What is the correlation between spherical trigonometry and Riemannian geometry?
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Well I'm not a geometer ... so I may err slightly here as i only had one non-euclidean geometry class and going over some links has only refreshed some of it.\n\nRiemannian geometry is otherwise known as elliptic geometry :\n"It can be visualized as the surface of a sphere on which "lines" are taken as great circles. In elliptic geometry, the sum of angles of a triangle is > 180 degrees"  [this last part taken from mathworld]. It is a "non-euclidean" geometry "with positive curvature" which replaces the "parallel postulate" with *through any point in the plane, there exist no lines parallel to a given line.*\n\ntaken from wikipedia :\nSpherical trigonometry is a part of spherical geometry that deals with polygons (especially triangles) on the sphere and explains how to find relations between the involved angles. This is of great importance for calculations in astronomy and earth-surface and orbital and space navigation.\n\nSo one could say ... paraphrase the above defn of spherical trigonometry mentioning how it is a study of the relations of polygons, angles etc on objects under (or in) a riemannian geometry system\n\nincluded below are some of the links so you can formulate a better answer than mine :) gain a better understanding and even give examples that you find in those links and related links :)\n\nenjoy :),\nTom
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Sometimes geometry is more than calculating numbers.  Geometry has to do with shapes in space.\nSimple geometry is used when driving an automobile.  The curve of the road, for example is constant, so you do not have to keep adjusting the steering wheel.\nChildren use geometry when stacking blocks.  Adults use the same techniques when stacking boxes in a warehouse (they might also calculate how many boxes can go into the space provided).\nGeometry is used to make sure that a door closes in the frame.  Right angles and the proper length and width.\nGeometry is use when slicing a pie or cake.  Sometime to get even pieces, sometimes to get different size pieces.\nGeometry is used when playing baseball.  The diamond is really a square and the batter must place the bat at just the right place to hit the ball.  This is usually done unconsciously.\nAny time that size and shape of objects matter, geometry is used.
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Non-Euclidean geometry is more general, spherical geometry is more specific.  There are several non-Euclidean geometries besides spherical - oblate spheroidal, cylindrical, etc.
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the radius of curvature of a plane mirror is infinity( iam dead sure)\n\n\nThe term radius of curvature has specific meaning and sign convention in optical design. A spherical lens or mirror surface has a center of curvature located in (x,y,z) either along or decentered from the system local optical axis. The vertex of the lens surface is located on the local optical axis at a distance equal to the radius of curvature from the center of curvature. The sign convention for the optical radius of curvature is as follows:\n\nIf the vertex lies to the left of the center of curvature, the radius of curvature is positive. \nIf the vertex lies to the right of the center of curvature, the radius of curvature is negative. \nThus when viewing a biconvex lens from the side, the left surface radius of curvature is positive, and the right surface has a negative radius of curvature.\n\nNote however that in some areas of optics other than design, other sign conventions are sometimes used.
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the geometrical center and center of gravity match only for object that have symmetrical shape (around an axe or point: like a cylinder or sphere) and a constant density
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Of course. Consider a single hill shaped almost spherically. This is the upper hemisphere of a sphere, so the surface area, given some radius r, will be half the surface area of the entire sphere, which is (1/2)*(4*Pi*r^2) = 2*Pi*r^2. \nThe projection of the sphere onto the plane is a disk, with area Pi*r^2, less than the surface area if the flat ground was instead covered by a spherical hill.
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The moment of inertia for a spherical shell is 2/3*M*R2. You might imagine the spherical shell to be made up of a series of tiny mass elements the mass of each being its volume times its density r. The volume would be the thickness of the shell times the difference in latitude covered by the mass element times the difference in longitude covered by the mass element. Lets call the differential longitude dq and the differential latitude dl. Then the mass of each element is r*dq*dl. The moment of inertia of the whole shell is the sum of the moments of inertia of each of the little mass elements. The moment of inertia, i, for a mass particle is m*r2 where m is the particle mass and r is its distance from the axis or rotation. In our case i=r*dq*dl*R*cos(l) where R is the radius of the spherical shell and l is the latitude measured from the equator of the sphere. To sum up all the moments of inertia i we must integrate over all q from 0 to 2*p and all l from -p to p. Carrying out this integration gives us the result.
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„input harvmac „overfullrule=0pt {„it Exam} (1) Prove via path \n„def„R{{„bf R}} „def„S{{„bf S}} (a) Work in four-dimensional (flat) Euclidean space $„R^4$ and show that conformal invariance implies...\nhttp://www.math.ias.edu/QFT/fall/exam1.tex \n\n \nhttp://www.math.ias.edu/QFT/spring/stringPS8.tex\n...def„det{{„rm det}} „def„Det{{„rm Det}} „def„tr ... B$ of the Wess-Zumino-Witten model, satisfy the equations for conformal invariance and for\nhttp://www.math.ias.edu/QFT/spring/stringPS8.tex \n\n \nComments: Latex(2e), 34 pages, 4 figures (eps) On some UNIX machines\nSome private shortcuts: „def„be{„begin{equation}} „def„ee{„end{equation}} „def„bea{„begin{eqnarray}} „def„eea{„end{eqnarray}} „def...\nhttp://www.ma.utexas.edu/mp_arc/e/96-436.latex.mpmix \n\n \n=4 19| >0 63| >1 65| >2 434| >4 525| 0 1773| 1 630| 1/2 2530| 10 \n\nhttp://www-library.desy.de/lists/spihep.k \n\n \nStardrive.org - - Article/Content Template\nStardrive.org is dedicated to fostering increased public awareness and education in the theoretical and technological aspects of interstellar travel.\nhttp://www.stardrive.org/Sarmail1-3-01.shtml \n\n \n„documentstyle[12pt]{article} %„renewcommand{„baselinestretch}{2.0}\n...oddsidemargin 0 mm „def„ee{„end{equation}} „def ... correlation functions have a power law behaviour, indicates that conformal invariance may be\nhttp://theory.ipm.ac.ir/papers/src/ipm63.tex \n\n \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%...\n...def „pa {„partial} „def „ti {„widetilde} „def „se {{„prime„prime}} „def ... „subsection{Pre-big bang inflation and conformal frames}\nhttp://www.ba.infn.it/~gasperin/tex/ba437.tex \n\n \nhttp://www.ba.infn.it/~gasperin/tex/cer96267.tex\n...def „pa {„partial} „def „ra {„rightarrow} „def ... discussed in „cite{6a}, to the conditions of $„sg$-model conformal invariance represented by\nhttp://www.ba.infn.it/~gasperin/tex/cer96267.tex \n\n \n%%%%%%%%% LATEX FILE FOR FORMATTING GUIDELINES \n...def„xp{X^{+}} „def„xm{X^{-}} „def„Yp{Y^+} „def„Ym ... must be obeyed (in a conformal gauge this is the same as maintaining conformal\nhttp://theory.tifr.res.in/group/colloq/das.tex \n\n \n„documentstyle[twoside,psfig]{article}\n...def„ds{„displaystyle} „parindent=0pt „def ... out that this model is advantageous for the study of the conformal invariance conjecture for\nhttp://www.math.utah.edu/~ejpecp/EcpVol1/paper8.tex
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This can be solved in several different ways.  One of course is by calculus, by developing a formula for the surface area of a "slice" of the remainder of the sphere and integrating it about a rotated axis, or something similar.  Another is by digging for the rather obscure formula for the "volume of the endcaps of a sphere" and subtracting two of those and the volume of a cylinder from the volume of the original sphere (and discovering that many of the terms will cancel out nicely).\n\nThe easiest method, however, is via the use of simple logic.  Unfortunately, this method also requires one more "given" in order for it to be valid, though this given is not an unreasonable one:  There must be a single, real answer (or more specifically, a unique answer on the real number line).  Such an assumption would be folly in a practical (real-world) context, since without some form of proof, one should never immediately assume that this is the case.  Perhaps there are multiple answers, or there is no answer at all?  But as luck would have it, there actually is one answer, and it may be derived logically due to its unique existence.  (*thumbs nose at any philosophers who may be reading*)\n\nSolution:\n\nLet R be the radius of the cylinder drilled into the sphere.  If and only if there is a single, real answer, then that answer would still be the same no matter the value of R... even if R=0.  Thus, the answer is the volume of a sphere of radius 3 (diameter 6), or 36*pi cubic units.\n\n(Yes, the volume is always the same, whether the original sphere were the size of a basketball, or the size of a planet.)\n\nIt's a surprising answer, along the lines of the more familiar puzzle: "If the Earth were a perfect sphere and a band were placed snugly about its equator, and then one more meter were added to the length of the band and it were stretched so that all points along it were equidistant from the surface of the Earth, how far above the Earth would the band rise?"  Many are surprised to find that the size of the Earth plays absolutely no part in the equation, and that the "rise" is exactly the same if the sphere were basketball-sized or, umm... Earth-sized (with a circumference greater than 40,000,000 meters).  It rises exactly the radius of a circle with a 1-meter circumference: 1 / (2*pi) meters.
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In general, there are two important types of curvature: extrinsic curvature and intrinsic curvature. The extrinsic curvature of curves in two- and three-space was the first type of curvature to be studied historically, culminating in the Frenet formulas, which describe a space curve entirely in terms of its "curvature," torsion, and the initial starting point and direction.\n\nAfter the curvature of two- and three-dimensional curves was studied, attention turned to the curvature of surfaces in three-space. The main curvatures that emerged from this scrutiny are the mean curvature, Gaussian curvature, and the Weingarten map. Mean curvature was the most important for applications at the time and was the most studied, but Gauss  was the first to recognize the importance of the Gaussian curvature.\n\nBecause Gaussian curvature is "intrinsic," it is detectable to two-dimensional "inhabitants" of the surface, whereas mean curvature and the Weingarten map are not detectable to someone who can't study the three-dimensional space surrounding the surface on which he resides. The importance of Gaussian curvature to an inhabitant is that it controls the surface area of spheres around the inhabitant.\n\nRiemann  and many others generalized the concept of curvature to sectional curvature, scalar curvature, the Riemann tensor, Ricci curvature, and a host of other intrinsic and extrinsic curvatures. General curvatures no longer need to be numbers, and can take the form of a map, group, groupoid, tensor field, etc.\n\nThe simplest form of curvature and that usually first encountered in calculus is an extrinsic curvature. In two dimensions, let a plane curve be given by Cartesian parametric equations.\nThe curvature of a two-dimensional curve is related to the radius of curvature of the curve's osculating circle.\n\nThe curvature at a point on a surface takes on a variety of values as the plane through the normal varies. As  varies, it achieves a minimum and a maximum (which are in perpendicular directions) known as the principal curvatures.The curvature  is sometimes called the first curvature and the torsion  the second curvature. In addition, a third curvature (sometimes called total curvature).
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You're already on a website.  What is it with people these days?  Look in your book.  \nInteresting though: the derivative of the area of a circle is equal to the circumference of it.  Works the same for a sphere too.
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Think what you are looking for is Euclidian Space:\n\nAround 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called "Euclidean geometry", which is the study of the relationships between angles and distances in space. Euclid first developed "plane geometry" which dealt with the geometry of two-dimensional objects on a flat surface. He then went on to develop "solid geometry" which analyzed the geometry of three-dimensional objects. All of the axioms of Euclid have been encoded into an abstract mathematical space known as a two- or three-dimensional Euclidean Space. These mathematical spaces may be extended to apply to any dimensionality, and such a space is called an n-dimensional Euclidean space or an n-space. This article is concerned with such mathematical spaces.\n\nIn order to develop these higher dimensional Euclidean spaces, the properties of the familiar Euclidean spaces must be very carefully expressed and then extended to an arbitrary dimension. Although the resulting mathematics is rather abstract, it nevertheless captures the essential nature of the Euclidean spaces we are all familiar with.\n\nAn essential property of a Euclidean space is its flatness. Other spaces exist that are not Euclidean. For example, the surface of a sphere is not a Euclidean space, nor is the four-dimensional spacetime described by the theory of relativity when gravity is present.
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assume u1 is angular path1, u2 is angular path2\nu1=w1*t, u2=w2*t, while u1+u2=2*pi rad \nthus 2*pi=(w1+w2)*t, hence t=2*pi/(w1+w2).
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Hi. Euclid did a lot more than that. : http://www.answers.com/topic/euclidean-geometry-1\nGetting folks to think rationally about geometry was a great contribution.
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U can see the whole world as Geo-metric figures. The Design of a house by civil Engineer is full of Geometry. Followed by lands Planned to use for building purpose and for Ariel view. And with the help of ice in winters if u go to the north and south pole u can find natural and Man made designs which r Geometric figures which u have mentioned above.\n\n  Geometry is not the different types of angles and triangles it includes figures like cone, circle, cylinders , Pentagon and u can go on  Infinity
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It depends on the surface and the triangle. For example, on a sphere, the amount the sum of the angles is more than 180 degrees is proportional to the area of the triangle. For other geometries, the sum is always less than 180 degrees. In Lobachevskian geometry, the deficit is proportional to the area.
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yes ofcoure it is meant to for an example i can tell you\n\n\nthe trigonometry is used in aviation\n\nu know how to calculate distance say a flight from \n\n30 deg north and 45 deg east is landing at 21 south and 36 west \n\n\n\nhere they use right angle  triangle the phythogers theorem ,\n\nmark the two location in a chart scaled say one deg per square,\nthen you can joint the lines to make a right angle triangle,\nno with no of degres in adjacent and opposite side of triange u can calculate the hypotunuse ( sum of suares of other two sides is the sum of squares of hypotenuse).\n\n\nthis is an ex: u have every thing use in one or more way in actual  applications.