00020002010012Shortest_Dist2.ACT0001020012Shortest_Dist2.EAC0100000033ebÉ Shortest_Dist2.EACŚACT†,†/†9ˆ<†@Š†S^E ˆŒ'A[ˆ4"DGeometric Motivation of Finding w ˆance Between Two Curves†LŠŒ\ŒTAut†Ă툈ŒíŽŹ„†ŹŒ. Wei-Chi Yang ŒBRadford University Ž/ e-mail: wyang@rŒ2.eduŽE!URL: http://www.–#/Š5Žţ Objec†÷es:[Ž˜R11. We shall present geometric motivation and nume† al appproximŒto fi†!2the %shortest distance between two curves.[Œ.P2–‡ also see whˆ.o switchˆe#an external computer algebra systemŒ6Œ… needed. [Œ•[†qˆ Note.ŒŤP T†ˇ†Ę†Üals‘ ed here can†ť fur†Ţr†€ plored from a paper published byIaut‡ at http://www.atcm.‡6h‡Ptech.org/EP/2006ˆ 6P149/fullŠY.pdf.–žŽżExample‡l: †(-Consider C1: x^2+(y-1)^2=1 and C2: y=-†-1. 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If we dragŠ0x-coordinate of †Lndš\back toŠ^Geometry Strip,ˆJŽ^Ushall see a scatˆ‘plotˆUˆ™’ŻfuncˆĄ†>hereˆŽcan†×nj†×uˆŠˆÎŒĂmi†Öumš— may occuré‡ umn 1=x valuŒĎŽ!Ž2=“7 between DŠňA['5 -0.4275510204I 1.850358557–& 163265306’&708ˆ<11 -0.4051020408 1.6203809Ž †% 393877551Œ$ 1.553568027Œ% †J 3826530612’&49922981Ž8Š&714285†’L 453454149–L6Œ”16’r†!09054Ž—Šr48979591Žź†˜379ˆťÎŠ˝3Šť0”—34972553˜˝źŽ&‡32321533–ↈŕ2”ź29982578˜–Œš3‘†– 279218025•.292Žö‡T26113460Žô‰T2ŠI‡W•z2‡/7831ˆ˜ -0.2704081633† 1.231798212ŒŠ& 591836735”&2028005˜&479Š)7”&10739866–LˆH46939Œ& †r031194ŽŠ— 255102041’—1973†„94–—14285714”˝1935†Ť1˜q03061224”˝19152947ŽĎ‰ 1Žâ4”˝19144358–˜18ŒKŹ†˝1933082@‰T1†Ý877†Â–%71†ň˜%5‰v265—y‡9707?‰S1‰)ŠLŸ1.211444221 -0.1357142857Œ †&22098263–& 244897959Œ& †L3536735–K13265306Ž_ˆK51517744–q0204081_ˆK708928Ž–†–09Š"ˆO”—93941938–'ˆ’18†ˆŽa†ž32126640Ž™ŠN6ˆ#3469–'†ž9538˜'‘ 1–N924218ÁŠœ4Žtü‡343926580ŠĂˆp38775–'9725557˜'2(6“5721722’ę -0.01224489796 1.678313929Œ-1.020408164îěë3’* 900660868[ˆ> Conjecture:†ˆPHWhen the angle FDA =Ž GHD= 90 degrees, †will achieve its minimum.Žc(S†lateŠTshortest distance function:\ŒĐ=Stat Editor (after†pleting some entries contains’L=0)ԉ,‡Ę‰(NFinaFormX$N†Graph2D†| 3Š ˆLISTSYSˆœ4ˆ< Modify ЈPˆ<STATCALC ôˆdˆŒ< ü\ˆx S9equenceˆŒ,ˆxSheetˆO„|’ŠŒˆiŒolveEqˆˆ´†wr€ˆ´(UpˆŒ’tupFLG1˜(Š<†Lis†{ŔD‰ˆPic†ÜViewWind(ˆŒ_osve†v‰EŠxy†^‰P–ˆ¸-†† a‡– MatDatab‡˘.EAC‹ŠŠ Š ‡˝ ¤×^††× 1Œj‰W4††Đ<ˆ5–<ŠDĐxö<ţxţ´ţđÓ,asystem`=@ˆ Œ†0'ä]Setu,ä^äö<˛<††† †ŒžŒ¤’ ’,† †0†!@†@PSheet1|đ‘|:˜^Š2ž3ž 4†5ž ä8ä7Ę|č†]–|œš" ˜| “i™ˆ.†ƒ! ™†ˆ’†Š( ^E †Š†††Ł†$’+’4˘œ†]D˛<’h†(œž+†Ë/ 3x12.60547198*x^2+5.271331881*x+1.71141939ŠcŽ?`– PýĆ$´$“`– p @9b6‡x Y™2e0Y`‡Š ˜‘jˆŘ5‡‡s —–2Œđ%28‰F†Ýq†’ ˆ(1…0qy`ˆ#Y‡uYƒ†* ˜–0– šHŽĆ$ĘH(W( †ş Y™p–R˘ŔšÍŽ°´âIâz⫓Ś'Œ¸‡¸ ‡ź$†0†<†H†T†`†l†x†„††œ†¨†´†Ŕ†̆؆ä†đ†Wü ,8DP\ht€Œ˜¤°źČ'U @Y™2e0`Ž †€Œ“‡uQˆ#†$‚e0a Œ0qB…qŽ<` @ˆ H—•‘<7uQ†&S†90a"Žx2pŒ„’…qBŒc&SŒœp@c0Œ¨Y6sPŒ´G•‘ƒ†J6sF“†Q%Q†X‰`†_$†v ‘ƒg4Žx€a"Dˆ i8wUŽxX‰ NF‡ PY™5qB…pŽ $H—•Ž&SŽ$0Œ0 2eŒ ˜•‘ƒg@ŠH˜ƒg4iŽ †Sq†38Y6sŒl˜F“‡uŒx˜4i8w`Š`˜"D‰y ˆlŒT—'†§† ††$†0†<†H†T†`†l†x†„††œ†¨†´†Ŕ†̆؆ä† đ†Jü†† †,†8†D†P†\†h†t†€†Œ†˜†¤†°†ź†Č…XU‹<p„'Q‹0ˆ b€˜‰_ˆU5h$I’)†E4T‹l†+#A@T€ŽH7˜ € 4—%S`223T)˜%x€ ‰k†U'’P‡k&4``‡k$Sx10‡k0#˜! Ž0"€ !9†0 1IˆSˆTsƒi@Ž`51T;)GlCXŽ„3#’<q““H 1—pŽ0!D"Ž´" ˜&’œSg5x%t†\'’)9A“€Žđ2f@đ56•8ü9$!„´C’e€œIrUW¨W!r#Hgƒ’Ŕ`†Ž`ˆˆ  '†† ††$†0†<†H†T†`†l†x†„††œ†¨†´†Ŕ†̆؆ä† đ†Tü†X† †,†8†D†P†\†h†t†€†Œ†˜†¤†°†Bź†@Č'U @Y™2e0`Ž †€Œ“‡uQˆ#†$‚e0a Œ0qB…qŽ<` @ˆ H—•‘<7uQ†&S†90a"Žx2pŒ„’…qBŒc&SŒœp@c0Œ¨Y6sPŒ´G•‘ƒ†J6sF“ <%Q<(W†$Ž<‘ƒg4Žx€a"Dˆi8wUŽxX‰ F‡ ˆ 5†ýŽ´$†ńˆ†ĺˆ†ŮŽ´ †ăŽH˜•†J‡t Y˜†Ti@ˆ qB…qŽ Y6sPŒF“‡u†4†j$`Œ0"D‰y ŒH—'† ††$†0†<†H†T†`†l†x†„††œ†¨†´†Ŕ†̆؆ä†yđ†Kü ,8DP\ht€Œ˜¤°źČ…XUpˆôp„'QŽ b€˜‰ ˆU5h$I’)$E4T‡e A@T€ŽH7˜ €†ę64—%S`223T)˜%x0'’‹`†„&4`‹T†$Sx10Žœ#˜! Ž¨#"€ !9†‰„  1Iˆ sƒi@†D51`)GP$CX03#’<q““H 1—pŽ0!D"Ž`" ˜&0Žl#Sg5x%t†'’)9A“€Žœ2f@„56•89$!„´C’e€œIrUW¨W!r#Hgƒ’Ŕ`†`&GR‡ '1ˆ‚ŒŔq8•ˆÚ  9EUeˆ‡. ™'I(a%‰9 —ŠŻ,Š´’ ŠőŚ`”23RHŒ<’e5‰yˆ” – œ˜`‡^‰[‰64We obtained the quadratic regression formula to be: Ž=12.60547198î’xŒT2Œ\G$+5.271331881î’x+1.71141939 and we fi†the minimum of†is function as follows:R†ŒfMin(12.60547198†gŒ12Œ9´‡ ,x)RŽP †OValue="Šu 1.16033†^90T…†ŒŽ,x=-Œ* 0.2090890325 2I–†ˇ ™[ŽŤVNote†ěatŠń number x= -˜: will be very close to‹#vˆ˜ c which‰=Š+ see†6‡ :[/†‰8Provˆx‡U Analytically˜(‰_C3Step 1. We define function f to be the lower half o††circle C1:Rˆ solve(xŒ2Ž+y’Œ+2î’x-† y+1=0,y)R†GKy=-ŽW-x ?ˆ:xŽ_+1,y=Â/1ŒľŒ´ ú(x)=-ÚEŒDdone[7‹U2™UgL representin†‡Ygraph‰TC2IŒOŽ“gŠ“x^2-¤m‰r9QStep 3. Suppose we call points A and D to be (c,g(c))Š(x,f(x† respectively, †X Swhich make†F†Ahe shortest distance. Thenˆt†uld haveˆ. perpendicularŠ}oth ŒZ=ˆK†? gent lines at’Šsimul†] eously. In †;er words, [Œ˘ ˆË-ˆĂŒ˛c-xŒź *g'(c)=-1ŠţÄ1/*f'(xˆ1ŠŇrefore,‰Lset upŠţ followings:R‡z*eq1:=ś•R*Œ'gŠˇxˆ|x=cŽ +1=0RŒ -2î’cî’ŽcŒ82Ž#-Ž -x’ŒˆAŽk+Žx-œs†2aeq2:=Œ g(c)-f(x)Œ~c†bŒ¸*ŽĂ”!’֘ːv+1Ž×ţҖҔ^-c”^Ď”˙ solve({eq1,‡},{c,x})‹”•–-2î’cî’cŽ 2-Ž -x’ŒˆAxŒ<+Žx-cŒ +1=0,+1Žjţv–v”^-c”^ÎśˆŁ[†,‰Note.  ˆ‰&U1) We notice that ClassPad can't solvˆese two equations but† is suggestsŒ@we rWIshould now switch to our favorite computer algebra system. Nevertheless, †  geometric Cintuition an†eumerical estimaŠpresented here wo†S m††6ing.[ŒK/;2) We obtain from Maple that c=-.1969444377; x†6335161712[Œ‘In o†łrˆhdŽśpo†§s Ž( ˜?ŒĚf(˜)ŽRŠÚž=”Œ g†=”Ą)Œ<  will makˆÇe5‡“rtest distance DA‘#3) The shortest distance isR†!@(-.6335161712+.1969444377)^2+(g†–-fš/)†$R"ŠL 1†?1244195DPG†j[Œk[Œv4) If we drag both Ž! ˜–ŒŞž|ŽK and ž=”ÇŒçžÉŒ<  back to t‡+ geometry Play†ćimŠ?!Έ[†_·%Š Šm‰.dI)W…† `5&F1&1Š Œ`† $sShsgŠ ŽD‰jv’K†r† Ž A† 'LŽ'‰žC˜ ux††ˆ yŽ †Ž ČČ € Ž'2† 2PŽ6Y™8wUŠ† '† @Ž2e0`Ž$ @€Œ0“‡uQŽ<‚e0aŽ<qB…qŽ<` @Ž<H—•‘<7uQŽx&SŽx0a"Žx2pŒœ’…qBŒ¨c&SŒ´p@c0ŒŔY6sŠĐY™G•‘ƒ<6sF“<%Q<(W<$Ž<‘ƒg4Žx€a"DŽxi‰!,X‰8F‡ Žx5†ýŽ´$†ńŽ´†ĺŽ´†ŮŽ´ 2e0Y˜•‘ƒg@Œ ƒg4iŽ qB…qŽY6sPŒ0F“‡uŽ 4i8w`ŒH"D‰y ŒT—Š Šl ˜(WŽ &S Œ8wUŽ Q†=Žc†%Œ<u†Ž ‡† ˆż†TˆÇŒ ™†2† †’” †ď…XUpˆ p„'QŠ\b€˜‰ˆU5h$I’)$E4TŽ<A@T€ŽH7˜ € 4—%S`223T)˜%x0'’‹L&4`‹@$Sx1‹”#˜!‹"€Ž !9†` 1Iˆ ˆ sƒi@Ž51$)GPŽ0CXŽ<3#’<q““H 1—pŽ0!D"Žl" ˜&0Žx#Sg5x%tx'’)9A“€Ž¨2f@„56•89$!„´C’e€œIrUW¨W!r#Hgƒ’Ŕ`†Ž`ˆˆ   Ŕ!ŢA‡~ČČ € 2‡2ŠÖ‰šY™8wUŠ† '† ‹šY™2e0`Y™ @€Œ “‡uQˆ†‚e0a Œ$qB…q@Œ0` @Ž<H—•‘<7uQ<&S<0a"<2pŒx’…qBŒ„c&SŒp@c0ŒœY6sPŒ¨G•‘ƒ<6sF“<%Q<(W<$Ž<‘ƒg4Žx€a"DŽxi8wUŽxX‰F‡ Žx5†ýŽ´$†ńŽ´†ĺŽ´†ŮŽ´ †ăŽH˜•‘ƒg8˜†ƒiD˜•P˜”䘔x˜4†¤˜"D‰y°˜ @Y—Š 0ˆ  ˜(WŽ &S Œ8wUŽ Q†=Žc†%Œ<u†Ž ‡† ˆ_†TˆgŒ ™†2† †’” †…XUpˆ p„'QŠ\b€˜ˆŻˆU5h$I’)$E4TŽ<A@T€ŽH7˜ € 4—%S`223T)˜%x0'’PŽ„&4``Ž$Sx1‹(#˜!‹"€ !9†0 1I´sƒi‹pˆ 51T)GlCXŔ3#đq““Žü 1—p!D"Ž " ˜&0Ž#Sg5ˆ#ˆ$%t@Ž0'’)9A“€ŽH2f@PŽT56•8 9$!„<C’e€ŽxIrUW W!r#Hgƒ’$`†Ž`ˆˆ   Ŕ!ŢA‡9ČŔ! ‰'3ˆţĐ6–°‡56a—ˆ ™I„v ˆÔ ™ž¤ Length DA: ˙˙ü†žf†‡Œ ĘČ!…Šn–m &I—”Šmv)T“r@ ™y”wtwŠ ††ÁH# †Dˆ%Đ6–°†,QBƒ–tP†+Y™'„Saˆ  ˜†7†:Š†S †XÄČ ŽCA˜C$H—ŠC3†‘†2`žB †›Ć@ Š††¨Œ˜† †]ĘŠoá9Ȏ †×ČČ …ŒV˜Á–”D7p†÷†Á‡‡A¨~ ‡ËČ#ŚB3Qaq‹‡•uW‰‰(Ł†„†„€’Ś2‡m2‰<‹.Y™8wUŽJ'† ‰‡Y2e0‹Y™ @€Œ°“‡uQŽ<‚e0a Y™qB…q@Ž ` @`ŽH—•‘€Ž$7uQˆ/ˆ0&S<0a"<2pŒT’…qBŒ`c&SŒlp@c0ŒxY6sPŒ„G•‘ƒ<6sF“<%Q<(W<$Ž<‘ƒg4Žx€a"DŽxi8wUŽxX2eŽxF“‡uŽx5†ýŽ´$†ńŽ´†ĺŽ´†ŮŽ´ †ăŽH˜•‘ƒg8˜†ƒiD˜•P˜”䘔x˜4†h˜"D‰y ˆlŒ—”x ˜”ü ˜&S ˜8wUŽ QŽc†%Œ$u†Ž ‡† ˆ;†<ˆCŒ ™†2† †’” †k…XUpˆ p„'QŠ\b€˜ˆ‹ˆU5h$I’)$E4TŽ<A@T€ŽH7˜ € 4—%S`223T)˜%x0'’PŽ„&4``Ž$Sx10Žœ#˜!‹"€ !9†0 1I´sƒi@ŽŔ51T)GlCXŔ3#đq““ü 1—‘ !D"‘ " ˜&Žœ#Sg5%t@Ž '’)9A“€Ž$2f@PŽ056•8 9$!„<C’e€ŽTIrUW W!r#Hgƒ’$`†Ž`ŠsŠx” ¨Đ)žQ†ű9ČŔ  ‰2ˆţĐ6–°‡”6a—ˆ ™#ŠÓY™ž¤ î DAG: ˙˙ü†žb ŽŰČČ"Œj–iv)T“r‰† Y™ &I—”Œuˆ)@6Š†3†7††ŠC‡ˇÇH  †† ÇH"ƒŠG@†Đ6–°†  #xyˆ* ™‰—†ƒ(† Y™<ŽCŽG†KÇ@!”]†fĘČ ƒ`ˆoF˜]WP’@  †iž]†ŠČČ ŒU–Ÿp1$€q0ŠŸ (rC(YŠŸ™!—#™†Nˆ…†F9ČŔŽř1˜ći6a—‰ ™I˜GaŐšč î FDA: ˙˙ü†Ľ †´° &I—”‰q Y™v)T“r5‹Oy”wtr‘[ˆň–Z‡Ťř´†ʈ! † @†Đ6–°†Š†$`” – †A†FĆH  †XŔH#†Œ–T)hu(p†yY™’ugˆD ˜’SŠ††††˘ÄH#œ\ˆ´ˆ † †††t2Ŕ@ ’*†ŢÁH#Š˜C–Ű@6†@•‡ŒŰ‘H—Adˆ‚ŠĎ˜‡†Šƒ@–߈ˆƒˆđˆ”sˆŽ"!‡]ĹHž‘ˆ8"Š Ťˆ8"#†Z5šZ †$‡ŤĘŔ!”ă-íëx^2-1Y6sF”Y™P†† ˆ  ™$H—•’†$ ˜†%†-!Ä@  /†?W–2g”‰f@†&†TŘ' †††'†r.ɘ(†„3ČŔ–$Ž4ˆ_)˘0ˆˆ4*†8.ČH!œhˆˆdˆ<ˆ"ˆ@ˆˆ+†š1˘šˆPH–vžƒˆˇ††††††††††† † †††^2ˆY#‘k[‹l('step1: construct two curves, C1 and C2.Ž/,step2: construct a tangent to† curve at C2.[Kˆ43œ4perpendicular lineˆ?the’NŠ†H a given pointSMˆS4: sele†„ˆ8inters† ion betweenŠQnormalŒdand C1 (**sometimes weŽ¨can†.t find this using CP)yRˆy 5: animateŠaŠš A (which†P•eˆHœÔ) along)•&?ˆZ6‰ZlŠÔ(x,y)‡hordinˆkof’jafterŠöŒˆ†ó‘m/9step7: collect distance DA after the animation.[[Œ VNote. If we dragŠ0x-coordinate of †Lndš\back toŠ^Geometry Strip,ˆJŽ^Ushall see a scatˆ‘plotˆUˆ™’ŻfuncˆĄ†>hereˆŽcan†×nj†×uˆŠˆÎŒĂmi†Öumš— may occuré‡ umn 1=x valuŒĎŽ!Ž2=“7 between DŠňA['5 -0.4275510204I 1.850358557–& 163265306’&708ˆ<11 -0.4051020408 1.6203809Ž †% 393877551Œ$ 1.553568027Œ% †J 3826530612’&49922981Ž8Š&714285†’L 453454149–L6Œ”16’r†!09054Ž—Šr48979591Žź†˜379ˆťÎŠ˝3Šť0”—34972553˜˝źŽ&‡32321533–ↈŕ2”ź29982578˜–Œš3‘†– 279218025•.292Žö‡T26113460Žô‰T2ŠI‡W•z2‡/7831ˆ˜ -0.2704081633† 1.231798212ŒŠ& 591836735”&2028005˜&479Š)7”&10739866–LˆH46939Œ& †r031194ŽŠ— 255102041’—1973†„94–—14285714”˝1935†Ť1˜q03061224”˝19152947ŽĎ‰ 1Žâ4”˝19144358–˜18ŒKŹ†˝1933082@‰T1†Ý877†Â–%71†ň˜%5‰v265—y‡9707?‰S1‰)ŠLŸ1.211444221 -0.1357142857Œ †&22098263–& 244897959Œ& †L3536735–K13265306Ž_ˆK51517744–q0204081_ˆK708928Ž–†–09Š"ˆO”—93941938–'ˆ’18†ˆŽa†ž32126640Ž™ŠN6ˆ#3469–'†ž9538˜'‘ 1–N924218ÁŠœ4Žtü‡343926580ŠĂˆp38775–'9725557˜'2(6“5721722’ę -0.01224489796 1.678313929Œ-1.020408164îěë3’* 900660868[ˆ> Conjecture:†ˆPHWhen the angle FDA =Ž GHD= 90 degrees, †will achieve its minimum.Žc(S†lateŠTshortest distance function:\ŒĐ=Stat Editor (after†pleting some entries contains’L=0)ԉ,‡Ę‰(NFinaFormX$N†Graph2D†| 3Š ˆLISTSYSˆœ4ˆ< Modify ЈPˆ<STATCALC ôˆdˆŒ< ü\ˆx S9equenceˆŒ,ˆxSheetˆO„|’ŠŒˆiŒolveEqˆˆ´†wr€ˆ´(UpˆŒ’tupFLG1˜(Š<†Lis†{ŔD‰ˆPic†ÜViewWind(ˆŒ_osve†v‰EŠxy†^‰P–ˆ¸-†† a‡– MatDatab‡˘.EAC‹ŠŠ Š ‡˝ ¤×^††× 1Œj‰W4††Đ<ˆ5–<ŠDĐxö<ţxţ´ţđÓ,asystem`=@ˆ Œ†0'ä]Setu,ä^äö<˛<††† †ŒžŒ¤’ ’,† †0†!@†@PSheet1|đ‘|:˜^Š2ž3ž 4†5ž ä8ä7Ę|č†]–|œš" ˜| “i™ˆ.†ƒ! ™†ˆ’†Š( ^E †Š†††Ł†$’+’4˘œ†]D˛<’h†(œž+†Ë/ 3x12.60547198*x^2+5.271331881*x+1.71141939ŠcŽ?`– PýĆ$´$“`– p @9b6‡x Y™2e0Y`‡Š ˜‘jˆŘ5‡‡s —–2Œđ%28‰F†Ýq†’ ˆ(1…0qy`ˆ#Y‡uYƒ†* ˜–0– šHŽĆ$ĘH(W( †ş Y™p–R˘ŔšÍŽ°´âIâz⫓Ś'Œ¸‡¸ ‡ź$†0†<†H†T†`†l†x†„††œ†¨†´†Ŕ†̆؆ä†đ†Wü ,8DP\ht€Œ˜¤°źČ'U @Y™2e0`Ž †€Œ“‡uQˆ#†$‚e0a Œ0qB…qŽ<` @ˆ H—•‘<7uQ†&S†90a"Žx2pŒ„’…qBŒc&SŒœp@c0Œ¨Y6sPŒ´G•‘ƒ†J6sF“†Q%Q†X‰`†_$†v ‘ƒg4Žx€a"Dˆ i8wUŽxX‰ NF‡ PY™5qB…pŽ $H—•Ž&SŽ$0Œ0 2eŒ ˜•‘ƒg@ŠH˜ƒg4iŽ †Sq†38Y6sŒl˜F“‡uŒx˜4i8w`Š`˜"D‰y ˆlŒT—'†§† ††$†0†<†H†T†`†l†x†„††œ†¨†´†Ŕ†̆؆ä† đ†Jü†† †,†8†D†P†\†h†t†€†Œ†˜†¤†°†ź†Č…XU‹<p„'Q‹0ˆ b€˜‰_ˆU5h$I’)†E4T‹l†+#A@T€ŽH7˜ € 4—%S`223T)˜%x€ ‰k†U'’P‡k&4``‡k$Sx10‡k0#˜! Ž0"€ !9†0 1IˆSˆTsƒi@Ž`51T;)GlCXŽ„3#’<q““H 1—pŽ0!D"Ž´" ˜&’œSg5x%t†\'’)9A“€Žđ2f@đ56•8ü9$!„´C’e€œIrUW¨W!r#Hgƒ’Ŕ`†Ž`ˆˆ  '†† ††$†0†<†H†T†`†l†x†„††œ†¨†´†Ŕ†̆؆ä† đ†Tü†X† †,†8†D†P†\†h†t†€†Œ†˜†¤†°†Bź†@Č'U @Y™2e0`Ž †€Œ“‡uQˆ#†$‚e0a Œ0qB…qŽ<` @ˆ H—•‘<7uQ†&S†90a"Žx2pŒ„’…qBŒc&SŒœp@c0Œ¨Y6sPŒ´G•‘ƒ†J6sF“ <%Q<(W†$Ž<‘ƒg4Žx€a"Dˆi8wUŽxX‰ F‡ ˆ 5†ýŽ´$†ńˆ†ĺˆ†ŮŽ´ †ăŽH˜•†J‡t Y˜†Ti@ˆ qB…qŽ Y6sPŒF“‡u†4†j$`Œ0"D‰y ŒH—'† ††$†0†<†H†T†`†l†x†„††œ†¨†´†Ŕ†̆؆ä†yđ†Kü ,8DP\ht€Œ˜¤°źČ…XUpˆôp„'QŽ b€˜‰ ˆU5h$I’)$E4T‡e A@T€ŽH7˜ €†ę64—%S`223T)˜%x0'’‹`†„&4`‹T†$Sx10Žœ#˜! Ž¨#"€ !9†‰„  1Iˆ sƒi@†D51`)GP$CX03#’<q““H 1—pŽ0!D"Ž`" ˜&0Žl#Sg5x%t†'’)9A“€Žœ2f@„56•89$!„´C’e€œIrUW¨W!r#Hgƒ’Ŕ`†`&GR‡ '1ˆ‚ŒŔq8•ˆÚ  9EUeˆ‡. ™'I(a%‰9 —ŠŻ,Š´’ ŠőŚ`”23RHŒ<’e5‰yˆ” – œ˜`‡^‰[‰64We obtained the quadratic regression formula to be: Ž=12.60547198î’xŒT2Œ\G$+5.271331881î’x+1.71141939 and we fi†the minimum of†is function as follows:R†ŒfMin(12.60547198†gŒ12Œ9´‡ ,x)RŽP †OValue="Šu 1.16033†^90T…†ŒŽ,x=-Œ* 0.2090890325 2I–†ˇ ™[ŽŤVNote†ěatŠń number x= -˜: will be very close to‹#vˆ˜ c which‰=Š+ see†6‡ :[/†‰8Provˆx‡U Analytically˜(‰_C3Step 1. We define function f to be the lower half o††circle C1:Rˆ solve(xŒ2Ž+y’Œ+2î’x-† y+1=0,y)R†GKy=-ŽW-x ?ˆ:xŽ_+1,y=Â/1ŒľŒ´ ú(x)=-ÚEŒDdone[7‹U2™UgL representin†‡Ygraph‰TC2IŒOŽ“gŠ“x^2-¤m‰r9QStep 3. Suppose we call points A and D to be (c,g(c))Š(x,f(x† respectively, †X Swhich make†F†Ahe shortest distance. Thenˆt†uld haveˆ. perpendicularŠ}oth ŒZ=ˆK†? gent lines at’Šsimul†] eously. In †;er words, [Œ˘ ˆË-ˆĂŒ˛c-xŒź *g'(c)=-1ŠţÄ1/*f'(xˆ1ŠŇrefore,‰Lset upŠţ followings:R‡z*eq1:=ś•R*Œ'gŠˇxˆ|x=cŽ +1=0RŒ -2î’cî’ŽcŒ82Ž#-Ž -x’ŒˆAŽk+Žx-œs†2aeq2:=Œ g(c)-f(x)Œ~c†bŒ¸*ŽĂ”!’֘ːv+1Ž×ţҖҔ^-c”^Ď”˙ solve({eq1,‡},{c,x})‹”•–-2î’cî’cŽ 2-Ž -x’ŒˆAxŒ<+Žx-cŒ +1=0,+1Žjţv–v”^-c”^ÎśˆŁ[†,‰Note.  ˆ‰&U1) We notice that ClassPad can't solvˆese two equations but† is suggestsŒ@we rWIshould now switch to our favorite computer algebra system. Nevertheless, †  geometric Cintuition an†eumerical estimaŠpresented here wo†S m††6ing.[ŒK/;2) We obtain from Maple that c=-.1969444377; x†6335161712[Œ‘In o†łrˆhdŽśpo†§s Ž( ˜?ŒĚf(˜)ŽRŠÚž=”Œ g†=”Ą)Œ<  will makˆÇe5‡“rtest distance DA‘#3) The shortest distance isR†!@(-.6335161712+.1969444377)^2+(g†–-fš/)†$R"ŠL 1†?1244195DPG†j[Œk[Œv4) If we drag both Ž! ˜–ŒŞž|ŽK and ž=”ÇŒçžÉŒ<  back to t‡+ geometry y˘÷ĸŒ‡ťÔôŽÝ[K¸\Öřk'/œXżű‘vb?vbO;ąˇœXΉýԉwbo;ą—Ř;NěU'ö3'ć\OK˙čÄžWŽę5óýěhß`Żş˜¨÷ľ^9:5—_’ú‘$ďyjzŹ‹V=âň•ŰÜjtŒžtĆe_¨:uĆe?¨÷@gĐÁ+ŻźňĘëżö×Ę{J@˝ĺîęÝŇ;ÚëRoľ›ś>¸ŮUmIeđ“žĎTmO+§ßsYË~†—őVîm$ŐŤăQ’ÓĘ\ŔçĚ”–•Ô!ŠËl_Ç\O_÷M?}˙ěă.s<^3ž¨O֌§jĆe[%i”Z?Ńś]ďť]ŽŞm/OԌ'kĆS5ăŐŰ~§ç*Ă+ć\šćÜ&Ę°:­ËťĂ~}üY—kë˛ňÍŁ]żM’œ–ă'VuÍXt÷ÂýšpR9w˛{ď^ź>%e% KšT”ô˛¤×$]•ô–¤÷$]÷H%Éňš,ŻÉňš,ŻÉňš,ŻÉňš,ŻçrâSĺv†˛UĂA—î´_ƒÚ4OeXŤŤŽĚ_:gÎÂa˙œ{˘jý7ŸżšŻ%ö|ôřxVĹ˝köŸ9š˙9ľžŮŞüTçmÁzź‹ÄoŐ<ˇĘbŰú×nąşŞÄł Đ3ŠÉqóŘp†ƒŽĹâŐëßÜ;Ôg¨gA™=.×ióŮľłEíeŰąů‘‡wm}XŽxýe VąŁƒŁÝă#2Şż*GÜË*śűäŃžŢ3_;ŐňjYő~lâKÔł–ĂfL=Wśbęyb‡íľ]ť–¨g´ŻZą“Ă›ťöË;ÚRsť^;śmçv{ľS9mßšăá‡ţË,“wďŃţcGőđˇHČ ¨Ç‰í÷Y3ŞďAT/g–Űoţ›ďƒář‹9´>RZűLiC¤”b‘R"RJFJŠHËÁŘƃąht}¤Ĺz/ö§ëośžDy}‰-˛Ćx¤ĺką_ł–ˇ?0Pů[°ÜÉ#GzOŒű[FŹĘtő$=ŹĽ;ĺ/ÝýLiM¤ôV¤Ľü|ýŘ)ëÁé óŞ<”‚Ď”VDJ?}Ćz*dÍł­ˇoôŘ Š9˘žáZąžŢ'‡üĽű"ĽH¤´1b?(ľŚm>vt°o$źő¸żt +"á/m”śFJŰ"ĽÎHiw¤Ô)őDJ;ž)=ié=şąď膸”ydŕř‡G¤v[ĘKÖîŸęcßíQO¸Uˇźbx˝ćÇ-S!-!éu#¤rĄ:ű“–\čöď-ü Ĺř’ŰřŃ˘5çÚUŸłŹ4öhőš–\(ßî[ÖŹ>`QÓ+ŸŻLíZ$7ć5wž1%šýUŮÖÓ3ĄŠźl™ ůŹĎSŒy-˜›×=ޢdÂü$ĺŠń#= [ô=Ăš6?Zę|zź*km™j7B#Ć×őŕűFżśeғ šťűŒżŇ—GŠďŞ>2é’mŽĚĺ˝ć'&ĆS’ĺCĹ1˝œúŻgzôĺcĹź^Z>ŇN_ÖK 2P,ÎëĽF¸\|S/…fB3ďęĽ; 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