Geometry   (Site not responding. Last check: 2007-11-07)
In particular this means that each edge of a polyhedron determines and arc of a great circle, since the two end points of an edge lie on the surface of the sphere.
The shortest path from one point to another on a spherical surface is along the arc of a great circle.
Each plane polygon that is a face of the polyhedron is thus transformed into a spherical polygon that is a face of the spherical polyhedron.
www.ul.ie /~cahird/polyhedronmode/newpage2.htm   (359 words)

  
 Anisotropic scattering in spherical geometry   (Site not responding. Last check: 2007-11-07)
Anisotropic scattering of light in the spherical geometry of the Earth's atmosphere
It is these situations, anisotropic scattering in a spherical geometry, ie anisotropic scattering during sun-rise or sun-set, that I am most interested in.
To solve the problem of anisostropic scattering in spherical geometry, I developed a new mathematical equation describing the physics of this process accurately in this context.
www.atm.damtp.cam.ac.uk /people/mgb/aniso.html   (389 words)

  
 Amazonite Hardness Information   (Site not responding. Last check: 2007-11-07)
Science crystal A material in which the atoms are arranged in amazonite hardness a rigid geometrical structure see geometry marked by symmetry.
Found in a cave in S Africa in 2004, they consist of peasized pierced shell beads that were probably strung into a necklace or bracelet.
The shortest distance between two points on a amazonite hardness spherical surface, measured on the surface, is the distance amazonite hardness along the great circle through those points.
www.amazonitehardness.jewelry-boxes-etc.com   (1183 words)

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