[PDF] Exploration of Spherical Geometry





Previous PDF Next PDF



Sphere theorems in geometry

The sphere theorem in differential geometry has a long history dating back to a paper by H.E. Rauch in 1951. In that paper [64]



Extremal Systems of Points and Numerical Integration on the Sphere

Oct 29 2001 Extremal Systems of Points and Numerical Integration on the Sphere. Ian H. Sloan. ? and Robert S. Womersley. †. School of Mathematics.



Volume of Prisms Cones

https://2fv5d843v9w22sxtto1ibxtu-wpengine.netdna-ssl.com/wp-content/uploads/2015/12/Geometry-H-Volume-of-Prisms-Cones-Pyramids-and-Spheres-v2-SOLUTIONS.pdf



The Viewable Sphere

14 September 2011 : : Math Horizons : : www.maa.org/mathhorizons. David Swart and Bruce Torrence radius of this imaginary sphere is not.



New upper bounds on sphere packings I

Annals of Mathematics 157 (2003)



Geometry of a qubit

Dec 22 2007 6 I will discuss the stereographic projection that provides a correspondence between the extended complex plane C? and the Bloch sphere. S2. I ...



QUADRATURE ON A SPHERICAL SURFACE CASPER H. L.

an integral over the unit sphere S2 = {x ? R3 : x2 = 1} E-mail address: beentjes@maths.ox.ac.uk. ... 2010 Mathematics Subject Classification.



Spherical Trigonometry

West Hills Institute of Mathematics. 1 Introduction side of the sphere the sides of a spherical triangle will be restricted between 0 and ? radians.



Spherical Geometry

{6} • Spherical Geometry AMSI module. A. B. O minor arc major arc. Let A and B be two different points on a circle with centre O. These two points.





[PDF] Spherical Geometry - Australian Mathematical Sciences Institute

Spherical Geometry – A guide for teachers (Years 11–12) Andrew Stewart Presbyterian Ladies' College Melbourne Page 3 Assumed knowledge



Spherepdf - Mathematics - Notes - Teachmint

' a ind the equation of the sphere which has the line segment joining the points (234) and (0—12) as diameter é 2 (a) Find the equation of the 



(PDF) SPHERE - ResearchGate

PDF Equation of a sphere Intersection of sphere and planes In solid geometry a sphere is the locus of all points equidistant from a fixed point



[PDF] SPHÈRE BOULE ET SECTIONS - maths et tiques

Yvan Monka – Académie de Strasbourg – www maths-et-tiques SPHÈRE BOULE ET SECTIONS I Sphères et boules Vidéo https://youtu be/YQF7CBY-uEk



[PDF] UNIT 2 THE SPHERE - eGyanKosh

In this unit you will see what a geometry calls a sphere We shall also obtain the general equation of a sphere Then we shall discuss linear and planar 



[PDF] Chapter 19 Mensuration of Sphere - PBTE

Applied Math Mensuration of Sphere Chapter 19 Mensuration of Sphere 19 1 Sphere: A sphere is a solid bounded by a closed surface every point of



[PDF] Sphere et boule - Cours - Collège Le Castillon

Soit O un point et r un nombre positif ? La sphère de centre O et de rayon r est l'ensemble des points de l'espace situés à la distance r du point O



[PDF] BSc II SEMESTER PAPER

%2520Mathematics-%2520Paper-II_%2520Unit-III.pdf



[PDF] Spheres and Cones - Maths Genie

Surface area of sphere = 4?r² Work out the total surface area of the hemisphere Give your answer in terms of ? Area of circle



[PDF] Surface Areas and Volumes of Spheres 118 - Big Ideas Math

a sphere with a radius of (a) 3 inches and (b) 2 centimeters USING TOOLS STRATEGICALLY To be proficient in math you need to identify relevant external

:

ExplorationofSphericalGeometry

MichaelBolin

September9,2003

parallellines:sphericalgeometry. computingareas. geometries. 1 antipodalpoints.

CBA,ifB0isbetweenA0andC0inp.

theirarcsarecongruent. theirtangentsarecongruent.

Hilbert'saxiomsofincidenceareasfollows:

(I2)Everylinecontainsatleasttwopoints. (I3)Thereexistthreenoncollinearpoints. 2 geometry. containsinnitelymanyS-points. equivalenttoCBAbydenition. notlieonlessthanhalfanS-line. andBCDhold,butABDandACDfail. butneitherofthesetwopointsisonn.

Hilbert'saxiomsofcongruence(

=)areasfollows: auniquepointDonrsuchthatAB =CD. (C2)IfAB =CDandAB =EF,thenCD =EF.Everylinesegmentiscongruent toitself. colinearpointsD,E,andFsatisfyingDEF,ifAB =DEandBC =EF, thenAC =DF. 3 sideofthelineDFsuchthat6 BAC =6EDF. (C5)Foranythreeangles,,and ,if =and ,then .Everyangle iscongruenttoitself. =DEandAC =DF,and that 6 BAC =EF, andalso6ABC =DEFand6ACB =6DFE. thissituationisclearlyimpossible. =EF.Additionally,because itselfinsphericalgeometry. AB =DEandBC OA =OB =OC =OD =OE =OF: =4ODEand 4OBC threedistinctpointsX;Y;andZ.Hence, 6

ABO+6CBO=6ABC;

=6

DEO+6FEO;

6 DEF:

Because

6 ABC =6DEF,itfollowsthat4ABC =4DEFbytheside-angle-side congruencetheorem(SAS).ItfollowsthatAC =DF.SoS-lineACiscongruentto

S-lineDF.

For(C4),takeangle6

projectedangle6

BACbecausetheirprojections

=,then 4

Similarly,if

Therefore,

anglesarecongruent,6 ABC =6DEFand6ACB =6DFE.Italsofollowsthat sidesdonotmeancongruentlunes. +area(T)=R2: G'T' RTB B R G g br

Figure4-1.SphericalTriangleT.

5 usedhere. summingtheareasofallsixlunes,weget L r+Lg+Lb+L0 r+L0 g+L0 b=(T+R)+(T+G)+(T+B) +(T0+R0)+(T0+G0)+(T+B0) =(T+R+G+B+T0+R0+G0+B0) +(T+T+T0+T0) =area(sphere)+2T+2T0:

2R2r+2R2g+2R2b+2R2r+2R2g+2R2b=4R2(r+g+b)

=4R2+2T+2T0: ofasphericaltriangle: area(T)=R2(r+g+b): thesamearea. BCA =6B0C0A BB' C' A C

Figure4-2.ProofofAAA.

6 and6ABC thesamearea.Hencetheycoincide. surprisingthatAASdoesnothold. P ABB'

Figure4-3.WhyAASandSSAFail.

6

4-3.NotethatPA

=PAandPB =PB0asbefore,andthattrianglesPABandPAB0 trianglePAB0.

Ffaces.Eulerprovedthat

VE+F=2:

7 X anglesofthen-gon=(n2)+1

R2areaofthen-gonn:(5-1)

Figure5-2.Polyhedroninsideasphere.

UsingFormula(5-1),weget

F X i=1 sumofanglesofQi=FX i=1(ei2)+1 R2F X i=1areaQi: 2V=FX i=1(ei2)+1 R2F X i=1areaQi: 2V=FX i=1(ei2)+4 FX i=1e iFX i=12+4: 8

2V=2EFX

i=12+4 =2E2F+4:

Thus,weobtainEuler'sfamousformula:

VE+F=2:

References

elements.html. 9quotesdbs_dbs10.pdfusesText_16
[PDF] jeux logico mathématiques ? imprimer

[PDF] jeux de société+objectifs pédagogiques

[PDF] sphère synonyme

[PDF] projet pédagogique jeux de société

[PDF] jeux mathématiques cycle 3 ? imprimer

[PDF] mathador

[PDF] jeux de calcul mental ? imprimer cycle 3

[PDF] jeu calcul mental cm1

[PDF] jeu calcul mental cm2 ? imprimer

[PDF] jeu du furet ce1 ce2

[PDF] jeu du furet table de multiplication

[PDF] jeux calcul mental ce2

[PDF] jeu du furet cm2

[PDF] jeux calcul mental cm2

[PDF] tdg test en ligne