{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "den" -1 256 "Times" 1 14 0 0 0 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "den2" -1 257 "Courier" 1 14 0 0 0 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" 256 258 "" 1 24 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 256 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 256 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 256 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Palatino" 1 12 0 0 0 1 2 2 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 255 1 2 2 2 0 0 0 0 0 0 1 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Palatino" 1 18 0 0 0 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 3" -1 258 1 {CSTYLE "" -1 -1 "Palat ino" 1 18 0 0 0 1 2 2 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 4" -1 259 1 {CSTYLE "" -1 -1 "Palatino" 1 12 0 0 0 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 \+ Font 5" -1 260 1 {CSTYLE "" -1 -1 "Palatino" 1 12 0 0 0 1 1 2 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 256 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 256 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 256 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 256 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 261 "" 0 "" {TEXT 258 36 "plot (parametric and po lar plotting)" }{TEXT 256 6 "\n\nThe " }{TEXT 257 4 "plot" }{TEXT 256 445 " command can be used for many things besides plotting graphs of f unctions. This section describes two of them. Both of these produce pl ots of curves in the plane, but the curves can be more general than pl ots of functions. They are plots of parametrically-defined curves (i.e ., each of x and y is given as a function of a third variable -- the p arameter) and polar plots -- where the polar coordinate r is given as \+ a function of the polar angle " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" } {TEXT 256 4 ". \n\n" }{TEXT 259 19 "PARAMETRIC PLOTTING" }{TEXT 256 610 " \nWe begin with parametric plots. We will use the letter t for t he parameter, but any variable name is permissible (as long as it has \+ not been previously given a value). A parametric curve is specified by giving x and y as functions of t. In Maple, three pieces of informati on are needed:\n (i) the expression which specifies x as a \+ function of the parameter\n (ii) the expression which specif ies y as a function of the parameter\n (iii) the range over w hich the parameter is to vary.\n\nFor example, to plot the unit circle , we can specify x=cos(t) and y=sin(t) as t ranges from 0 to " }{TEXT 257 4 "2*Pi" }{TEXT 256 110 ". In Maple, these three pieces of informa tion are organized as follows (the square brackets are obligatory ):\n " }{TEXT 257 46 " [cos(t), sin(t), t=0..2*Pi]" } {TEXT 256 54 "\nSo to get a plot of the circle, we use the statement: " }}{PARA 268 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}{PARA 262 "> " 0 " " {MPLTEXT 1 0 52 "plot([cos(t),sin(t),t=0..2*Pi],scaling=CONSTRAINED) ;" }}{PARA 13 "" 1 "" {GLPLOT2D 137 121 121 {PLOTDATA 2 "6&-%'CURVESG6 $7S7$$\"\"\"\"\"!$F*F*7$$\"3w\"4hRPij!**!#=$\"3Ikwb#=y_O\"F/7$$\"3E8J# ))4-Qn*F/$\"3%[#\\ff*)GLDF/7$$\"3-N5')yke[#*F/$\"3gLj&[K5J!QF/7$$\"3?g oz=42`')F/$\"3j9NXR4U7]F/7$$\"3Sb](G._U!zF/$\"3t_H-qceDhF/7$$\"3_\\$R' oTd#3(F/$\"3!GO#*3qU&fqF/7$$\"3?H$>jubk6'F/$\"3!\\@@&\\!>8\"zF/7$$\"3V GuIc:x5]F/$\"3-$>p(Qh-a')F/7$$\"37C3nW'R,#QF/$\"3aK+16bcT#*F/7$$\"3$oY @iF'QDDF/$\"3s&Q\"[M\"oen*F/7$$\"3OB^hAo8X8F/$\"33)fVPO<\"4**F/7$$!3+q B/u(p5(f!#@$\"2%HhJ<#)******!#<7$$!33)\\T#fB[i8F/$\"3ap%>wGZn!**F/7$$! 3)e:d.:#=YEF/$\"3PA>=\\D`V'*F/7$$!3%*yV1J*3Jx$F/$\"3IzF#e`m3E*F/7$$!3V (\\AOF:B/&F/$\"3vU'yI2&oN')F/7$$!3[igNw%*[RgF/$\"3!G;)RQ+BqzF/7$$!3/XU wYjJ*3(F/$\"3AfvOg?x_qF/7$$!3&=e_b?/U!zF/$\"3u3HvXQF/7$$!3!p!RS4Nij' *F/$\"3o$p9m#Q%=d#F/7$$!3#p(pT>+.2**F/$\"31V?00]Ug8F/7$$!3/gKG4>&***** F/$\"3vCA\\O%485$!#?7$$!3__Is=!Q%3**F/$!3q#4*>c=8]8F/7$$!3)>b`sYlSn*F/ $!3UNbFGIGKDF/7$$!37_J.8AEj#*F/$!3q4&=j`Bsw$F/7$$!3%\\F)4Kh=u')F/$!3kE NX')3zv\\F/7$$!3EB*z7xng%zF/$!37W(H!pUCrgF/7$$!3XD84Pfc5rF/$!3Y?#o?\"y MJqF/7$$!3!Q%y6Ike[gF/$!3)4j2yeGL'zF/7$$!3g@UD=%)o!*\\F/$!3I_Mh6Mil')F /7$$!3o+1s\"[\"ysPF/$!3#\\IN,%***4E*F/7$$!30yCH\"el'3EF/$!3=V>(fp[Pl*F /7$$!3g67Cvnc\"H\"F/$!3;$RoI*>C;**F/7$$!3EzyOHMQdIFas$!3W1O#>E`*****F/ 7$$\"3GIAj`Ks(G\"F/$!3lYZ3X=u;**F/7$$\"3EKRqX8/bDF/$!3U$[o%H'z!o'*F/7$ $\"3@%)yb&Q)zNQF/$!36.2<#>x]B*F/7$$\"3O7w4w0I.]F/$!3wHgb5wMe')F/7$$\"3 @\\#)[*R]$4hF/$!3/a*[=H2o\"zF/7$$\"33/wY'Q+$*4(F/$!3)4d)eg?sUqF/7$$\"3 [f=s$Ry,!zF/$!3D1$H)zD(p_v\\F/7$$\"3!G $e@`4+D#*F/$!3i&>2ndo*fQF/7$$\"31mGAp))oa'*F/$!3%)f]nRQ=0EF/7$$\"3@zb' f`bp!**F/$!3/up91t'4O\"F/7$F($\"36YKhSr8/#)!#F-%'COLOURG6&%$RGBG$\"#5! \"\"F+F+-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6$Q!6\"Fg[l-%%VIEWG6$ %(DEFAULTGF\\\\l" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 263 "" 0 "" {TEXT 256 4 "The " }{TEXT 257 19 "scaling=CONSTRAINED" }{TEXT 256 206 " option in the plot state ment tells Maple to use the same scale on the x and y axes -- otherwis e the circle would look like an ellipse. \n\nNotice that parametric pl otting changes the \"what\" in the standard\n\n" }{TEXT 257 15 "plot(w hat,how);" }{TEXT 256 393 "\n\nsyntax. \"What\" now is a parametric ex pression (or possibly a set of them, enclosed in braces -- you must be careful to keep the grouping symbols straight!), and \"how\" is used \+ only for special options such as scaling, or colors or titles (see the Maple help on plot options for all the variations that can go here). \n\nHere is an example with two curves plotted on the same axes (and a title):\n" }}{PARA 264 "> " 0 "" {MPLTEXT 1 0 85 "plot(\{[t^2,t^3,t=- 2..2],[t*cos(t),t*sin(t),t=0..10]\}, title=`Two parametric curves`);" }}{PARA 13 "" 1 "" {GLPLOT2D 207 169 169 {PLOTDATA 2 "6'-%'CURVESG6$7W 7$$\"\"%\"\"!$!\")F*7$$\"3&*GdQIK_FQ!#<$!3sEI\"HB)=)[(F07$$\"3A!e4#)QZ )eOF0$!3w?n/M\\o)*pF07$$\"3xVC;(zv^^$F0$!3MqoPKN`!f'F07$$\"3'*[e7a'F07$$\"3I_xvy7AoIF0$!3#R^x(o.Su`F07$$\"3AdyQ%yLZx#F0$!3 V-BOJ<-AYF07$$\"3eCFF07$$\"31:$3_Ay9N\"F0$!3G\"flSFN 6d\"F07$$\"3UlK?Jn;z6F0$!3aA8G&f]/G\"F07$$\"3mq\"))\\%))R#***!#=$!3sD_ TM/g))**F\\p7$$\"3ND5!QdEbL)F\\p$!3m+v\"*e6E5wF\\p7$$\"3yo4Oxt$3)oF\\p $!3V-)Gda5xq&F\\p7$$\"3[Bt(zL,1o&F\\p$!3g&=[p#eX\"G%F\\p7$$\"3%37I_u2I S%F\\p$!3tcMg&*Hi@HF\\p7$$\"39$[s$3d(yW$F\\p$!3pU[Q^[aC?F\\p7$$\"3W.`j Pjd$[#F\\p$!3UdHo0DqP7F\\p7$$\"3'*4!*RKWoh7$$\"3. _6\"F\\p$!3MR*HP'\\s2PFbr7$$\"3CH$*>9#z`J'Fbr$!3ia=g573(e\"Fbr7 $$\"38sw$4j*)>u#Fbr$!3^9sTi![/a%!#?7$$\"37a/+x'ova(Fbs$!3Ia6KI>3dl!#@7 $$\"3gBHSG34)*Q!#B$!3)GpZ66Gip(!#E7$$\"3[xsBc\")3LuFbs$\"361\\2^OY3kFh s7$$\"3)*)*GqVMScEFbr$\"3hd6ewH`HVFbs7$$\"3SBo$=Oul/'Fbr$\"3GKpeC&Ro[ \"Fbr7$$\"3+&>/'3\")G*4\"F\\p$\"3=*)f[FiuWOFbr7$$\"3q1NSD.>D#pF\\p$\"3G#[#H.1'*edF\\p7 $$\"3:3Cjqg!*=%)F\\p$\"3-,t\"pLLZs(F\\p7$$\"3))y\"p6+56'**F\\p$\"3qt$p Zv@<%**F\\p7$$\"3EMgH-E:Xn7F07$$\"3K8jd&Q?gN\"F0$\"3K$o a%GD1z:F07$$\"3k%GUH\"\\/k:F0$\"3%GjvwUAg&>F07$$\"3qtR@7\\Uy UY(zu]hF07$$\"3)zaASz?o]$F0$\"3Q7P(\\b]qc'F07$$\"3?')3\"\\D2*fOF0$\"3i k^*=9E<+(F07$$\"3`@xA@_1GQF0$\"39g+gH*[\">]]F\\p$\"3!e$4iO*fBh$F\\p7$ $\"3oK\"RMFl_V&F\\p$\"37#Hn!p(or%[F\\p7$$\"3c;\"p7y^Yg&F\\p$\"3OiQCD$e y>'F\\p7$$\"3h=qh)[!RF_F\\p$\"3G9!*)f!)*=)4*F\\p7$$\"3'\\)[\"p0ES'RF\\ p$\"3CR%)4pvu#=\"F07$$\"36gr&4^GPr\"F\\p$\"3&fI* )zy=/0?F0$\"3_TO0jeGF0$\"3_W)fiQ7MD'F\\p7$$!3%R(G78gr9JF0$\"3G77D#z1#\\!)Fbr7$$!37m Pl/P$[F$F0$!3pyIV9WLAmF\\p7$$!3?$)yb0q@mKF0$!3IMZT#Q#GW8F07$$!3qR=2f;i rIF0$!3y^&[LTc$e@F07$$!3fIO>vwdEFF0$!3qo5S!)>(*eGF07$$!3_*opC^lN;#F0$! 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Can you tell which curve is which ?\n\n" }{TEXT 260 14 "POLAR PLOTTING" }{TEXT 256 144 "\nTo plot the po lar coordinate (radius) r as a function of the polar angle (usually ca lled theta, but any variable name can be used), the option " }{TEXT 257 12 "coords=polar" }{TEXT 256 325 " must be specified in the \"How \" part of the plot command. In the \"what\" part, the same syntax as \+ parametric equations is used -- except the name of the angle variable \+ is the second function specified between the brackets. In polar coordi nates, the equation r=1+cos(theta) is a cardioid (we will use \"th\" i nstead of \"theta\"):\n" }}{PARA 266 "> " 0 "" {MPLTEXT 1 0 45 "plot([ 1+cos(th),th,th=0..2*Pi],coords=polar);" }}{PARA 13 "" 1 "" {GLPLOT2D 164 158 158 {PLOTDATA 2 "6$-%'CURVESG6$7`o7$$\"\"#\"\"!$F*F*7$$\"3U/Dh JW(H*>!#<$\"3IReVL=)oO\"!#=7$$\"3gH%QAR'*>(>F/$\"3_(eDVAsxr#F27$$\"3!* fzByihU>F/$\"3aU>^vT!z'QF27$$\"3;cUN!o/K!>F/$\"3c0/-lD%R)\\F27$$\"3))z &QsY?u%=F/$\"3EwU@z#>j='F27$$\"3)[\"eXm@A!y\"F/$\"35UYm(z\\/K(F27$$\"3 FN!*[$G_>q\"F/$\"31K!GP*48\"Q)F27$$\"3Zlp8bM19;F/$\"3!*z)[eK/(\\$*F27$ $\"3wL'4T@&>=:F/$\"34+f!\\L!G@5F/7$$\"3!)*fz_?(>:9F/$\"3[(e]oHSn4\"F/7 $$\"3'f8A#f!)[98F/$\"3e(eW*>>Yc6F/7$$\"3x3nS&)f))47F/$\"3d.n%yh^f?\"F/ 7$$\"3wS#)e\\Sn)4\"F/$\"3E3z`f&fhC\"F/7$$\"3?.w)f$)ev&)*F2$\"3kx7F/7$$\"3=JHP*RV:>%F2$\"3Av]TVHf\\7F/7$$\"3ORUjgQ9jJF2$\"3`<;Uf)R> @\"F/7$$\"3MM1xHh2E:F2$\"3%RAIG#H?C6F/7$$!39Axh1W]nf!#@$\"3=Slg[r-%*** F27$$!3?qqJTl%o<\"F2$\"3[g\\$fWqpb)F27$$!3qt-2`T&f%>F2$\"3ox&y$z\"y;4( F27$$!3k+m\\IQZ\\BF2$\"3_i>D72kmdF27$$!3GD=wT4#)*\\#F2$\"3E<3'))R+8G%F 27$$!3'**\\aG;Y>R#F2$\"3)[*QH%*zhcJF27$$!3#3P9.svM1#F2$\"31r4hM)QG0#F2 7$$!3ep]C$zfll\"F2$\"3uI3j-1\"QG\"F27$$!3W0zR=u%4;\"F2$\"392-!pelcq'!# >7$$!3=@#eEDy\"*4(F_u$\"3/!47K2Hw&HF_u7$$!3_(=NDw:1D$F_u$\"3qv(yv+y5l) !#?7$$!3rIiS&)Qc5#*Fju$\"3=vM/joyk7Fju7$$!3Z9U#oa[!4[!#B$\"3K&oKj$>W\" \\\"!#D7$$!3SY6aj@Os!*Fju$!3%o6(Qxu?O7Fju7$$!3]My#y+7J:$F_u$!3squ-c[e` #)Fju7$$!3%3nBHI&fCoF_u$!3Y@n:JfXvFF_u7$$!3!=npmiN+:\"F2$!3i.\"R5cspf' F_u7$$!3muDqo%o?j\"F2$!3gD#fsU#*pC\"F27$$!3k\")*=D9^X0#F2$!3&)H[j(eh;. #F27$$!3srs&)\\m/!R#F2$!3Ge$HVYSm9$F27$$!3WRXWH8***\\#F2$!3YjU)=x!)3M% F27$$!3o6#R3Z$R\\BF2$!3m(\\*\\Kq-ndF27$$!3(eZ!zp>:G>F2$!3VG!41]3a8(F27 $$!3I80v,BvC6F2$!3_4&=P7$\\N')F27$$!351P$f$e.[IFju$!3)y9E0dz$p**F27$$ \"3C[nUrja`9F2$!3#*3.DvVP>6F/7$$\"3L)eHN(\\'y?$F2$!3jZ`\"[RJQ@\"F/7$$ \"3XJF-6L82`F2$!3iy$f5mYxF\"F/7$$\"3Qg*zYA/0R'F2$!3'p)p>[8m$H\"F/7$$\" 3&H[v:C-m](F2$!3u'3Np\"z.*H\"F/7$$\"3%*\\FmAZqp')F2$!3!*f)Qd2OJH\"F/7$ $\"3%=(4')HmwT)*F2$!3S%='4FiMv7F/7$$\"3%H'[QI#e**4\"F/$!3SAvt\"ojdC\"F /7$$\"35^(>%)pIR@\"F/$!3ogS_+iD/7F/7$$\"3#*p&fGL)*eJ\"F/$!3'>+X^D6d:\" F/7$$\"3/$[)*e-YTT\"F/$!3M>1zP5V(4\"F/7$$\"3(GkS3\"*f3_\"F/$!3=&**32E3 \">5F/7$$\"3/@B2l]()>;F/$!3]yip+qY\"H*F27$$\"3w(*R*3O!y+F/$!3j+9h\"pSX$RF2 7$$\"3O[D^LKF/$!3*o1P\\cr#4FF27$$\"3O@Ad`*=I*>F/$!3K,eMBdbi8F27$F($ \"3A\\E7Gu#3k\"!#E-%'COLOURG6&%$RGBG$\"#5!\"\"F+F+-%%VIEWG6$%(DEFAULTG Feal" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " }}}}{EXCHG {PARA 267 "" 0 "" {TEXT 261 6 "Errors" }{TEXT 256 177 ": \+ The kinds of errors that can happen in parametric and polar plotting a re the same as those that occur in regular plotting. See the section o n basic plotting for more details.\n" }}}}{MARK "0 1 0" 8 }{VIEWOPTS 1 0 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }