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:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::>:;B:::::::::::::::::::::: ::::::::::::::::j:vC:;:::::::::N;?B:yyyxI:;Z::::::j;JDJ:::::::::::::ya y=J:B:::::::::::::::::::jysy:>:<:::::::::::::::::::vYxI:;Z:::::::::::: :::::::3:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 44 "Verification that the Two Formulas are Equal" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 109 "The denominators of th e two expressions are the same so we need only show that the two numer ators are equal. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "We will show that for any two vectors " }{TEXT 263 1 "a" }{TEXT -1 5 " and " }{TEXT 262 1 "b" }{TEXT -1 41 " the following alg ebraic identity holds:" }{XPPEDIT 19 1 "norm(crossprod(a,b),2)^2 = nor m(a,2)^2*norm(b,2)^2-dotprod(a,b)^2;" "6#/*$-%%normG6$-%*crossprodG6$% \"aG%\"bG\"\"#F-,&*&-F&6$F+F-F--F&6$F,F-F-\"\"\"*$-%(dotprodG6$F+F,F-! \"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "interface(warnlevel=0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "interface(showassumed=0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "interface(version);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%[pMaple~Worksheet~Interface,~Maple~8.00,~IBM~INTE L~NT,~Mar~19~2002~Build~ID~109153G" }}}{PARA 0 "" 0 "" {TEXT -1 19 " \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(li nalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "for i from 1 to 3 do \nassume( cat(a,i) , real): assume( cat(b,i) , real);\nend do;" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "a := vector([a1,a2,a3]); b \+ := vector([b1,b2,b3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG-%'vec torG6#7%%$a1|irG%$a2|irG%$a3|irG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"bG-%'vectorG6#7%%$b1|irG%$b2|irG%$b3|irG" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 48 "numeratorFormula_1 := norm( crossprod(a,b) , 2);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%3numeratorFormula_1G*$,(*$)-%$absG6 #,&*&%$a2|irG\"\"\"%$b3|irGF/F/*&%$a3|irGF/%$b2|irGF/!\"\"\"\"#F/F/*$) -F*6#,&*&F2F/%$b1|irGF/F4*&%$a1|irGF/F0F/F/F5F/F/*$)-F*6#,&*&F>F/F3F/F /*&F.F/F " 0 "" {MPLTEXT 1 0 67 "numeratorFormula_2 := sqrt(norm(a,2)^2*norm(b,2)^2-dotprod(a,b)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%3numeratorFormula_2G*$,&*&,(*$)-%$ absG6#%$a1|irG\"\"#\"\"\"F0*$)-F,6#%$a2|irGF/F0F0*$)-F,6#%$a3|irGF/F0F 0F0,(*$)-F,6#%$b1|irGF/F0F0*$)-F,6#%$b2|irGF/F0F0*$)-F,6#%$b3|irGF/F0F 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:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::>:;B::::::::::::::::: :::::::::::::::::::::::::::::::::::::::j:vC:;:::::::::N;?B:yyyxI:;Z::: :::j;JDJ:::::::::::::yay=J:B:::::::::::::::::::jysy:>:<::::::::::::::: ::::vYxI:;Z:::::::::::::::::::3:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 265 71 "Verification that the Third Formula is E qual to The First Two Formulas:" }{TEXT 327 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "numeratorFormula_3 := norm(matadd(scalarmul(b,no rm(a,2)^2), scalarmul(a,-dotprod(a,b))),2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%3numeratorFormula_3G*$,(*$)-%$absG6#,&*&,(*$)-F*6#%$a 1|irG\"\"#\"\"\"F5*$)-F*6#%$a2|irGF4F5F5*$)-F*6#%$a3|irGF4F5F5F5%$b1|i rGF5F5*&,(*&F3F5F@F5!\"\"*&F:F5%$b2|irGF5FD*&F?F5%$b3|irGF5FDF5F3F5F5F 4F5F5*$)-F*6#,&*&F.F5FFF5F5*&FBF5F:F5F5F4F5F5*$)-F*6#,&*&F.F5FHF5F5*&F BF5F?F5F5F4F5F5#F5F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "tes teq(numeratorFormula_3^2 = norm(a,2)^2*numeratorFormula_2^2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 " Code for " }{TEXT 266 9 "curv ature" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1694 "curvature := proc()\n\nlocal j, N, s, t, speed, vel ocity, acceleration, r0, r, cross_Product, crossProductLength;\n\nif n args <> 2 then \n error \"curvature expects two arguments.\";\nelif \+ not type(args[1], \{name,procedure\} ) then \n error \"curvature ex pects its first argument to be a name or procedure.\";\nelif not type( args[2],algebraic) then\n error \"curvature expects its second argum ent to be an algebraic expression.\";\nend if:\n \nr0 := args[1]:\nt : = args[2]:\n\nif not type(r0(s), \{ algebraic,list\} ) then\n error \+ \"curvature expects its first argument to be a scalar or vector-valued function.\";\nend if;\n\nif type(r0(s), algebraic) then\n N := 1;\n else\n N := nops(r0(s)):\nend if;\n\n\nif N < 1 or N > 3 then \n e rror \"curvature expects its first argument to be a scalar function or a vector-valued function with two or three components.\"\nelif N = 1 \+ then\n r := s -> [s,r0(s),0]:\nelif N = 2 then\n r := s -> [r0(s)[ 1],r0(s)[2],0]:\nelse\n r := r0: \nend if;\n\nN := 3:\n\nvelocity := []:\nacceleration := []:\n\n\nspeed := 0:\n for j from 1 to N do \n speed := speed + diff(r(s)[j],s)^2:\n velocity := [op(velocity), simplify(diff(r(s)[j],s))]:\n acceleration := [op(acceleration),simp lify(diff(r(s)[j],s$2))]:\n end do:\n\nspeed := simplify(sqrt(speed) ):\n if subs(s=t0,speed) = 0\n then error \"singular point encount ered.\";\n end if:\n\ncross_Product := convert(linalg[crossprod](con vert(velocity,vector),convert(acceleration,vector)),list);\n\ncrossPro ductLength := 0:\n\nfor j from 1 to N do\ncrossProductLength := crossP roductLength + simplify(cross_Product[j]^2):\nend do:\n\ncrossProduct Length := sqrt(simplify(crossProductLength));\n\nreturn simplify(subs( s=t,crossProductLength/speed^3));\nend proc;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%*curvatureGf*6\"6-%\"jG%\"NG%\"sG%\"tG%&speedG%)veloc ityG%-accelerationG%#r0G%\"rG%.cross_ProductG%3crossProductLengthGF&F& C4@(09#\"\"#YQAcurvature~expects~two~arguments.F&4-%%typeG6$&9\"6#\"\" \"<$%%nameG%*procedureGYQjncurvature~expects~its~first~argument~to~be~ a~name~or~procedure.F&4-F<6$&F?6#F7%*algebraicGYQ_ocurvature~expects~i ts~second~argument~to~be~an~algebraic~expression.F&>8+F>>8'FJ@$4-F<6$- FP6#8&<$%%listGFLYQiocurvature~expects~its~first~argument~to~be~a~scal ar~or~vector-valued~function.F&@%-F<6$FWFL>8%FA>F\\o-%%nopsG6#FW@)52F \\oFA2\"\"$F\\oYQarcurvature~expects~its~first~argument~to~be~a~scalar ~function~or~a~vector-valued~function~with~two~or~three~components.F&/ F\\oFA>8,f*6#F*F&6$%)operatorG%&arrowGF&7%9$-T#6#Fap\"\"!F&F&6$F/FP/F \\oF7>Fjof*F\\pF&F]pF&7%&FbpF@&FbpFKFepF&F&Ffp>FjoFP>F\\oFeo>8)7\">8*F aq>8(Fep?(8$FAFAF\\o%%trueGC%>Feq,&FeqFA*$)-%%diffG6$&-FjoFX6#FgqFYF7F AFA>F`q7$-%#opG6#F`q-%)simplifyG6#F^r>Fcq7$-Fgr6#Fcq-Fjr6#-F_r6$Far-% \"$G6$FYF7>Feq-Fjr6#-%%sqrtG6#Feq@$/-%%subsG6$/FY%#t0GFeqFepYQ8--%(convertG6$-&%'linalgG6#%*crossprodG6$-Fit6 $F`q%'vectorG-Fit6$FcqFcuFen>8.Fep?(FgqFAFAF\\oFhq>Fgu,&FguFA-Fjr6#*$) &FgtFcrF7FAFA>Fgu-F[t6#-Fjr6#FguO-Fjr6#-F`t6$/FYFR*&FguFAFeq!\"$F&F&F& " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 1 " \+ " }{TEXT 309 14 "Description of" }{TEXT -1 2 " " }{TEXT 267 9 "curvat ure" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 310 39 "c urvature - curvature of a space curve" }}{PARA 4 "" 0 "usage" {TEXT 311 16 "Calling Sequence" }}{PARA 0 "" 0 "" {TEXT 312 21 " curvatu re(r , t)" }}{PARA 4 "" 0 "" {TEXT 313 10 "Parameters" }}{PARA 0 "" 0 "" {TEXT 314 27 " r - name or procedure" }}{PARA 0 "" 0 "" {TEXT 315 30 " t - algebraic expression" }}{SECT 1 {PARA 4 "" 0 "info" {TEXT 316 11 "Description" }}{PARA 15 "" 0 "" {TEXT 268 9 "curvature" }{TEXT 317 53 " computes the curvature of a planar or space curve " }{TEXT 271 4 "s ->" }{TEXT 318 1 " " }{TEXT 269 4 "r(s)" }{TEXT 319 17 " at the point " }{TEXT 270 4 "r(t)" }{TEXT 320 2 ". " }}{PARA 15 "" 0 "" {TEXT 321 65 "The first argument may be a scalar valued fun ction in which case " }{TEXT 272 10 "curvature(" }{TEXT 322 1 "r" } {TEXT 273 3 " , " }{TEXT 323 1 "t" }{TEXT 274 1 ")" }{TEXT 324 78 " re turns the curvature of the graph of \ny = r(x) at the point (t , r( t)). " }}}{SECT 0 {PARA 4 "" 0 "examples" {TEXT 325 8 "Examples" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "f := x -> exp(x): curvature (f,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$expG6#,$*&\"\"#\"\"\"% \"xGF*F*#F*F),&F*F*F$F*#!\"$F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "f := x -> x^2: curvature(f,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#\"\"\",&F&F&*&\"\"%F&)%\"xGF%F&F&#!\"$F%F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "curvature(sin,Pi/2);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot(curvature(cos,x),x = -2*Pi .. 2*Pi); " }}{PARA 13 "" 1 "" {GLPLOT2D 563 218 218 {PLOTDATA 2 "6%-%'CURVESG6$7_s7$$!3)****>YH&=$G' !#<$\"2c***************F*7$$!3ANa4O(Hi9'F*$\"3BgynJ))pN'*!#=7$$!3dp3dx TF4gF*$\"3u]IyL]ff')F27$$!3a`M)z-#p\\fF*$\"3R$zxT+956)F27$$!3]PgRy)4,* eF*$\"3(GFy=+fD_(F27$$!3Z@'3)Gx_IeF*$\"3Z.nFLmN;pF27$$!3W07Azb%4x&F*$ \"3dsn#p-q3J'F27$$!3Agw]?#RRq&F*$\"3')QXA?dT[cF27$$!3,:TzhG$pj&F*$\"3c e'p*\\%3!>]F27$$!3!)p03.l#*pbF*$\"3DAvs!o`F*$\"3![8xpFET$HF27$$!3()o*[H1=JB&F*$\"3&) 3?NL2\\W@F27$$!3GX$QL=e))4&F*$\"3W7?@-1N)[\"F27$$!3gAxs.$)fk\\F*$\"3Q5 #)yx3T^#*!#>7$$!3(ozn)3A7S[F*$\"3qm::(Hb\"fXF^p7$$!3/sy+9hk:ZF*$\"3Wt8 kixd^6!#?7$$!3O=srF$ene%F*$\"3E)\\,Za?F[%F^p7$$!3ellUT0(yX%F*$\"35H2= \"o(QU$*F^p7$$!3v)))o%3YdCVF*$\"3')=1w8QS%\\\"F27$$!3!>@6bny7>%F*$\"3i [,)fm?l9#F27$$!3/U5IC-TeSF*$\"3GcjtG\\_BHF27$$!3/t34tat7T%F27$$!3SmDTR-())y$F*$\"3r/'))yP'o4]F 27$$!338MdsW`?PF*$\"3Y%zO!ovN^cF27$$!3wfUt0()>_OF*$\"31#*o!fVFtK'F27$$ !3W+k5D&3?f$F*$\"3[$3$RMVDRpF27$$!3:T&yWM==`$F*$\"3'*)*f\\\"Hr7b(F27$$ !3S\"o]Q;G;Z$F*$\"3fY-Q'[oT9)F27$$!35AGA$)zV6MF*$\"3a6(4\\;n\\p)F27$$! 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\"$F%F&" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 " Code for " }{TEXT 277 17 "unitTangentVector" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1104 "unitTangentVector := proc()\nlocal r, \+ t0, j, N, speed, velocity,unitTangentV;\n\n\nif nargs <> 2 then \n e rror \"unitTangentVector expects two arguments.\";\nelif not type(args [1], \{name,procedure\} ) then \n error \"unitTangentVector expects its first argument to be a name or procedure.\";\nelif not type(args[ 2],algebraic) then\n error \"unitTangentVector expects its second ar gument to be an algebraic expression.\";\nend if:\n \nr := args[1]:\nt 0 := args[2]:\n\nif not type(r(s), list) then\n error \"unitTangentV ector expects its first argument to be a vector-valued function.\";\ne nd if;\n\nN := nops(r(s)):\n\nif N <> 3 then \n error \"unitTangentV ector expects its first argument to be a vector-valued function with t hree components.\" \nend if;\n\n\nvelocity := []:\n\n\nspeed := 0:\n \+ for j from 1 to N do \n speed := speed + diff(r(s)[j],s)^2:\n ve locity := [op(velocity),diff(r(s)[j],s)]:\n end do:\nspeed := simpli fy(sqrt(speed)):\n \n\n if subs(s=t0,speed) = 0\n then error \"s ingular point encountered.\";\n end if:\n\n\nunitTangentV := simplif y(subs(s=t0, map(z->z/speed,velocity))):\nreturn unitTangentV;\nend pr oc;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%2unitTangentVectorGf*6\"6)%\" rG%#t0G%\"jG%\"NG%&speedG%)velocityG%-unitTangentVGF&F&C/@(09#\"\"#YQI unitTangentVector~expects~two~arguments.F&4-%%typeG6$&9\"6#\"\"\"<$%*p rocedureG%%nameGYQbounitTangentVector~expects~its~first~argument~to~be ~a~name~or~procedure.F&4-F86$&F;6#F3%*algebraicGYQgounitTangentVector~ expects~its~second~argument~to~be~an~algebraic~expression.F&>8$F:>8%FF @$4-F86$-FL6#%\"sG%%listGYQgounitTangentVector~expects~its~first~argum ent~to~be~a~vector-valued~function.F&>8'-%%nopsG6#FS@$0FZ\"\"$YQ]qunit TangentVector~expects~its~first~argument~to~be~a~vector-valued~functio n~with~three~components.F&>8)7\">8(\"\"!?(8&F=F=FZ%%trueGC$>Fao,&FaoF= *$)-%%diffG6$&FS6#FdoFUF3F=F=>F^o7$-%#opG6#F^oF[p>Fao-%)simplifyG6#-%% sqrtG6#Fao@$/-%%subsG6$/FUFNFaoFboYQ8*- Fgp6#-F_q6$Faq-%$mapG6$f*6#%\"zGF&6$%)operatorG%&arrowGF&*&9$F=T#!\"\" F&F&6$F,FaoF^oOFeqF&F&F&" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 1 " " } {TEXT 328 14 "Description of" }{TEXT 279 19 " unitTangentVector" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 329 57 "unitTang entVector - unit tangent vector of a space curve" }}{PARA 4 "" 0 "usa ge" {TEXT 330 16 "Calling Sequence" }}{PARA 0 "" 0 "" {TEXT 331 29 " \+ unitTangentVector(r , t)" }}{PARA 4 "" 0 "" {TEXT 332 10 "Parameter s" }}{PARA 0 "" 0 "" {TEXT 333 27 " r - name or procedure" }} {PARA 0 "" 0 "" {TEXT 334 30 " t - algebraic expression" }}{SECT 1 {PARA 4 "" 0 "info" {TEXT 335 11 "Description" }}{PARA 15 "" 0 "" {TEXT 280 17 "unitTangentVector" }{TEXT 336 53 " computes the unit tan gent vector of a space curve " }{TEXT 283 4 "s ->" }{TEXT 337 1 " " }{TEXT 281 4 "r(s)" }{TEXT 338 17 " at the point " }{TEXT 282 4 "r( t)" }{TEXT 339 2 ". " }}}{SECT 1 {PARA 4 "" 0 "examples" {TEXT 340 9 " Examples\n" }{TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT 658 8 "Curve 1." } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "r := t -> [t^2,t,t ]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "unitTangentVector(r,t );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,$*(\"\"#\"\"\"%\"tGF',&F&F'*& \"\"%F')F(F&F'F'#!\"\"F&F'*&F'F'*$F)#F'F&F.F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT 341 15 "Curve 2. Helix" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "r := s -> [cos(s),sin(s),s]:\nunit TangentVector(r,t);\nunitTangentVector(r,Pi/3);\nunitTangentVector(r,P i/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,$*&#\"\"\"\"\"#F'*&-%$sinG 6#%\"tGF'F(#F'F(F'!\"\",$*&F.F'*&-%$cosGF,F'F(F.F'F',$*&F(F/F(F.F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%,$*(\"\"%!\"\"\"\"$#\"\"\"\"\"#F+F)F ',$*&F&F'F+F)F*,$*&F+F'F+F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#! \"\"\"\"##\"\"\"F&,$*&F&F%F&F'F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 190 "arrowPlot := plots[arrow](r (Pi/4),r(Pi/4)+unitTangentVector(r,Pi/4),difference,width=0.015):\ncur vePlot := plots[spacecurve](r(s),s=0..Pi/2,thickness=3):\nplots[displa y](arrowPlot,curvePlot); " }}{PARA 13 "" 1 "" {GLPLOT3D 519 283 283 {PLOTDATA 3 "6'-%)POLYGONSG6-7&7%$\"+]KF'>#!\"*$\"+F!fJb%!#5$\"+Na\\&y (F-7%$\"+KA]6HF*$\"+\\SMmIF-$\"+1(\\Y5\"F*7%$\"+\\L.5HF*$\"+=7PCIF-$\" 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By\"=()*F3$\"+FVns[F97%$\"+wp'z'HF3$!+%*oS\\&*F3$\"+Eas8]F97%$\"+SQ)4G %F3$!+g8KP!*F3$\"+Dlxa^F97%$\"+\"))p*3bF3$!+$\\KdM)F3$\"+Cw#eH&F97%$\" +!HMvi'F3$!+VpP)[(F3$\"+B(yoV&F97%$\"+R'fWh(F3$!+))QG#['F3$\"+A)Hzd&F9 7%$\"+tJ9]%)F3$!+.oVZ`F3$\"+@4)*=dF97%$\"+rh)z6*F3$!+FlP1TF3$\"+??.geF 97%$\"+OOs/'*F3$!+eNv$y#F3$\"+>J3,gF97%$\"+qxo+**F3$!+F%QeS\"F3$\"+=U8 UhF97%F($\"+rNT?)*!#=$\"+<`=$G'F9-%*THICKNESSG6#\"\"'" 1 2 0 1 10 0 2 1 1 4 2 1.000000 43.000000 48.000000 0 0 "Curve 1" "Curve 2" "Curve 3 " "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 1 0" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" " Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT 342 9 "Curve 3.\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "r := u -> [exp(u), exp(-u), \+ u]: \nunitTangentVector(r,t);\nunitTangentVector(r,0);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%*&-%$expG6#%\"tG\"\"\",(-F&6#,$*&\"\"#F)F(F)F )F)-F&6#,$*&F/F)F(F)!\"\"F)F)F)#F4F/,$*&-F&6#,$F(F4F)F*F5F4*&F)F)*$F*# F)F/F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,$*&\"\"$!\"\"F&#\"\"\"\" \"#F),$*&F&F'F&F(F'F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 167 "a rrowPlot := plots[arrow](r(0),r(0)+unitTangentVector(r,0),difference, \+ shape=arrow):\ncurvePlot := plots[spacecurve](r(s),s=-1..1):\nplots[di splay](arrowPlot,curvePlot);" }}{PARA 13 "" 1 "" {GLPLOT3D 455 257 257 {PLOTDATA 3 "6$-%'CURVESG6%7$7%$\"\"\"\"\"!F($F*F*7%$\"+p-Nx:!\"*$ \"+2t\\EU!#5$\"+$p-Nx&F27%7%$\"+abB(\\\"F/$\"+P7vMdF2$\"+a@!)=YF2F,7%$ \"+w[_E9F/$\"+cWkF]F2F;-%&STYLEG6#%,PATCHNOGRIDG-F$6#7T7%$\"+7WzyOF2$ \"+G=G=FF/$!\"\"F*7%$\"+#zb?$QF2$\"+Y`c4EF/$!+Nn$=f*F27%$\"+e@q\"*RF2$ \"+Kp>0DF/$!+qMn$=*F27%$\"+9&**z:%F2$\"+'o-]S#F/$!+0-^v()F27%$\"+W\\AJ VF2$\"+lc\")3BF/$!+SpMn$)F27%$\"+yqm6XF2$\"+-cZ;AF/$!+vO=fzF27%$\"+oli *p%F2$\"+Q'Gy7#F/$!+5/-^vF27%$\"+&f;a*[F2$\"+rqsU?F/$!+Xr&G9(F27%$\"+$ Qj$*4&F2$\"+.\"H5'>F/$!+!)QpMnF27%$\"+\\n!=J&F2$\"+4')f#)=F/$!+:1`EjF2 7%$\"+p15LbF2$\"+4\\I2=F/$!+]tO=fF27%$\"+iQhjdF2$\"+ZD-NF/$\"+!Rj$*4& F2$\"+mQpMnF27%$\"+oqsU?F/$\"+,mT&*[F2$\"+Jr&G9(F27%$\"+N'Gy7#F/$\"+vl i*p%F2$\"+'R?5b(F27%$\"+)fvk@#F/$\"+%3n;^%F2$\"+hO=fzF27%$\"+ic\")3BF/ $\"+]\\AJVF2$\"+EpMn$)F27%$\"+$o-]S#F/$\"+?&**z:%F2$\"+\">5bx)F27%$\"+ Gp>0DF/$\"+k@q\"*RF2$\"+cMn$=*F27%$\"+V`c4EF/$\"+(zb?$QF2$\"+@n$=f*F27 %$\"+D=G=FF/$\"+ " 0 "" {MPLTEXT 1 0 1473 "principalUnitNormalVector := proc()\nlocal r, t0, \+ j, N, speed, velocity, principalUnitNormalV,\nunitTangentVector,normal Vector,normalLength;\n\n\nif nargs <> 2 then \n error \"principalUni tNormalVector expects two arguments.\";\nelif not type(args[1], \{name ,procedure\} ) then \n error \"principalUnitNormalVector expects it s first argument to be a name or procedure.\";\nelif not type(args[2], algebraic) then\n error \"principalUnitNormalVector expects its seco nd argument to be an algebraic expression.\";\nend if:\n \nr := args[1 ]:\nt0 := args[2]:\n\nif not type(r(s), list) then\n error \"princip alUnitNormalVector expects its first argument to be a vector-valued fu nction.\";\nend if;\n\nN := nops(r(s)):\n\nif N <> 3 then \n error \+ \"principalUnitNormalVector expects its first argument to be a vector- valued function with three components.\" \nend if;\n\nvelocity := []: \n\n\nspeed := 0:\n for j from 1 to N do \n speed := speed + diff (r(s)[j],s)^2:\n velocity := [op(velocity),diff(r(s)[j],s)]:\n end do:\nspeed := simplify(sqrt(speed)):\n \n\n if subs(s=t0,speed) = 0\n then error \"singular point encountered.\";\n end if:\n\n\nun itTangentVector := map(z->z/speed,velocity):\nnormalVector := map(z->d iff(z,s),unitTangentVector):\n\nnormalLength := 0:\nfor j from 1 to N \+ do\nnormalLength := normalLength + (normalVector[j])^2:\nend do:\nnorm alLength := sqrt(normalLength);\n\nprincipalUnitNormalV := subs(s=t0,m ap(z->z/normalLength,normalVector)):\n\nreturn map(simplify,principalU nitNormalV)\nend proc;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%:principal UnitNormalVectorGf*6\"6,%\"rG%#t0G%\"jG%\"NG%&speedG%)velocityG%5princ ipalUnitNormalVG%2unitTangentVectorG%-normalVectorG%-normalLengthGF&F& C4@(09#\"\"#YQQprincipalUnitNormalVector~expects~two~arguments.F&4-%%t ypeG6$&9\"6#\"\"\"<$%*procedureG%%nameGYQjoprincipalUnitNormalVector~e xpects~its~first~argument~to~be~a~name~or~procedure.F&4-F;6$&F>6#F6%*a lgebraicGYQ_pprincipalUnitNormalVector~expects~its~second~argument~to~ be~an~algebraic~expression.F&>8$F=>8%FI@$4-F;6$-FO6#%\"sG%%listGYQ_ppr incipalUnitNormalVector~expects~its~first~argument~to~be~a~vector-valu ed~function.F&>8'-%%nopsG6#FV@$0Fgn\"\"$YQeqprincipalUnitNormalVector~ expects~its~first~argument~to~be~a~vector-valued~function~with~three~c omponents.F&>8)7\">8(\"\"!?(8&F@F@Fgn%%trueGC$>Fdo,&FdoF@*$)-%%diffG6$ &FV6#FgoFXF6F@F@>Fao7$-%#opG6#FaoF^p>Fdo-%)simplifyG6#-%%sqrtG6#Fdo@$/ -%%subsG6$/FXFQFdoFeoYQ8+-%$mapG6$f*6#% \"zGF&6$%)operatorG%&arrowGF&*&9$F@T#!\"\"F&F&6$F,FdoFao>8,-Fjq6$f*F]r F&F_rF&-F_p6$FcrFXF&F&F&Fhq>8-Feo?(FgoF@F@FgnFho>F_s,&F_sF@*$)&FhrFbpF 6F@F@>F_s-F]q6#F_s>8*-Fbq6$Fdq-Fjq6$f*F]rF&F_rF&FbrF&F&6$F1F_sFhrO-Fjq 6$FjpFjsF&F&F&" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 343 15 " Description of" }{TEXT 284 27 " principleUnitNormalVector" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 344 74 "principalUnitNormalVecto r - principal unit normal vector of a space curve" }}{PARA 4 "" 0 "us age" {TEXT 345 16 "Calling Sequence" }}{PARA 0 "" 0 "" {TEXT 346 37 " \+ principalUnitNormalVector(r , t)" }}{PARA 4 "" 0 "" {TEXT 347 10 " Parameters" }}{PARA 0 "" 0 "" {TEXT 348 27 " r - name or procedur e" }}{PARA 0 "" 0 "" {TEXT 349 30 " t - algebraic expression" }} {SECT 1 {PARA 4 "" 0 "info" {TEXT 350 11 "Description" }}{PARA 15 "" 0