7(9JX=,dd +(I)   1Courier New1Symbol New1Courier New1Courier NewTimesr NewTimes New Roman1Courier New1Courier New1Courier New1Courier NewX x)VX x)VX x)VX x)V ?/ <```@@@?~~~~~ԱޱޱԱ޿޿ph (((Pdd{  d<  p {2 yz<?OoS @p^fG@I" !]@???Q@|.$J HFz@cfff0&Xafff0@cfff1o<@LB[}@?033333??@^ߛOw?̌??? Ic@" V@@?@ @@@`?R dt)@@@@@@e8 yea)i.@U%}@?@dddddddddddddddddddddddddddddddddddddddd /2??Uy@$$$Uy@r3܀@e8 v??F?????== ===== =@D@ARqR sin(u)cos(t) sin(u)sin(t)cos(u)Riemann Sphere52??@$$$WDi!pz???????????)))))!)")XAAA001Projection Origin52??@$$$M%7N???HTi?`Uظ??TI݌?0xPl?g????j??0xPl?g?HU\?Az????/,?#) $)%)&)')() ))$B@8PQ001 NX(O)/DN(O) NY(O)/DN(O) NZ(O)/DN(O)Vecr Pr Orig to Pt 52??@$$$>v?N}??(\M?TU*@V.zAҦ?TU*@????V.zAҦ?TU*@X]A\^A`_Ad`AhaAlbApcA'B Q4RA*cos(G)+R*cos(O)A*sin(G)+R*sin(O)0Pt Z on Circle in C42???$$$??ĽOWʊd=^@b 8y?;jJ?????/,?$*)(+),,)0-)4.)8/)<0)(QHEPP+Q000 A*cos(G) A*sin(G)0Z vector Z0circle42???$$$?????? \8)`9)d:)h;)l<)p=)t>)BCPRP`SFClCȀF4aF NX(t)/DN(t) NY(t)/DN(t) NZ(t)/DN(t)0.0;2piCircle on Sphere42???$$$??ZՂ?N㚣?+&?jϏl@????/,?rAsAtAuAvAwAxA Dl"S@OS A*cos(G) A*sin(G)0A*cos(G)+R*cos(O)A*sin(G)+R*sin(O)0Vec: Zc to Z on Cir52??@$$$۱ȱK???&gu?'w?4 Z?p?铀?f-????p݀?铀ś?f-⍌?x?)|@)A)B)C)D)E)SSPS XC(R,A,G) YC(R,A,G) CC(R,A,G)Sph Circle Center42???$$$??VԒU?=0?ۇ?kst7?W܈|?=*4????/,?yAzA{A|A}A~AA'D_c~lZ9WCa?yA5&? oJ????/,? F)G)H)I)J)K)L)LI`UpaV000XC(R,A,G)/NN(R,A,G)YC(R,A,G)/NN(R,A,G)CC(R,A,G)/NN(R,A,G)U Normal Sph Circlexyzgffffff@JXT@gf#Dy0?J[9F?Rffffff@3333333?QQ@h!@@@@@@@@h!@@@@@@@@h!@@@@@@@@@@@@w@"$w@"$w@"$ R*R+A*A+2*R*A*cos(x-G)+1#p?#?#?#?#?#?#?#DN?O)2(A*cos(G)+R*cos(x))@M @X Rb@)-O@Q*@E\6i@UGЁ@9x@es"@MCiì@c i{@[,,@ Y"@47@\J10@#7?@RZc@mؔw@YC(@XQ@g:@-̓@$=@90!2@k NX@aG,P)2(A*sin(G)+R*sin(x))@T|փ@뒃@TW-FL@r_hb@X&3lz@jMI@l`@_bA@\gH@4Q7#@\8K`@߳@QKFՃ@=I @9_3>@Yu3l@sAA@&|/@s@UXބ@@dxb.p@*#@x'-h]/@NY5@@Q)DN(x)-2     !"#     NZTR)(x*x+1-y*y)*(x*x+1-y*y)+4*y*yy*yDS3XhS)2*y*cos(z)*(y*y-x*x+1)/DS(x,y)XC|T)2*y*sin(z)*(y*y-x*x+1)/DS(x,y)YCU)((x*x-y*y)(x*x-y*y)-1)/DS(x,y)2A3333333@2-gt@MtM%%42gt?M%%|?CCV)sqrt(XC(x,y,z)*XC(x,y,z)+YC(x,y,z)*YC(x,y,z)+CC(x,y,z)*CC(x,y,z))J)J)J)J)J)J)JNN)Riemann Standard Projection Circle ---> Sphere PU 6sS?/vٙ=?쎱a?Times New Roman(x,y,z) = (sin(u)cos(t),sin(u)sin(t),cos(u)))TɸfeZ*aA7?Æb᱁?1Courier New TA(x,y,z) = (0,0,1)ownloads\@@@TɸfeZ r,H!xV?Ig$?1Courier New A(x,y,z) = (NX(O)/DN(O),NY(O)/DN(O),NZ(O)/DN(O))*s-1)/(s*s+1TɸfeZyk'+ubHmY]@҅0 ?1Courier New ăAsegment (0,0,1)--(NX(O)/DN(O),NY(O)/DN(O),NZ(O)/DN(O))s*s-1TɸfeZpήHa?ZTA@Zb?1Courier New |A(x,y,z) = (A*cos(G)+R*cos(O),A*sin(G)+R*sin(O),0)))sin(O),0TɸfeZa9a<l11@Tf?1Courier New 4Asegment (0,0,0)-->(A*cos(G),A*sin(G),0)in(G)+R*sin(O),0)))TɸfeZwxYlF%A K@=iv&?1Courier New Asegment (NX(O)/DN(O),NY(O)/DN(O),NZ(O)/DN(O))<--(A*cos(G)+R*?TɸfeZ & 1:x?@ TɸfeZ6:A?f=w?1Courier New ̈A segment (A*cos(G),A*sin(G),0)-->(A*cos(G)+R*cos(O),A*sin(G)+TɸfeZw"oW%? AQ?1Courier New A (x,y,z) = (XC(R,A,G),YC(R,A,G),CC(R,A,G))TɸfeZ]"+ݲB?ލ?1Courier New (NX(O)/DN(O),NY(O)TɸfeZ]"+Q%V?r;J8t?1Courier New A segment (0,0,0)-->(XC(R,A,G)/NN(R,A,G),YC(R,A,G)/NN(R,A,G),CTɸfeZEf }R(? gyqM?1Courier New @@@h!@