I9Odd   1Courier NewSymbolExtendedBxyx1Courier New1Courier NewTimes New RomanTimes New Romanxyx1Courier New1Courier New1Courier New1Courier NewIPDIPDIPDIPD { d< 22pˊ@ˊˢ@˖@˖??X?h!?@@@@@@033333??@^ߛOw????p`oL:@LPZPZ@@?@ @@@GimN?@@ 2??@Nw!@??????? T $4wl1/.5(ka(r)+kb(r)+kc(r))*(t-sin(t))*h(theta(r)-t)11-.5(ka(r)+kb(r)+kc(r))*(1-cos(t))*h(theta(r)-t)(x,y) = (.5(ka(r)+kb(r)+kc(r))*(t-sin(t))*h(theta(r)-t),1-.5(ka(r)+kb(r)+kc(r))*(1-cos(t))*h(theta(r)-t)); 0.000000 <= t <= 6.283185OO2??@l, PS@????????l, PS@Нѝҝӝԝ՝֝r0pi/4(x,y) = (r,0)22??@?????????? $(,048$|01pi/4(x,y) = (0,1)QQ2??@l, PS˶????????l, PS˶ם؝ٝڝ۝ܝݝa-0-(r/4-1)*hvs(r-4)pi/4(x,y) = (0,-(r/4-1)*hvs(r-4))QQ2??@l, PS˶???????<@DHLPTĴo,sl010-(r/4-1)*hvs(r-4)#seg (0,1) to (0,-(r/4-1)*hvs(r-4))QQ2??@l, PS˶l, PS@l, PS˶??????ޝߝ<䘫`0-(r/4-1)*hvs(r-4)r-(r/4-1)*hvs(r-4)3seg (0,-(r/4-1)*hvs(r-4)) to (r,-(r/4-1)*hvs(r-4))QQ2??@l, PS@l, PS˶????????l, PS@l, PS˶X\`dhlp`Ԩar-(r/4-1)*hvs(r-4)pi/4(x,y) = (r,-(r/4-1)*hvs(r-4))QQ2??@l, PS@l, PS˶l, PS@?????? ,xì|.r-(r/4-1)*hvs(r-4)r0#seg (r,-(r/4-1)*hvs(r-4)) to (r,0)QQ2??@??????tx|++$BDDJ0*hvs(1/sqrt(3)-r)1*hvs(1/sqrt(3)-r)r*hvs(1/sqrt(3)-r)0*hvs(1/sqrt(3)-r)Wseg (0*hvs(1/sqrt(3)-r),1*hvs(1/sqrt(3)-r)) to (r*hvs(1/sqrt(3)-r),0*hvs(1/sqrt(3)-r))QQ2??@?????? 8ptк$0*hvs(r-1/sqrt(3))*hvs(2+sqrt(3)-r)$1*hvs(r-1/sqrt(3))*hvs(2+sqrt(3)-r),1/sqrt(3)*hvs(r-1/sqrt(3))*hvs(2+sqrt(3)-r)$0*hvs(r-1/sqrt(3))*hvs(2+sqrt(3)-r)seg (0*hvs(r-1/sqrt(3))*hvs(2+sqrt(3)-r),1*hvs(r-1/sqrt(3))*hvs(2+sqrt(3)-r)) to (1/sqrt(3)*hvs(r-1/sqrt(3))*hvs(2+sqrt(3)-r),0*hvs(r-1/sqrt(3))*hvs(2+sqrt(3)-r))QQ2??@??????@*8-ti׵.(1/sqrt(3))*hvs(r-1/sqrt(3))*hvs(2+sqrt(3)-r)$0*hvs(r-1/sqrt(3))*hvs(2+sqrt(3)-r)$r*hvs(r-1/sqrt(3))*hvs(2+sqrt(3)-r)$0*hvs(r-1/sqrt(3))*hvs(2+sqrt(3)-r)seg ((1/sqrt(3))*hvs(r-1/sqrt(3))*hvs(2+sqrt(3)-r),0*hvs(r-1/sqrt(3))*hvs(2+sqrt(3)-r)) to (r*hvs(r-1/sqrt(3))*hvs(2+sqrt(3)-r),0*hvs(r-1/sqrt(3))*hvs(2+sqrt(3)-r))QQ2??@e?@??????? $(,0444t|`0*hvs(r-2-sqrt(3))1*hvs(r-2-sqrt(3))!r*(2/sqrt(3)-1)*hvs(r-2-sqrt(3))$-(r*(2-sqrt(3))-1)*hvs(r-2-sqrt(3))vseg (0*hvs(r-2-sqrt(3)),1*hvs(r-2-sqrt(3))) to (r*(2/sqrt(3)-1)*hvs(r-2-sqrt(3)),-(r*(2-sqrt(3))-1)*hvs(r-2-sqrt(3)))QQ2??@?@ek, PS@e??????Ĺ۵Ğdݵ䂯լ!r*(2/sqrt(3)-1)*hvs(r-2-sqrt(3))$-(r*(2-sqrt(3))-1)*hvs(r-2-sqrt(3))r*hvs(r-2-sqrt(3))$-(r*(2-sqrt(3))-1)*hvs(r-2-sqrt(3))seg (r*(2/sqrt(3)-1)*hvs(r-2-sqrt(3)),-(r*(2-sqrt(3))-1)*hvs(r-2-sqrt(3))) to (r*hvs(r-2-sqrt(3)),-(r*(2-sqrt(3))-1)*hvs(r-2-sqrt(3)))QQ2??@l, PS@el, PS@??????8<@DHLPg}|Rr*hvs(r-2-sqrt(3))$-(r*(2-sqrt(3))-1)*hvs(r-2-sqrt(3))r*hvs(r-2-sqrt(3))0*hvs(r-2-sqrt(3))hseg (r*hvs(r-2-sqrt(3)),-(r*(2-sqrt(3))-1)*hvs(r-2-sqrt(3))) to (r*hvs(r-2-sqrt(3)),0*hvs(r-2-sqrt(3)))QQ2 2??@$$$@???????@?Ⱥ̻мԽؾܿ!V@ݶ%(1+m(r)/r*x^2-x/r*(1+m(r)*r))*h(r-x)Ey = (1+m(r)/r*x^2-x/r*(1+m(r)*r))*h(r-x); 0.000000 <= x <= 20.000000QQxyzJXT@l, PS@@@@@@@@@@@@@@@@@@@K7A@@@@@@@@@n Ln Ln LPI+X-PI/8*X^2+(PI^2/32-1/6)X^3+(PI/12-5PI^3/512)*X^4TTWO8¡(7/120-5PI^2/128+7PI^4/2048)*X^5 8TTWOA¡(-311PI/5760+7PI^3/384-21PI^5/16384)*X^6-0.00374019394*X^7TTWOB¡.00223180547*X^8-.00096099957*X^9+.0002715983*X^10٭TTWOC4áTTWO(X-PI/2)+TTWOA(X-PI/2)+TTWOC(X-PI/2)THETBáHVS(X-PI/4)*HVS(PI-X)*THETB(X)X)THETABá2PI-2*SQRT(PI/X)-PI*SQRT(PI/X)/(3X)+2PI/(3X^2)TTHREE0ġ(3X -9/10*X^3+729/1400*X^5)*HVS(PI/4-X)THETAAġTHETAA(X)+THETAB(X)+HVS(X-PI)*TTHREE(X)471(x-pi/2)^3-0.04099THETA-pi/ġ4/(9X^2)+3/5+81/700*X^2-27/1400*X^4+22599/3080000*X^6t <= 6KASfWr,š1+X^2/4-PI*X^3/16+.1510990443X^4-.1137821991X^5KBAš.08402736939X^6-.06102302726X^7+.04369367357X^88@KBBš-.03091563799X^9+.02165728736X^10KBC(ơ(KBA(X-PI/2)+KBB(X-PI/2)+KBC(X-PI/2))KBS|ơX/PI+2/(3SQRT(PI*X))+SQRT(PI/X)/(5X)2/(9X^2)KCSơKA(X)+KB(X)+KC(X)KB(X-PI/2)+HVS(X)-PI/4)*KB(X-PI/2)K$ǡKBS(X)*HVS(3PI/4-X)*HVS(X-PI/4)KBxǡHVS(PI/4-X)*KAS(X)KAǡHVS(X-3PI/4)*KCS(X)KC ȡSQRT(X)/SQRT(ABS(X))Htȡ1.5323669440479-0.66016927346735/SQRT(X)-0.22362231884588/XMLȡ0.540205760122/X+0.522690675747*X-0.046079190492*X^3?MSɡ1.02766988-0.106475891X+0.0882426631X^2@MA<pɡ -0.01842006891X^3+0.00128246266X^4,*R@ ףp=@MBɡMS(X)*HVS(1.25-X)MSAʡ(MA(X)+MB(X))*HVS(X-1.25)*HVS(5.25-X)MAAlʡMSA(X)+MAA(X)+ML(X)*HVS(X-5.25)/SQRT(X)-0.22362231884588/XMa/La Max IuYA!QpMHM>@vI*$@GETATimes New RomanHub  xˡk as a function of r = a/Lr = a/L 朗l<@b؊!@.Go?be԰|@Times New RomanHub 0̡"L" Trajectory Path of Least Time: Beta = 0@h&?ơh@Qb35)^xfWTimes New RomanHub ̡Cycloid Path of Least Timee!U&|PA?dm6 @e36#eOTimes New RomanHub ͡Parabolic Path of Least TimeOOZ^5O/v?T.ɬ@nOTimes New RomanHub XΡ"Left Ramp" Path of Least Time: Beta > 0@4N?N9@(Op,ʼTimes New RomanHub Y(x,y) = (.5(ka(r)+kb(r)+kc(r))*(t-sin(t))*h(theta(r)-t),1-.5M1b@s*g@b'\@B6 ϡu(x,y) = (r,0)Mh!@ٽ@С(x,y) = (0,1)(1.1-r)*h(r-x)*(1-x(1/r+1.016712932)+1.0167129M˖@˖T08ѡ(x,y) = (0,-(r/4-1)*hvs(r-4))M˖@˖wğѡseg (0,1) to (0,-(r/4-1)*hvs(r-4))M˖@˖T0ҡseg (0,-(r/4-1)*hvs(r-4)) to (r,-(r/4-1)*hvs(r-4))?M˖@˖wğ`ӡ(x,y) = (r,-(r/4-1)*hvs(r-4))?M˖@˖T0ԡseg (r,-(r/4-1)*hvs(r-4)) to (r,0)/4-1)*hvs(r-4))?M˖@˖T0ԡseg (0*hvs(1/sqrt(3)-r),1*hvs(1/sqrt(3)-r)) to (r*hvs(1/sqrtM˖@˖BfVա seg (0*hvs(r-1/sqrt(3))*hvs(2+sqrt(3)-r),1*hvs(r-1/sqrt(3))*M˖@˖"S+@֡ seg ((1/sqrt(3))*hvs(r-1/sqrt(3))*hvs(2+sqrt(3)-r),0*hvs(r-1M˖@˖e%֡ seg (0*hvs(r-2-sqrt(3)),1*hvs(r-2-sqrt(3))) to (r*(2/sqrt(3)M˖@˖"S+ס seg (r*(2/sqrt(3)-1)*hvs(r-2-sqrt(3)),-(r*(2-sqrt(3))-1)*hvsM˖@˖e%hء seg (r*hvs(r-2-sqrt(3)),-(r*(2-sqrt(3))-1)*hvs(r-2-sqrt(3)))M˖@˖"S+ ١y = (1+m(r)/r*x^2-x/r*(1+m(r)*r))*h(r-x); 0.000000 <= x <= 2M˖@˖sl0I?