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À�Êøý¿ÃŒõÏ”Áí�þ?™qÔÂ�vªþ?xD t”.`;YläQ Èø’ pü" Ø" d^’ DWÀ€p xP(p)yP(p)zp(P)Image Pt on Sphere4ÿÿÿÿ2ÐÌÌÌÌÌÌü?€ÿ?€ÿ?$$$––€ÿ?€ÿ?FW`¶ž±ÄÑù¿š($6ê"Èü¿'vî·':i–þ¿H€k¾6Xh‘ü?bz>HU\�ú?åô¤Aøz›þ?€ÿ?€ÿ?€ÿ?´êÝ’Æg€ü?j% XFYp†.ÀS ¬¢’ <>eôˆ<œ¾ ü–4tA4001xP(p)yP(p)zp(P)Projection Part 25ÿÿÿÿ2ÐÌÌÌÌÌÌü?€ÿ? À @$$$––CÚlxÅÅd¸ý??ÆWB"TŠþ¿€ÿ?€ÿ?$¥%ZæÍ%Úÿ¿°[Þà¸ÿ?€ý¿$¥%ZæÍ%šÿ¿°[Þàøÿ?€ý?€ÿ?€ÿ?€ÿ?$¥%ZæÍ%ºÿ¿°[ÞàØÿ?¸Yì†<lx.œ…’ ˆt2 hj.4ÓSdU\~D¬×Í p*cos(S)+Q*sin(S)*sin(S-O)p*sin(S)-Q*cos(S)*sin(S-O)0Source pt in Clane4ÿÿÿÿ2ÐÌÌÌÌÌÌü?€ÿ?€ÿ?$$$––€ÿ?€ÿ?ã½…âNÛ†ÿ¿ií'¦¦Tô–ý¿š™™™™™™þ¿·ÿ“¦e·'Äû?~ãv$eô�Ï@š™™™™™™þ?€ÿ?€ÿ?€ÿ?´êÝ’Æg€ü?È<<¨èT`–-(u’ ˜Bf"Á ÈCXì0 Pû˜4 000Qcos(O)Qsin(O)0C Vector Z0 Cge4ÿÿÿÿ2ÐÌÌÌÌÌÌü?€ÿ?€ÿ?$$$––€ÿ?€ÿ?Òhñ�îŠÆÿ¿€”Ä�îÕèý?]Ìv5Î^ˆû¿Zú�eôš5Çý¿ò±äÙ,ýPçÿ?9ýXƒ‰[‚»þ?€ÿ?€ÿ?€ÿ?´êÝ’Æg€ü?\ˆ-0OfXz-ôh’ Tl-l&<,›,ðÑDô Eà  p*cos(S)+Qsin(S)sin(S-O)p*sin(S)-Q*cos(S)*sin(S-O)0xP(p)yP(p)zP(p)Projection Part 10ÿÿÿÿ2ÐÌÌÌÌÌÌü?€ÿ?€À€@$$$––€À€@€ÿ?8ÉËxQq޲Àc-³3š’€ü¿áR¸…ëQ¸þ¿ ZÛ>@Žé¦@qà¯Pà‚@áR¸…ëQ¸þ?€ÿ?€ÿ?€ÿ?€@´êÝ’Æg€ü? ô9g(�,œ'g$,Üb’ q,Œ!<t„DàæD€§Dè§DŒSò ¸éD´§Dt*cos(S)+Q*sin(S)*sin(S-O)t*sin(S)-Q*cos(S)*sin(S-O)0-4;44Source Line in C 0ÿÿÿÿ2ÐÌÌÌÌÌÌü?€ÿ?Àh!¢ÚÉ@$$$––Àh!¢ÚÉ@€ÿ?±!5Î÷ZÅÝþ¿ød=Ê¿�Åý¿'P—>R(�þ?Äš¡�š·Ãïû?=§vßJ·®þ?{¸ð>ZY8…ÿ?€ÿ?€ÿ?€ÿ?ˆ<d”+a’ S’ `†+°Og\x+@òD`Æ<\Œ 8@DdI4h.Œ (8Œ .rs(0)*(bc(0)*ca(S,O)(cos(t)+1)-sa(S,O)sin(t)).rs(0)*(bc(0)*sa(S,O)(cos(t)+1)+ca(S,O)sin(t))bc(0)*bc(0)-rs(0)*rs(0)cos(t)0.0;2piParamet Image Circl5ÿÿÿÿ2ÐÌÌÌÌÌÌü?€ÿ? À @$$$––×’ßÉøÔËÌû?¸ƒ$¨pã´œþ?€ÿ?€ÿ?7w@=ræŸþ¿–„Y¢T>Ðû¿̘Šy‚Ì‹þ?¸¹ê�3ÿû¿ÚžiWújðËý?fLÅ»ü?¸é^ŽAûþ?€ÿ?€ÿ?€ÿ?´êÝ’Æg€ü? Lˆ*Hz*Dl*›)�))q)€EDÜù¾ øç÷ 000ca(S,O)*rs(0)sa(S,O)*rs(0)bc(0)Unit Norm Cir Plane4ÿÿÿÿ2ÐÌÌÌÌÌÌü?€ÿ?€ÿ?$$$––€ÿ?€ÿ?:ò_}¶ }þý¿I § o0Úû?´°(J!§þ?GáÝ÷ºý¿ÓÈÆ0韛þ?ì!h6òè!Ïþ?€ÿ?€ÿ?€ÿ?´êÝ’Æg€ü?T”(P†(Lx(Hj(t–'pˆ'Tu àÙ Ä½Ô uõ xC(S,O)yC(S,O) bc(0)*bc(0)xP(p)yP(p)zP(p)Center to Image Pt0ÿÿÿÿ2ÐÌÌÌÌÌÌü?€ÿ?€À€@$$$––€À€@€ÿ?8ÉËxQq޲Àc-³3š’€ü¿áR¸…ëQ¸þ¿ ZÛ>@Žé¦@qà¯Pà‚@áR¸…ëQ¸þ?€ÿ?€ÿ?€ÿ?€@´êÝ’Æg€ü? p‡Z H´klz'$†khl' ›&ðp<$ ® ´ˆj\·� lO‹ 4N‹ èë� 8O‹ -t*cos(S)+Q*sin(S)*sin(S-O)-t*sin(S)-Q*cos(S)*sin(S-O)0-4;44Source Line in C 4ÿÿÿÿ2ÐÌÌÌÌÌÌü?€ÿ?€ÿ?$$$––€ÿ?€ÿ?Ш뭶uU�ÀÓr(Äã)÷”û¿š™™™™™™þ¿OU…+Û6¼ü?ùš·-™ÙÓÌþ?š™™™™™™þ?€ÿ?€ÿ?€ÿ?´êÝ’Æg€ü?ô~<øŒ<üš<Xl=,…_HˆfL–f0*� ¤P‹ w� 000Q*sin(S-O)*sin(S)-Q*sin(S-O)*cos(S)0C Vector Eta0xyz À À À À À À À À À À À€À À À À À À À Ààw„+«ô´@õ(\�Âõ(Üþ?›J»¢XTø­@3sóÛë›Î˜ÿ?ͬ)V,É”@H¿}8gDñÿ?ª…ëQ¸…ÿ?gffffffÆ@Àÿ?3sóÛë›Î˜ÿ?€ÿ? @Àh!¢ÚÉ@€ÿ? @ @ @Àh!¢ÚÉ@Àh!¢ÚÉ@ @ @ @ @ @ @ @¼t“VÉ@€@ @ @Àh!¢ÚÉ@ @ @ @ @ @ @ @ÀJêd.ÒdÀJêd.ÒdÀJêd.Òd (Q*sin(S-O))^2+1+x*x\Math521\ @ @ @Àh!¢ÚÉ@Àh!¢ÚÉ@ @ @ @ @ @ @ @Àh!¢ÚÉ@ @ @ @€@ @ @ @ @ @DenE„Ü3äÑ2(x*cos(S)+Q*sin(S-O)*sin(S))/Den(x)xPüHÝ 2(x*sin(S)-Q*sin(S-O)*cos(S))/Den(x)      !"#yPJÝ 1-2/Den(x)p!ã>qIO½iëý§²× fÉa-¶–„"“o�üŠÙöHâ¼êN®ÒpA ¦=DÈ ,•Ön!h±ƒü`’zPeG Q*sin(x-y)sin(S)/Den(0)xC( -Q*sin(x-y)cos(S)/Den(0)yC< sin(x)*sgn(sin(x-y))€ÿ? @ @ @Àh!¢ÚÉ@Àh!¢ÚÉ@ @ @ @ @ @ @ @¼t“VÉ@€@ @ @Àh!¢ÚÉ@ @ @ @ @ @ca@<`D-cos(x)*sgn(sin(x-y))Math521\ @ @ @Àh!¢ÚÉ@Àh!¢ÚÉ@ @ @ @ @ @ @ @¼t“VÉ@€@ @ @Àh!¢ÚÉ@ @ @ @ @ @sa@ý‘åPaD1/sqrt(Den(x))rsdbDsqrt(1-1/Den(x))7©3�@$sÞ�Vª3�@”KL—»«3�@ìn–I­3�@c†Ö…ý®3�@1©-Ô°3�@eß§¬É²3�@‡ ~Ú´3�@f¢ç�·3�@-Þ„=¹3�@-X`.‡»3�@³8` Û½3�@ƒ¿óŒ4À3�@eé*%�Â3�@çûú;æÄ3�@ɬ¾ìb¹3�@ê[ýjHµ3�@øG‰ED±3�@ãèŠO^­3�@¤Ò[!ž©3�@˜kº ¦3�@>�ÿ«¢3�@€#P”‡Ÿ3�@b9J磜3�@,bcš3�@…ØProjection Polar C Line ---> Circle on Sphere šCéµîó#ûÀpç€BE‚@ wó;2§âÿ?gÔ®©¼é…@èÿÿÿ¼Times New Romanÿÿÿÿ(x,y,z) = (sin(u)cos(t),sin(u)sin(t),cos(u)))ÿÿ¾×8Oa»Q�Àÿ�‡ä*æ˜ÀaA¢Ç7ªÿ?ŒÌw€Šäˆý?óÿÿÿ�1Courier New ˆ% þÿÿÿ(x,y,z) = (0,0,1)ownloads\ @ @ @ÿ¾×8Oa»Q�À �r,÷ÿšÔÀêH!xºVÄâÿ?]-:òžš{ƒý?óÿÿÿ�1Courier New @& þÿÿÿ(x,y,z) = (xP(p),yP(p),zp(P))a)*(cos(O)+1)-sin(a)*sin(O)),s¾×8Oa»Q�Àyk'+‚u…�ÀbHmY]œ±¢@{ùÇfK&üü?óÿÿÿ�1Courier New ø& þÿÿÿsegment (0,0,1)<--(xP(p),yP(p),zp(P)))*(cos(O)+1)-sin(a)*siÿ¾×8Oa»Q�ÀÇpήçHa‘À�?ZTAŸ@2Þ}«�aUñü?óÿÿÿ�1Courier New °' þÿÿÿ(x,y,z) = (p*cos(S)+Q*sin(S)*sin(S-O),p*sin(S)-Q*cos(S)*sin(¾×8Oa»Q�Àa9Ña<þÀ‘l1Ð1‰@ÞŸ�¸w„æü?óÿÿÿ�1Courier New h( þÿÿÿsegment (0,0,0)-->(Qcos(O),Qsin(O),0)),b*sin(a)/sqrt(1-b*b)ÿ¾×8Oa»Q�ÀwxYlF%ˆÀA K´„@E# V ¤âÐü?óÿÿÿ�1Courier New ) þÿÿÿsegment (p*cos(S)+Qsin(S)sin(S-O),p*sin(S)-Q*cos(S)*sin(S-O)¿?¾×8Oa»Q�Àð & Ì1:ÛÀÀ˜ˆx­¯ÿ?xŒ;Na³¡ý?óÿÿÿ�1Courier New Ø) þÿÿÿ(x,y,z) = (t*cos(S)+Q*sin(S)*sin(S-O),t*sin(S)-Q*cos(S)*sin(Ã×8Oa»Q�À —Wš$ÄÊ�ÀÙê UÕÑÿ?xŒ;Na³¡ý?óÿÿÿ�1Courier New �* þÿÿÿ(x,y,z) = (rs(0)*(bc(0)*ca(S,O)(cos(t)+1)-sa(S,O)sin(t)),rs( ÿÃ×8Oa»Q�Àðõº6 :ØÀ“�‘èA¸ñŸÿ?Ìû^? b �ý?óÿÿÿ�1Courier New H+ þÿÿÿ (x,y,z) = (xC(S,O),yC(S,O),1-1/Den(0))1-b*b)*sin(A),b*b)Æ×8Oa»Q�À8Ô ž,›Àþø4ÚW™ÿ?KÖ߈ÅÂüü?óÿÿÿ�1Courier New , þÿÿÿ segment (0,0,0)-->(ca(S,O)*rs(0),sa(S,O)*rs(0),bc(0))en(0))ÿÃ×8Oa»Q�Àx{Ô"Ìå˜À͈}b9Ž ‰ÿ?­E(DÆëü?óÿÿÿ�1Courier New ¸, þÿÿÿ segment (xC(S,O),yC(S,O),bc(0)*bc(0))-->(xP(p),yP(p),zP(p))ÿÃ×8Oa»Q�Àz„ÒBaC¢À~x öÛƒÑ@·vJòRžYÎý?óÿÿÿ�1Courier New p- þÿÿÿ (x,y,z) = (-t*cos(S)+Q*sin(S)*sin(S-O),-t*sin(S)-Q*cos(S)*siÃ×8Oa»Q�ÀÄ.-£Ûïô÷À–a‹éB­‘@Ä"sÖª ×ÿþ?óÿÿÿ�1Courier New (. þÿÿÿ segment (0,0,0)-->(Q*sin(S-O)*sin(S),-Q*sin(S-O)*cos(S),0),yÃ×8Oa»Q�À¢“cX¸‡ë†À 8ƒÀlPû“ÿ?g4ïð9Ÿìþ?óÿÿÿ�1Courier New þÿÿÿ À @ À @ @Àh!¢ÚÉ@