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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$FjpFjsF&F&F&" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 343 15 " Description of" }{TEXT 284 27 " principleUnitNormalVector" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 344 74 "principalUnitNormalVecto r - principal unit normal vector of a space curve" }}{PARA 4 "" 0 "us age" {TEXT 345 16 "Calling Sequence" }}{PARA 0 "" 0 "" {TEXT 346 37 " \+ principalUnitNormalVector(r , t)" }}{PARA 4 "" 0 "" {TEXT 347 10 " Parameters" }}{PARA 0 "" 0 "" {TEXT 348 27 " r - name or procedur e" }}{PARA 0 "" 0 "" {TEXT 349 30 " t - algebraic expression" }} {SECT 1 {PARA 4 "" 0 "info" {TEXT 350 11 "Description" }}{PARA 15 "" 0 "" {TEXT 285 25 "principalUnitNormalVector" }{TEXT 351 62 " computes the principal unit normal vector of a space curve " }{TEXT 288 4 "s ->" }{TEXT 352 1 " " }{TEXT 286 4 "r(s)" }{TEXT 353 17 " at the poi nt " }{TEXT 287 4 "r(t)" }{TEXT 354 2 ". " }}}{SECT 1 {PARA 4 "" 0 "e xamples" {TEXT 355 9 "Examples\n" }}{PARA 5 "" 0 "" {TEXT 659 8 "Curve 1." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "r := t -> [t ^2,t,t]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "principalUnitNo rmalVector(r,t);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7%*(-%%csgnG6#*&,& *$)%\"tG\"\"#\"\"\"F.*&F-F.)-%$absG6#F,\"\"%F.F.F.)-%*conjugateGF3F-F. F.,&F.F.*&F-F.F+F.F.F.*$)F8\"\"$F.#!\"\"F-,$**F,F.F%F.F8F.F:F=F>F?" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT 356 14 "Curve 2. Helix" }}{PARA 0 "" 0 "" {TEXT 357 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "r := s -> [cos(s),sin(s),s/10]:\nprincipalUnitNormalV ector(r,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,$-%$cosG6#%\"tG!\"\" ,$-%$sinGF'F)\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 358 94 "Verification that the unit tangent vector and the prin cipal unit normal vector are orthogonal:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "linalg[dotprod](prin cipalUnitNormalVector(r,t),unitTangentVector(r,t)):\nsimplify(%) assum ing t::real;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT 359 0 "" }}{PARA 0 "" 0 "" {TEXT 361 69 "Plot of the unit t angent vector and the principal unit normal vector:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 376 "principal NormalVectorPlot := plots[arrow](r(Pi/4),r(Pi/4)+principalUnitNormalVe ctor(r,Pi/4),difference,width=0.03):\nunitTangentVectorPlot := plots[a rrow](r(Pi/4),r(Pi/4)+unitTangentVector(r,Pi/4),difference,width=0.03) :\ncurvePlot := plots[spacecurve](r(s),s=-Pi..2*Pi,thickness=3,scaling =constrained):\nplots[display](principalNormalVectorPlot,unitTangentVe ctorPlot,curvePlot);" }}{PARA 13 "" 1 "" {GLPLOT3D 365 365 365 {PLOTDATA 3 "6,-%)POLYGONSG6-7&7%$\"+5y1rq!#5F($\"+N;)RN'!#67%$\"+hN@9 9F*$\"+jN@99F*$\"+>;)RN'F-7%$\"+?&p=N\"F*$\"+/wbw9F*$\"+GhXSmF-7%$\"+p Ps3qF*$\"+^=TLrF*$\"+WhXSmF-7&FqpF*$\"+(eU><(F*$\"+WhX!R(F-7&FKFD7%FEFG$\"+4r]<$)F-7%FLFN $\"+Er]<$)F-7&FVFS7%F6F8$\"+5r]n!*F-7%F=F?$\"+Er]n!*F-7&FgnFZ7%F/F1$\" +>;)RN*F-7%F(F($\"+N;)RN*F-7&F^oF[o7%$\"+-wbw9F*$\"+A&p=N\"F*Fen7%F?F= Fhn7&FgoFbo7%$\"+Q$)3::F*$\"+'yQLJ\"F*FT7%FNFLFW7&F^pFio7%FjoF\\pFI7%F NFLFP7&FapF`p7%FcoFeoF:7%F?F=FA7&FdpFcpF.F'-%&STYLEG6#%,PATCHNOGRIDG-F $6-7%7%F/F1$\"+>;)R&[F-7%$!\"\"F*$\"\"\"F*$\"+9;)R&yF-7%$\"+![D&*G\"F* $\"+W;!*Q:F*$\"+O1$pU&F-7%FgqF`q7%$\"+2SY77F*$\"+9(F*$\"+S'ykk'F-7&FbvF[v7%$\"+x3!*R:F*$\"+I%)R!G\"F hu$\"+3aIN:F*7%$\"+L7oorF*$\"+TR?vrF*$\"+Llv#R(F-7&FawFjv7%$\"+&eBka\" F*$\"+fhuz7Fhu$\"+G/bF;F*7%FdwFbw$\"+Pn?:$)F-7&F`xFiw7%$\"+I)pJ^\"F*$ \"+(Ql`F\"Fhu$\"+<#y@q\"F*7%FevFcv$\"+IY[h!*F-7&F[yFdx7%$\"+<9%GX\"F*$ \"+?%H*o7Fhu$\"+'\\$oIt #*R8F*7%F_]lFh\\l7%$\"++V^P;F*$\"+V?\"3H\"Fhu$\"+)*G=*[\"F*7%Fg]lFh\\l 7%$\"+;(f0l\"F*$\"+,v]*G\"Fhu$\"+QHnt;F*7%F_^lFh\\l7%$\"+1A0%e\"F*$\"+ efu!G\"Fhu$\"+=&GH#=F*7%Fg^lFh\\l7%$\"+!Q&Rj9F*$\"+CS(yE\"Fhu$\"+w!R*z =F*7%F__lFh\\l7%$\"+WgnM8F*$\"+T$3eD\"FhuF\\_l7%Fg_lFh\\l7%$\"+311Z7F* $\"+!fd\"\\7FhuFd^l7%F]`lFh\\l7%$\"+#>:SB\"F*$\"+K@Y]7FhuF\\^l7%Fc`lFh \\l7%$\"+-F_+8F*$\"+wOAf7FhuFd]l7%Fi`lFh\\lFa\\lFfp-F$6$7,Fa\\lF_]lFg] lF_^lFg^lF__lFg_lF]`lFc`lFi`lFfp-F$6$7,F\\uFbvFawF`xF[yFfyF_zFjzFe[lF \\\\lFfp-%'CURVESG6$7T7%$Fbq\"\"!$!+\\\"*3JK!#C$!+aEfTJF*7%$!+r:f:)*F* $!+zie6>F*$!+Z-D\\HF*7%$!+yv;p#*F*$!+Q+n_PF*$!+Sy!pv#F*7%$!+d5)3Q)F*$! ++!\\`X&F*$!+LackDF*7%$!+9N\\$=(F*$!+$\\Do&pF*$!+EIAsBF*7%$!+>m;@dF*$! +LDs,#)F*$!+>1))z@F*7%$!+bM$y/%F*$!+?i7W\"*F*$!+7#Qv)>F*7%$!+o$4_A#F*$ !+:\"z#\\(*F*$!+0e>&z\"F*7%$!+)3e^?$F-$!+h@'[***F*$!+)R`Gg\"F*7%$\"+9* )*ff\"F*$!+Sy\"=()*F*$!+\"*4^59F*7%$\"+10l`MF*$!+MUo%Q*F*$!+%eo\"=7F*7 %$\"+Xc#R=&F*$!+`wU^&)F*$!+xh#e-\"F*7%$\"+q)3Is'F*$!+++y-uF*$!+'pP[L)F -7%$\"+$>OT,)F*$!+S`5\")fF*$!+AOT6kF-7%$\"+g')o4!*F*$!+Iu$)QVF*$!+[&*) z[%F-7%$\"+?'[Hn*F*$!+!)eaODF*$!+uackDF-7%$\"+DRXz**F*$!+=C-2kF-$!++ST 6k!#77%$\"+W,!z\"**F*$\"+v:xy7F*$\"+uEG#G\"F-7%$\"+$[d0\\*F*$\"+T@3^JF *$\"+[nq0KF-7%$\"+hq=8()F*$\"+%[vr!\\F*$\"+A38H^F-7%$\"+5'fWh(F*$\"+AR G#['F*$\"+'*[b_qF-7%$\"+]!)*[B'F*$\"++[J=yF*$\"+q*yf(*)F-7%$\"+THQDYF* $\"+VI*f'))F*$\"+/.%**3\"F*7%$\"+7fFXGF*$\"+<&yme*F*$\"+6FG#G\"F*7%$\" ++JI-'*F-$\"+D6z`**F*$\"+=^iu9F*7%$!+Y?I-'*F-$\"+N6z`**F*$\"+Dv'pm\"F* 7%$!+5eFXGF*$\"+Z&yme*F*$\"+K*4$f=F*7%$!+ZGQDYF*$\"+#4$*f'))F*$\"+RBl^ ?F*7%$!+nz*[B'F*$\"+m[J=yF*$\"+YZ*RC#F*7%$!+R&fWh(F*$\"+1SG#['F*$\"+`r LOCF*7%$!+0q=8()F*$\"+%evr!\\F*$\"+g&z'GEF*7%$!+Yub!\\*F*$\"+`A3^JF*$ \"+n>-@GF*7%$!+G,!z\"**F*$\"+'pr(y7F*$\"+uVO8IF*7%$!+LRXz**F*$!+i6-2kF -$\"+\"y1d?$F*7%$!+`'[Hn*F*$!+bdaODF*$\"+)=\\!)R$F*7%$!+>()o4!*F*$!+4t $)QVF*$\"+&f\"R!f$F*7%$!+vi89!)F*$!+H_5\")fF*$\"+-St#y$F*7%$!+v*3Is'F* $!+0*zFS(F*$\"+4k2vRF*7%$!+qd#R=&F*$!+xvU^&)F*$\"+;)=u;%F*7%$!+V1l`MF* $!+%=%o%Q*F*$\"+B7wfVF*7%$!+e!**ff\"F*$!+F*$ \"+')G%34'F*7%$\"+++++5Fhu$!+k'ezT\"!#<$\"+$H&=$G'F*-%*THICKNESSG6#\" \"$-%(SCALINGG6#%,CONSTRAINEDG" 1 2 0 1 10 0 2 1 1 1 1 1.000000 56.000000 20.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" }}}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 5 "" 0 "" {TEXT 360 9 "Curve 3. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "r := u -> [e xp(u), exp(-u), u]:\nprincipalUnitNormalVector(r,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,$*&\"\"#!\"\"F&#\"\"\"F&F)F$\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 362 94 "Verification that t he unit tangent vector and the principal unit normal vector are orthog onal:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "linalg[dotprod](principalUnitNormalVector(r,t),unitT angentVector(r,t)):\nsimplify(%) assuming t::real;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "l inalg[dotprod](principalUnitNormalVector(r,1),unitTangentVector(r,1)); simplify(%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*,-%$expG6#\"\"\"F (,&\"\"#F(-F&6#F*F(F(,(-F&6#\"\"%F(F(F(F+F(!\"\",(F.F(*&F0F(F+F(F(F(F( #F1F*F+F(F(**,&*&F*F(F+F(F(F(F(F(F-F1F2F4F%F(F1**,&F.F(F(F1F(F-F1F2F4F %F(F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT 363 0 "" }}{PARA 0 "" 0 "" {TEXT 364 69 "Plot of the unit tangen t vector and the principal unit normal vector:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 358 "principalNo rmalVectorPlot := plots[arrow](r(0),r(0)+principalUnitNormalVector(r,0 ),difference,width=0.03):\nunitTangentVectorPlot := plots[arrow](r(0), r(0)+unitTangentVector(r,0),difference,width=0.03):\ncurvePlot := plot s[spacecurve](r(s),s=-1/2..1/2,thickness=3,scaling=constrained):\nplot s[display](principalNormalVectorPlot,unitTangentVectorPlot,curvePlot); " }}{PARA 13 "" 1 "" {GLPLOT3D 365 365 365 {PLOTDATA 3 "6,-%)POLYGONSG 6-7&7%$\"\"\"\"\"!F($!#:!\"$7%$\"+Daol:!\"*F/$!+;+++:!#67%$\"+H)>>d\"F 1$\"+@5Xf:F1$!+2b_87F47%$\"+/WB15F1$\"+fflP**!#5$!+\"\\DN@\"F47&F7%$\"+LU:y:F1$\"+`FA$\"+nwK'\\%FA7%$\"+3+gg9F1$\"+O=Jp_FA$\"+8#=(>XFA7%$\" +Iz>()**FA$\"+*)R6)))*FA$!+UTR3**FL7&FbuF[u7%$\"+t]o`9F1$\"+@(*Rh_FA$ \"+ca&4e%FA7%$\"+s&[!=**FA$\"+u=?!))*FA$!+$ypYy$FL7&FavFju7%$\"+./!*\\ 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71.000000 168.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curv e 6" "Curve 7" "Curve 8" "Curve 9" }}}}{PARA 0 "" 0 "" {TEXT -1 1 "\n " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 " Code for " }{TEXT 278 14 "binormalVector" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1647 "binormalVector := proc()\nlocal r, t, t0, j, N, sp eed, velocity, principalUnitnormalVector,\nunitTangentVector,normalVec tor,normalLength, binormalV;\n\nif nargs <> 2 then \n error \"binorm alVector expects two arguments.\";\nelif not type(args[1], \{name,proc edure\} ) then \n error \"binormalVector expects its first argument to be a name or procedure.\";\nelif not type(args[2],algebraic) then \n error \"binormalVector expects its second argument to be an algeb raic expression.\";\nend if:\n \nr := args[1]:\nt0 := args[2]:\n\nif n ot type(r(s), list) then\n error \"binormalVector expects its first \+ argument to be a vector-valued function.\";\nend if;\n\nN := nops(r(s) ):\n\nif N <> 3 then \n error \"binormalVector expects its first arg ument to be a vector-valued function with three components.\" \nend if ;\n\nvelocity := []:\n\nspeed := 0:\n for j from 1 to N do \n spee d := speed + diff(r(s)[j],s)^2:\n velocity := [op(velocity),diff(r( s)[j],s)]:\n end do:\nspeed := simplify(sqrt(speed)):\n if subs(s= t0,speed) = 0\n then error \"singular point encountered.\";\n end \+ if:\nunitTangentVector := map(z->z/speed,velocity):\n\n\nnormalLength \+ := 0:\nnormalVector := map(z->diff(z,s),unitTangentVector):\nfor j fro m 1 to N do\nnormalLength := normalLength + (normalVector[j])^2:\nend \+ do:\nnormalLength := sqrt(normalLength);\n\nprincipalUnitnormalVector \+ := map(z->z/normalLength,normalVector):\nprincipalUnitnormalVector := \+ subs(s = t0,principalUnitnormalVector):\nunitTangentVector := subs(s = t0, unitTangentVector):\n\n\nbinormalV := convert(linalg[crossprod](c onvert(unitTangentVector,vector),convert(principalUnitnormalVector,vec tor)),list);\n\n\nreturn simplify(binormalV);\nend proc;" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%/binormalVectorGf*6\"6.%\"rG%\"tG%#t0G%\"jG%\" NG%&speedG%)velocityG%:principalUnitnormalVectorG%2unitTangentVectorG% -normalVectorG%-normalLengthG%*binormalVGF&F&C7@(09#\"\"#YQFbinormalVe ctor~expects~two~arguments.F&4-%%typeG6$&9\"6#\"\"\"<$%*procedureG%%na meGYQ_obinormalVector~expects~its~first~argument~to~be~a~name~or~proce dure.F&4-F=6$&F@6#F8%*algebraicGYQdobinormalVector~expects~its~second~ argument~to~be~an~algebraic~expression.F&>8$F?>8&FK@$4-F=6$-FQ6#%\"sG% %listGYQdobinormalVector~expects~its~first~argument~to~be~a~vector-val ued~function.F&>8(-%%nopsG6#FX@$0Fin\"\"$YQjpbinormalVector~expects~it s~first~argument~to~be~a~vector-valued~function~with~three~components. F&>8*7\">8)\"\"!?(8'FBFBFin%%trueGC$>Ffo,&FfoFB*$)-%%diffG6$&FX6#FioFZ F8FBFB>Fco7$-%#opG6#FcoF`p>Ffo-%)simplifyG6#-%%sqrtG6#Ffo@$/-%%subsG6$ /FZFSFfoFgoYQ8,-%$mapG6$f*6#%\"zGF&6$%) operatorG%&arrowGF&*&9$FBT#!\"\"F&F&6$F-FfoFco>8.Fgo>8--F\\r6$f*F_rF&F arF&-Fap6$FerFZF&F&F&Fjq?(FioFBFBFinFjo>Fjr,&FjrFB*$)&F\\sFdpF8FBFB>Fj r-F_q6#Fjr>8+-F\\r6$f*F_rF&FarF&FdrF&F&6$F2FjrF\\s>F\\t-Fdq6$FfqF\\t>F jq-Fdq6$FfqFjq>8/-%(convertG6$-&%'linalgG6#%*crossprodG6$-Fjt6$Fjq%'ve ctorG-Fjt6$F\\tFduFenO-F\\q6#FhtF&F&F&" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 365 14 "Description of" }{TEXT 289 16 " binorma lVector" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 366 51 "binormalVector - binormal vector of a space curve" }}{PARA 4 "" 0 "usage" {TEXT 367 16 "Calling Sequence" }}{PARA 0 "" 0 "" {TEXT 368 26 " binormalVector(r , t)" }}{PARA 4 "" 0 "" {TEXT 369 10 "Parame ters" }}{PARA 0 "" 0 "" {TEXT 370 27 " r - name or procedure" }} {PARA 0 "" 0 "" {TEXT 371 30 " t - algebraic expression" }}{SECT 1 {PARA 4 "" 0 "info" {TEXT 372 11 "Description" }}{PARA 15 "" 0 "" {TEXT 290 14 "binormalVector" }{TEXT 373 49 " computes the binormal ve ctor of a space curve " }{TEXT 293 4 "s ->" }{TEXT 374 1 " " }{TEXT 291 4 "r(s)" }{TEXT 375 17 " at the point " }{TEXT 292 4 "r(t)" } {TEXT 376 2 ". " }}}{SECT 1 {PARA 4 "" 0 "examples" {TEXT 377 9 "Examp les\n" }}{PARA 5 "" 0 "" {TEXT 378 14 "Curve 1. Helix" }}{PARA 0 "" 0 "" {TEXT 379 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "r := s - > [cos(s),sin(s),s/10]:\nbinormalVector(r,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,$*&#\"\"\"\"$,\"F'*&-%$sinG6#%\"tGF'F(#F'\"\"#F'F',$ *&#F'F(F'*&-%$cosGF,F'F(F.F'!\"\",$*(\"#5F'F(F6F(F.F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 380 119 "Verification that \+ the binormal vector is orthogonal to the unit tangent vector and to th e principal unit normal vector:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "linalg[dotprod](binormalVect or(r,t), unitTangentVector(r,t)):\nsimplify(%) assuming t::real;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "linalg[dotprod](binormalVector(r,t), principalUnitNor malVector(r,t)):\nsimplify(%) assuming t::real;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT 381 0 "" }}{PARA 0 "" 0 "" {TEXT 385 28 "Plot of the binormal vector:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 226 "binormalVec torPlot := plots[arrow](r(Pi/4),r(Pi/4)+binormalVector(r,Pi/4),differe nce,width=0.03):\ncurvePlot := plots[spacecurve](r(s),s=-Pi..2*Pi,thic kness=3,scaling=constrained):\nplots[display](binormalVectorPlot, curv ePlot);" }}{PARA 13 "" 1 "" {GLPLOT3D 365 365 365 {PLOTDATA 3 "6(-%)PO LYGONSG6-7&7%$\"+Uugwr!#5$\"+y\"Gb'pF*$\"+cgs/x!#67%$\"+ya[RxF*$\"+U,l -kF*$\"+E,xI()F*7%$\"+h^)pl(F*$\"+yBYgjF*$\"+a1iL()F*7%$\"+Dr5%4(F*$\" +9/MBpF*$\"+N8BLxF/7&F>F77%$\"+4YolvF*$\"+d9qujF*$\"+MM3T()F*7%$\"+tl! 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NESSG6#\"\"$-%(SCALINGG6#%,CONSTRAINEDG" 1 2 0 1 10 0 2 1 1 1 1 1.000000 51.000000 6.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4 " "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve \+ 11" "Curve 12" "Curve 13" }}}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 5 "" 0 "" {TEXT 382 9 "Curve 2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 386 119 "Verification that the binormal vector i s orthogonal to the unit tangent vector and to the principal unit norm al vector:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 91 "linalg[dotprod](binormalVector(r,t), unitTangentVec tor(r,t)):\nsimplify(%) assuming t::real;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "lin alg[dotprod](binormalVector(r,t), principalUnitNormalVector(r,t)):\nsi mplify(%) assuming t::real;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT 384 0 "" }}{PARA 0 "" 0 "" {TEXT 387 28 "Plot of the binormal vector:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 217 "binormalVectorPlot := plots[arrow] (r(0),r(0)+binormalVector(r,0),difference,width=0.03):\ncurvePlot := p lots[spacecurve](r(s),s=-0.7..0.7,thickness=3,scaling=constrained):\np lots[display](binormalVectorPlot, curvePlot);" }}{PARA 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%$\"+(*p]'p\"F-$\"+UVY%*eF*$\"+'H9dG&F*7%$\"+?xnX!>F-$\"+Q-)yD&F*$\"+S9dGkF*7%$\"+'3Jq&>F-$\"+o3y4^F *$\"+EdG9nF*7%$\"+5Fv8?F-$\"+KI&e'\\F*$\"+7+++qF*-%*THICKNESSG6#\"\"$- %(SCALINGG6#%,CONSTRAINEDG" 1 2 0 1 10 0 2 1 1 1 1 1.000000 79.000000 173.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curv e 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "C urve 13" }}}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{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 389 11 "tange ntLine" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1265 "tangentLine := proc()\nlocal r, t0, t, j, N, speed , velocity, unitTangentV, returnList;\n\n\nif nargs <> 3 then \n err or \"tangentLine expects three arguments.\";\nelif not type(args[1], \+ \{name,procedure\} ) then \n error \"tangentLine expects its first \+ argument to be a name or procedure.\";\nelif not type(args[2],algebrai c) or not type(args[3],algebraic) then\n error \"tangentLine expects its second and third arguments to be an algebraic expression.\";\nend if:\n \nr := args[1]:\nt0 := args[2]:\nt := args[3]:\n\nif not type(r (s), list) then\n error \"tangentLine expects its first argument to \+ be a vector-valued function.\";\nend if;\n\nN := nops(r(s)):\n\nif N < > 3 then \n error \"tangentLine expects its first argument to be a v ector-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,spe ed) = 0\n then error \"singular point encountered.\";\n end if:\n \n\nunitTangentV := map(z->z/speed,velocity):\n\n\nreturnList := []:\n for j from 1 to N do\nreturnList := map(simplify,\n [op(returnList),r (t0)[j] + t*subs(s=t0,unitTangentV[j])]):\nend do:\nreturn simplify(re turnList);\nend proc;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%,tangentLin eGf*6\"6+%\"rG%#t0G%\"tG%\"jG%\"NG%&speedG%)velocityG%-unitTangentVG%+ returnListGF&F&C2@(09#\"\"$YQEtangentLine~expects~three~arguments.F&4- %%typeG6$&9\"6#\"\"\"<$%*procedureG%%nameGYQ\\otangentLine~expects~its ~first~argument~to~be~a~name~or~procedure.F&43-F:6$&F=6#\"\"#%*algebra icG-F:6$&F=6#F5FLYQ\\ptangentLine~expects~its~second~and~third~argumen ts~to~be~an~algebraic~expression.F&>8$F<>8%FI>8&FO@$4-F:6$-FT6#%\"sG%% listGYQaotangentLine~expects~its~first~argument~to~be~a~vector-valued~ function.F&>8(-%%nopsG6#Fgn@$0F^oF5YQgptangentLine~expects~its~first~a rgument~to~be~a~vector-valued~function~with~three~components.F&>8*7\"> 8)\"\"!?(8'F?F?F^o%%trueGC$>Fjo,&FjoF?*$)-%%diffG6$&Fgn6#F]pFinFKF?F?> Fgo7$-%#opG6#FgoFdp>Fjo-%)simplifyG6#-%%sqrtG6#Fjo@$/-%%subsG6$/FinFVF joF[pYQ8+-%$mapG6$f*6#%\"zGF&6$%)operat orG%&arrowGF&*&9$F?T#!\"\"F&F&6$F-FjoFgo>8,Fho?(F]pF?F?F^oF^p>F^s-F`r6 $F`q7$-F\\q6#F^s,&&-FT6#FVFhpF?*&FXF?-Fhq6$Fjq&F^rFhpF?F?O-F`qFesF&F&F &" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 391 14 "Description of" }{TEXT 390 13 " tangentLine" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 405 44 "tangen tLine - normal line to a space curve" }}{PARA 4 "" 0 "usage" {TEXT 406 16 "Calling Sequence" }}{PARA 0 "" 0 "" {TEXT 407 26 " tangent Line(r , t0 , " }{TEXT 396 2 "t " }{TEXT 408 1 ")" }}{PARA 4 "" 0 "" {TEXT 409 10 "Parameters" }}{PARA 0 "" 0 "" {TEXT 410 27 " r - na me or procedure" }}{PARA 0 "" 0 "" {TEXT 411 62 " t0 - algebraic e xpression\n t - algebraic expression\n" }}{SECT 1 {PARA 4 "" 0 "i nfo" {TEXT 412 11 "Description" }}{PARA 15 "" 0 "" {TEXT 392 11 "tange ntLine" }{TEXT 413 58 " computes points on the line that is (1) throug h a point " }{TEXT 397 5 "r(t0)" }{TEXT 414 21 " on a space curve \+ " }{TEXT 395 4 "s ->" }{TEXT 415 1 " " }{TEXT 393 4 "r(s)" }{TEXT 416 59 " and (2) has the direction of the unit tangent vector to " } {TEXT 398 1 "r" }{TEXT 417 15 " at the point " }{TEXT 394 5 "r(t0)" } {TEXT 418 18 ". \nSpecifically, " }{TEXT 404 19 "tangentLine(r,t0,t) " }{TEXT 419 21 " returns the point " }{TEXT 399 2 "P " }{TEXT 420 27 " such that the vector from " }{TEXT 400 5 "r(t0)" }{TEXT 421 4 " t o " }{TEXT 401 1 "P" }{TEXT 422 62 " is the scalar multiplication of \+ the unit tangent vector at " }{TEXT 403 5 "r(t0)" }{TEXT 423 5 " by \+ " }{TEXT 402 1 "t" }{TEXT 424 1 "." }}}{SECT 0 {PARA 4 "" 0 "examples " {TEXT 425 10 "Examples\n\n" }}{PARA 5 "" 0 "" {TEXT 426 19 "Curve 1. The Helix\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "r := u -> [cos(u), sin(u), u]:\ntangentLine(r, Pi/ 4, 1);\ntangentLine(r, Pi/4, t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7% ,$*(\"\"%!\"\"\"\"##\"\"\"F(,&F(F'*$F(F)F*F*F',&F)F**&F(F'F(F)F*,&*&F( F'F(F)F**&F&F'%#PiGF*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,$*(\"\"% !\"\"\"\"##\"\"\"F(,&F(F'*&%\"tGF*F(F)F*F*F',&*&F(F'F(F)F**&F(F'F-F*F* ,&*&F&F'%#PiGF*F**(F(F'F-F*F(F)F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 219 "sc1 := plots[spacecurve](r( t),t=-2*Pi..2*Pi,color=GRAY,thickness=2,tickmarks=[3,3,3]):\nsc2 := pl ots[spacecurve](tangentLine(r, 3*Pi/4, t),t=-3/2..3/2,color=NAVY,thick ness=2,tickmarks=[3,3,3]):\nplots[display](sc1,sc2);\n\n" }}{PARA 13 " " 1 "" {GLPLOT3D 372 364 364 {PLOTDATA 3 "6%-%'CURVESG6%7T7%$\"\"\"\" \"!$\"+)Hy@Y'!#C$!+2`=$G'!\"*7%$\"+I'[Hn*!#5$\"+ReaODF4$!+k(Gn-'F07%$ \"+Uq=8()F4$\"+>b<2\\F4$!+@AFqdF07%$\"+.N\\$=(F4$\"+/b#o&pF4$!+yc\"Q^& 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:\nplots[display](sc1,sc2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,&\" \"\"F%*(\"\"%F%\"\"*!\"\"%\"tGF%F%,&F%F%*(\"\")F%F(F)F*F%F%,&#F%\"\"#F %*&F(F)F*F%F)" }}{PARA 13 "" 1 "" {GLPLOT3D 365 365 365 {PLOTDATA 3 "6 %-%'CURVESG6%7T7%$\"\"\"\"\"!F($\"+++++]!#57%$\"+dEi0!*F-$\"+VR75\")F- $\"+L3gh_F-7%$\"+apIj!)F-$\"+/>p,lF-$\"+_[3ObF-7%$\"+$*G0trF-$\"+!yo_9 &F-$\"+dk2BeF-7%$\"+s/'[L'F-$\"+@d/8SF-$\"+%fw=7'F-7%$\"+$pH([bF-$\"+@ ,%)yIF-$\"+fNRJkF-7%$\"+b0m9[F-$\"+Ec4=BF-$\"+&Gq+v'F-7%$\"+eIlKTF-$\" +I@)yq\"F-$\"+wA\"e2(F-7%$\"+-sq-NF-$\"+ud*oA\"F-$\"+sA#fS(F-7%$\"+()H #[#HF-$\"+/&*ea&)!#6$\"+p)\\qt(F-7%$\"+8/+*R#F-$\"+\")H?bdFeo$\"+xi;l! )F-7%$\"+![R_#>F-$\"+bqa1PFeo$\"+Med&Q)F-7%$\"+)=SN]\"F-$\"+(4L1E#Feo$ \"+2h(Hp)F-7%$\"+PD!R8\"F-$\"+k\\t&G\"Feo$\"+Nnd\")*)F-7%$\"+x_Ej\")Fe o$\"+)***)Qm'!#7$\"+@IGX#*F-7%$\"+!f@\"3bFeo$\"+X.%R.$Fiq$\"+cL%zZ*F-7 %$\"+9VftLFeo$\"+gQ6Q6Fiq$\"+%G]On*F-7%$\"+\\MofHuSW!#9$\"+NA!Q$**F-7%$\"+**\\4r$*F_s$\"+ 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expects three arguments.\";\nelif not type(args[1], \{name,procedure\} ) then \n error \"principalNormalLine expects i ts first argument to be a name or procedure.\";\nelif not type(args[2] ,algebraic) or not type(args[3],algebraic) then\n error \"principalN ormalLine expects its second and third arguments to be an algebraic ex pression.\";\nend if:\n \nr := args[1]:\nt0 := args[2]:\nt := args[3]: \n\nif not type(r(s), list) then\n error \"principalNormalLine expec ts its first argument to be a vector-valued function.\";\nend if;\n\nN := nops(r(s)):\n\nif N <> 3 then \n error \"principalNormalLine exp ects its first argument to be a vector-valued function with three comp onents.\" \nend if;\n\nvelocity := []:\n\n\nspeed := 0:\n for j from 1 to N do \n speed := speed + diff(r(s)[j],s)^2:\n velocity := [ op(velocity),diff(r(s)[j],s)]:\n end do:\nspeed := simplify(sqrt(spe ed)):\n \n\n if subs(s=t0,speed) = 0\n then error \"singular poi nt encountered.\";\n end if:\n\n\nunitTangentV := map(z->z/speed,vel ocity):\nnormalVector := map(z->diff(z,s), unitTangentV):\n\nnormalLen gth := 0:\nfor j from 1 to N do\nnormalLength := normalLength + (norma lVector[j])^2:\nend do:\nnormalLength := sqrt(normalLength);\n\nprinci palUnitNormalV := subs(s=t0,map(z->z/normalLength,normalVector)):\n\nr eturnList := []:\nfor j from 1 to N do\nreturnList := map(simplify,\n \+ [op(returnList),r(t0)[j] + t*subs(s=t0,principalUnitNormalV[j])]):\ne nd do:\nreturn simplify(returnList);\nend proc;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%4principalNormalLineGf*6\"6.%\"rG%#t0G%\"tG%\"jG%\"NG %&speedG%)velocityG%5principalUnitNormalVG%-unitTangentVG%-normalVecto rG%-normalLengthG%+returnListGF&F&C7@(09#\"\"$YQMprincipalNormalLine~e xpects~three~arguments.F&4-%%typeG6$&9\"6#\"\"\"<$%*procedureG%%nameGY QdoprincipalNormalLine~expects~its~first~argument~to~be~a~name~or~proc edure.F&43-F=6$&F@6#\"\"#%*algebraicG-F=6$&F@6#F8FOYQdpprincipalNormal Line~expects~its~second~and~third~arguments~to~be~an~algebraic~express ion.F&>8$F?>8%FL>8&FR@$4-F=6$-FW6#%\"sG%%listGYQioprincipalNormalLine~ expects~its~first~argument~to~be~a~vector-valued~function.F&>8(-%%nops G6#Fjn@$0FaoF8YQ_qprincipalNormalLine~expects~its~first~argument~to~be ~a~vector-valued~function~with~three~components.F&>8*7\">8)\"\"!?(8'FB FBFao%%trueGC$>F]p,&F]pFB*$)-%%diffG6$&Fjn6#F`pF\\oFNFBFB>Fjo7$-%#opG6 #FjoFgp>F]p-%)simplifyG6#-%%sqrtG6#F]p@$/-%%subsG6$/F\\oFYF]pF^pYQ8,-%$mapG6$f*6#%\"zGF&6$%)operatorG%&arrowG F&*&9$FBT#!\"\"F&F&6$F-F]pFjo>8--Fcr6$f*FfrF&FhrF&-Fhp6$F\\sF\\oF&F&F& Far>8.F^p?(F`pFBFBFaoFap>Fhs,&FhsFB*$)&FasF[qFNFBFB>Fhs-Ffq6#Fhs>8+-F[ r6$F]r-Fcr6$f*FfrF&FhrF&F[sF&F&6$F2FhsFas>8/F[p?(F`pFBFBFaoFap>F[u-Fcr 6$Fcq7$-F_q6#F[u,&&-FW6#FYF[qFB*&FenFB-F[r6$F]r&FctF[qFBFBO-FcqFbuF&F& F&" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 448 14 "Descript ion of" }{TEXT -1 3 " " }{TEXT 306 19 "principalNormalLine" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 428 43 "normalLine - \+ normal line to a space curve" }}{PARA 4 "" 0 "usage" {TEXT 429 16 "Ca lling Sequence" }}{PARA 0 "" 0 "" {TEXT 430 34 " principalNormalLi ne(r , t0 , " }{TEXT 298 2 "t " }{TEXT 431 1 ")" }}{PARA 4 "" 0 "" {TEXT 432 10 "Parameters" }}{PARA 0 "" 0 "" {TEXT 433 27 " r - na me or procedure" }}{PARA 0 "" 0 "" {TEXT 434 62 " t0 - algebraic e xpression\n t - algebraic expression\n" }}{SECT 1 {PARA 4 "" 0 "i nfo" {TEXT 435 11 "Description" }}{PARA 15 "" 0 "" {TEXT 294 19 "princ ipalNormalLine" }{TEXT 436 58 " computes points on the line that is (1 ) through a point " }{TEXT 299 5 "r(t0)" }{TEXT 437 21 " on a space \+ curve " }{TEXT 297 4 "s ->" }{TEXT 438 1 " " }{TEXT 295 4 "r(s)" } {TEXT 439 61 " and (2) has the direction of the principal unit norma l to " }{TEXT 300 1 "r" }{TEXT 440 15 " at the point " }{TEXT 296 5 " r(t0)" }{TEXT 441 18 ". \nSpecifically, " }{TEXT 307 27 "principalNor malLine(r,t0,t)" }{TEXT 442 21 " returns the point " }{TEXT 301 2 "P " }{TEXT 443 27 " such that the vector from " }{TEXT 302 5 "r(t0)" } {TEXT 444 4 " to " }{TEXT 303 1 "P" }{TEXT 445 71 " is the scalar mul tiplication of the principal unit normal vector at " }{TEXT 305 5 "r( t0)" }{TEXT 446 5 " by " }{TEXT 304 1 "t" }{TEXT 447 1 "." }}}{SECT 1 {PARA 4 "" 0 "examples" {TEXT 449 8 "Examples" }{TEXT -1 1 "\n" }} {PARA 5 "" 0 "" {TEXT 450 8 "Curve 1." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 328 "r := u -> [cos(u), sin( u), u]: \nprincipalNormalLine(r, Pi/4, 1);\nprincipalNormalLine(r, Pi /4, t);\nsc1 := plots[spacecurve](r(t),t=-Pi/2..3*Pi/2,color=GRAY,thic kness=2,tickmarks=[3,3,3]):\nsc2 := plots[spacecurve](principalNormalL ine(r, 3*Pi/4, t),t=-3/2..3/2,color=NAVY,thickness=2,tickmarks=[3,3,3] ):\nplots[display](sc1,sc2);\n\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7% \"\"!F$,$*&\"\"%!\"\"%#PiG\"\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# 7%,&*&\"\"#!\"\"F&#\"\"\"F&F)*(F&F'%\"tGF)F&F(F'F$,$*&\"\"%F'%#PiGF)F) " }}{PARA 13 "" 1 "" 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N, speed, velocity, principalUnitnormalVector ,\nunitTangentVector,normalVector,normalLength, binormalV, returnList; \n\nif nargs <> 3 then \n error \"binormalLine expects two arguments .\";\nelif not type(args[1], \{name,procedure\} ) then \n error \"b inormalLine expects its first argument to be a name or procedure.\";\n elif not type(args[2],algebraic) or not type(args[3],algebraic) then\n error \"binormalLine expects its second and third arguments to be a lgebraic expressions.\";\nend if:\n \nr := args[1]:\nt0 := args[2]:\nt := args[3]:\n\nif not type(r(s), list) then\n error \"binormalLine \+ expects its first argument to be a vector-valued function.\";\nend if; \n\nN := nops(r(s)):\n\nif N <> 3 then \n error \"binormalLine expec ts its first argument to be a vector-valued function with three compon ents.\" \nend if;\n\nvelocity := []:\n\nspeed := 0:\n for j from 1 t o N do \n speed := speed + diff(r(s)[j],s)^2:\n velocity := [op(v elocity),diff(r(s)[j],s)]:\n end do:\nspeed := simplify(sqrt(speed)) :\n if subs(s=t0,speed) = 0\n then error \"singular point encounte red.\";\n end if:\nunitTangentVector := map(z->z/speed,velocity):\n \n\nnormalLength := 0:\nnormalVector := map(z->diff(z,s),unitTangentVe ctor):\nfor j from 1 to N do\nnormalLength := normalLength + (normalVe ctor[j])^2:\nend do:\nnormalLength := sqrt(normalLength);\n\nprincipal UnitnormalVector := map(z->z/normalLength,normalVector):\nprincipalUni tnormalVector := subs(s = t0,principalUnitnormalVector):\nunitTangentV ector := subs(s = t0, unitTangentVector):\n\n\nbinormalV := convert(li nalg[crossprod](convert(unitTangentVector,vector),convert(principalUni tnormalVector,vector)),list);\n\n\nreturnList := []:\nfor j from 1 to \+ N do\nreturnList := map(simplify,\n [op(returnList),r(t0)[j] + t*subs (s=t0,binormalV[j])]):\nend do:\nreturn simplify(returnList);\nend pro c;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%-binormalLineGf*6\"6/%\"rG%#t0 G%\"tG%\"jG%\"NG%&speedG%)velocityG%:principalUnitnormalVectorG%2unitT angentVectorG%-normalVectorG%-normalLengthG%*binormalVG%+returnListGF& F&C:@(09#\"\"$YQDbinormalLine~expects~two~arguments.F&4-%%typeG6$&9\"6 #\"\"\"<$%*procedureG%%nameGYQ]obinormalLine~expects~its~first~argumen t~to~be~a~name~or~procedure.F&43-F>6$&FA6#\"\"#%*algebraicG-F>6$&FA6#F 9FPYQbobinormalLine~expects~its~second~argument~to~be~an~algebraic~exp ression.F&>8$F@>8%FM>8&FS@$4-F>6$-FX6#%\"sG%%listGYQbobinormalLine~exp ects~its~first~argument~to~be~a~vector-valued~function.F&>8(-%%nopsG6# F[o@$0FboF9YQhpbinormalLine~expects~its~first~argument~to~be~a~vector- valued~function~with~three~components.F&>8*7\">8)\"\"!?(8'FCFCFbo%%tru eGC$>F^p,&F^pFC*$)-%%diffG6$&F[o6#FapF]oFOFCFC>F[p7$-%#opG6#F[pFhp>F^p -%)simplifyG6#-%%sqrtG6#F^p@$/-%%subsG6$/F]oFZF^pF_pYQ8,-%$mapG6$f*6#%\"zGF&6$%)operatorG%&arrowGF&*&9$FCT#! \"\"F&F&6$F-F^pF[p>8.F_p>8--Fdr6$f*FgrF&FirF&-Fip6$F]sF]oF&F&F&Fbr?(Fa pFCFCFboFbp>Fbs,&FbsFC*$)&FdsF\\qFOFCFC>Fbs-Fgq6#Fbs>8+-Fdr6$f*FgrF&Fi rF&F\\sF&F&6$F2FbsFds>Fdt-F\\r6$F^rFdt>Fbr-F\\r6$F^rFbr>8/-%(convertG6 $-&%'linalgG6#%*crossprodG6$-Fbu6$Fbr%'vectorG-Fbu6$FdtF\\vF^o>80F\\p? 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}{TEXT -1 21 " on a space curve " }{TEXT 457 4 "s ->" }{TEXT -1 1 " " }{TEXT 455 4 " r(s)" }{TEXT -1 48 " and (2) has the direction of the binormal to " }{TEXT 461 1 "r" }{TEXT -1 15 " at the point " }{TEXT 456 5 "r(t0)" } {TEXT -1 18 ". \nSpecifically, " }{TEXT 459 20 "binormalLine(r,t0,t) " }{TEXT -1 21 " returns the point " }{TEXT 462 2 "P " }{TEXT -1 31 " in such that the vector from " }{TEXT 463 5 "r(t0)" }{TEXT -1 4 " t o " }{TEXT 464 1 "P" }{TEXT -1 65 " is the scalar multiplication of t he (unit) binormal vector at " }{TEXT 466 5 "r(t0)" }{TEXT -1 5 " by \+ " }{TEXT 465 1 "t" }{TEXT -1 1 "." }}}{SECT 1 {PARA 4 "" 0 "examples " {TEXT -1 8 "Examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 303 "r := u -> [cos(u), sin(u), u]: \nbinormalLine(r, Pi/4, 1);\nbinormalLi ne(r, Pi/4, t);\nsc1 := plots[spacecurve](r(t),t=-Pi/2..3*Pi/2,color=G RAY,thickness=2,tickmarks=[3,3,3]):\nsc2 := plots[spacecurve](binormal Line(r, 3*Pi/4, t),t=-4..4,color=NAVY,thickness=2,tickmarks=[3,3,3]): \nplots[display](sc1,sc2);\n\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,$ *(\"\"%!\"\"\"\"##\"\"\"F(,&F(F**$F(F)F*F*F*,&*&F(F'F(F)F*#F*F(F',&*&F &F'%#PiGF*F**&F(F'F(F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,$*(\"\" %!\"\"\"\"##\"\"\"F(,&F(F**&%\"tGF*F(F)F*F*F*,&*&F(F'F(F)F**&F(F'F-F*F ',&*&F&F'%#PiGF*F**(F(F'F-F*F(F)F*" }}{PARA 13 "" 1 "" {GLPLOT3D 639 263 263 {PLOTDATA 3 "6%-%'CURVESG6%7T7%$!+R8[\"\\$!#C$!\"\"\"\"!$!+Fjz q:!\"*7%$\"+>;xy7!#5$!+Q,!z\"**F4$!+b!oDW\"F07%$\"+XeaODF4$!+H'[Hn*F4$ !+$yRVJ\"F07%$\"+f+n_PF4$!+pv;p#*F4$!+6:6'=\"F07%$\"+Lb<2\\F4$!+Mq=8() F4$!+RK)y0\"F07%$\"+<`5\")fF4$!+5i89!)F4$!+u'\\lH*F47%$\"+t'F47%$\"+QwU^&)F 4$!+pc#R=&F4$!+E:q\\aF47%$\"+Pi7W\"*F4$!+;M$y/%F4$!+5)=u;%F47%$\"+N&ym e*F4$!+^eFXGF4$!+%4O^)GF47%$\"+Py\"=()*F4$!+M*)*ff\"F4$!+yL&Gg\"F47%$ \"+j@'[***F4$!+)ed^?$!#6$!+?mq0KF^q7%$\"+G6z`**F4$\"+iFI-'*F^q$\"+S07< '*F^q7%$\"+=\"z#\\(*F4$\"+d$4_A#F4$\"+qZ*RC#F47%$\"+9Uo%Q*F4$\"+h0l`MF 4$\"+'[xi_$F47%$\"+bI*f'))F4$\"+=HQDYF4$\"+--c3[F47%$\"+ODs,#)F4$\"+;m 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(F-$\"+*)*[Ci#Fa]l7%$\"+kik:@Fa]l$\"+TV)3@(F-$\"+FDHJFFa]l7%$\"+K!o+<# Fa]l$\"+>*H[2(F-$\"+lg8SGFa]l7%$\"+,)*[CAFa]l$\"+(\\v(QpF-$\"+.'z*[HFa ]l7%$\"+q:\"*yAFa]l$\"+u5s-oF-$\"+SJ#y0$Fa]l7%$\"+RLLLBFa]l$\"+_mmmmF- $\"+ymmmJFa]l-Ffal6&Fhal$\")!\\DP\"F[blFchm$\")viobF[blF\\bl-%*AXESTIC KSG6%\"\"$FjhmFjhm" 1 2 0 1 10 0 2 1 1 4 1 1.000000 -49.000000 -177.000000 0 0 "Curve 1" "Curve 2" }}}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 " Code for " }{TEXT 496 11 "frenet Frame" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3069 "frenetFrame := proc()\nlocal r, t0, t, j, N, a, b, speed, velocity, principalUnitnormalVector,\nunitTangentVector,normal Vector,normalLength, binormalV, Tangent_Line,\nPrincipal_Normal_Line, \+ Binormal_Line, sc, tl, nl, bl;\n\nif nargs <> 3 then \n error \"fren etFrame expects two arguments.\";\nelif not type(args[1], \{name,proce dure\} ) then \n error \"frenetFrame expects its first argument to \+ be a name or procedure.\";\nelif not type(args[2],algebraic) then\n \+ error \"frenetFrame expects its second argument to be an algebraic exp ression.\";\nelif not type(args[3],range) then\n error \"frenetFrame expects its third argument to be a range.\";\nend if:\n \nr := args[1 ]:\nt0 := args[2]:\n\n\nif not type(op(1,args[3]),realcons) or not typ e(op(2,args[3]),realcons) then\nerror \"frenetFrame expects its third \+ argument to be a range of the form a .. b with a and b real consta nts.\";\nelse\na := min(op(1,args[3]) , op(2,args[3])):\nb := max(op(1 ,args[3]) , op(2,args[3])):\nend if;\n\n\nif not type(r(s), list) then \n error \"frenetFrame expects its first argument to be a vector-val ued function.\";\nelse\nN := nops(r(s)):\nif N <> 3 then \n error \" frenetFrame expects its first argument to be a vector-valued function \+ with three components.\" \nend if:\nend if:\n\nvelocity := []:\n\nspee d := 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:\nspee d := simplify(sqrt(speed)):\n if subs(s=t0,speed) = 0\n then error \"singular point encountered.\";\n end if:\nunitTangentVector := ma p(z->z/speed,velocity):\n\n\nnormalLength := 0:\nnormalVector := map(z ->diff(z,s),unitTangentVector):\nfor j from 1 to N do\nnormalLength := normalLength + (normalVector[j])^2:\nend do:\nnormalLength := sqrt(no rmalLength);\n\nprincipalUnitnormalVector := map(z->z/normalLength,nor malVector):\nprincipalUnitnormalVector := subs(s = t0,principalUnitnor malVector):\nunitTangentVector := subs(s = t0, unitTangentVector):\n\n \nbinormalV := convert(linalg[crossprod](convert(unitTangentVector,vec tor),convert(principalUnitnormalVector,vector)),list);\nbinormalV := s ubs(s=t0,binormalV):\n\n\nTangent_Line := s -> [r(t0)[1]+s*unitTangent Vector[1],\n r(t0)[2]+s*unitTangentVector[2],\n \+ r(t0)[3]+s*unitTangentVector[3]]:\nPrincipal_Norma l_Line := s -> [r(t0)[1]+s*principalUnitnormalVector[1],\n \+ r(t0)[2]+s*principalUnitnormalVector[2],\n \+ r(t0)[3]+s*principalUnitnormalVector[3]]:\nBinor mal_Line := s -> [r(t0)[1]+s*binormalV[1],\n \+ r(t0)[2]+s*binormalV[2],\n r(t0)[3] +s*binormalV[3]]: \nsc := plots[spacecurve](r(t),t=a..b,color=GRAY,thi ckness=3,tickmarks=[0,0,0],scaling=constrained):\ntl := plots[spacecur ve](Tangent_Line(t),t= 0..1,color=navy,thickness=3,scaling=constrained ):\nnl := plots[spacecurve](Principal_Normal_Line(t),t= 0..1,color=blu e,thickness=3,scaling=constrained):\nbl := plots[spacecurve](Binormal_ Line(t),t= 0..1,color=plum,thickness=3,scaling=constrained):\nplots[di splay](sc,tl,nl,bl);\n\nend proc;\n\n\n\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%,frenetFrameGf*6\"67%\"rG%#t0G%\"tG%\"jG%\"NG%\"aG%\" bG%&speedG%)velocityG%:principalUnitnormalVectorG%2unitTangentVectorG% -normalVectorG%-normalLengthG%*binormalVG%-Tangent_LineG%6Principal_No rmal_LineG%.Binormal_LineG%#scG%#tlG%#nlG%#blGF&F&C>@*09#\"\"$YQCfrene tFrame~expects~two~arguments.F&4-%%typeG6$&9\"6#\"\"\"<$%*procedureG%% nameGYQ\\ofrenetFrame~expects~its~first~argument~to~be~a~name~or~proce dure.F&4-FF6$&FI6#\"\"#%*algebraicGYQaofrenetFrame~expects~its~second~ argument~to~be~an~algebraic~expression.F&4-FF6$&FI6#FA%&rangeGYQVfrene tFrame~expects~its~third~argument~to~be~a~range.F&>8$FH>8%FT@%43-FF6$- %#opG6$FKFgn%)realconsG-FF6$-Ffo6$FVFgnFhoYQcqfrenetFrame~expects~its~ third~argument~to~be~a~range~of~the~form~a~..~b~with~~a~~and~~b~~real~ constants.F&C$>8)-%$minG6$FeoF[p>8*-%$maxGFdp@%4-FF6$-F]o6#%\"sG%%list GYQaofrenetFrame~expects~its~first~argument~to~be~a~vector-valued~func tion.F&C$>8(-%%nopsG6#F]q@$0FeqFAYQgpfrenetFrame~expects~its~first~arg ument~to~be~a~vector-valued~function~with~three~components.F&>8,7\">8+ \"\"!?(8'FKFKFeq%%trueGC$>Far,&FarFK*$)-%%diffG6$&F]q6#FdrF_qFVFKFK>F^ r7$-Ffo6#F^rF[s>Far-%)simplifyG6#-%%sqrtG6#Far@$/-%%subsG6$/F_qF_oFarF brYQ8.-%$mapG6$f*6#%\"zGF&6$%)operatorG %&arrowGF&*&9$FKT#!\"\"F&F&6$F/FarF^r>80Fbr>8/-Fft6$f*FitF&F[uF&-F\\s6 $F_uF_qF&F&F&Fdt?(FdrFKFKFeqFer>Fdu,&FduFK*$)&FfuF_sFVFKFK>Fdu-Fis6#Fd u>8--Fft6$f*FitF&F[uF&F^uF&F&6$F4FduFfu>Ffv-F^t6$F`tFfv>Fdt-F^t6$F`tFd t>81-%(convertG6$-&%'linalgG6#%*crossprodG6$-Fdw6$Fdt%'vectorG-Fdw6$Ff vF^xF`q>Fbw-F^t6$F`tFbw>82f*F^qF&F[uF&7%,&&-T%6#F`uFJFK*&F_uFK&T'FJFKF K,&&FjxFUFK*&F_uFK&F_yFUFKFK,&&FjxFhnFK*&F_uFK&F_yFhnFKFKF&F&6(F)F_oF( F]oF2Fdt>83f*F^qF&F[uF&FgxF&F&6(F)F_oF(F]oF1Ffv>84f*F^qF&F[uF&FgxF&F&6 (F)F_oF(F]oF5Fbw>85-&%&plotsG6#%+spacecurveG6(-F]o6#8&/F[[l;FapFfp/%&c olorG%%GRAYG/%*thicknessGFA/%*tickmarksG7%FbrFbrFbr/%(scalingG%,constr ainedG>86-Fdz6'-FexFjz/F[[l;FbrFK/F_[l%%navyGFa[lFf[l>87-Fdz6'-FjyFjzF ^\\l/F_[l%%blueGFa[lFf[l>88-Fdz6'-F^zFjzF^\\l/F_[l%%plumGFa[lFf[l-&Fez 6#%(displayG6&FbzFj[lFc\\lFj\\lF&F&F&" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 17 " Description of " }{TEXT 497 11 "frenetFrame" }}{PARA 3 "" 0 "" {TEXT -1 11 "frenetFrame" }{TEXT 30 34 " - normal plane to a space curve" }}{PARA 4 "" 0 "usage" {TEXT -1 16 "Calling Sequence" }} {PARA 0 "" 0 "" {TEXT -1 17 " frenetFrame(" }{TEXT 35 1 "r" } {TEXT -1 3 " , " }{TEXT 35 2 "t0" }{TEXT -1 3 " , " }{TEXT 501 7 "a .. b " }{TEXT -1 2 ")\n" }}{PARA 4 "" 0 "" {TEXT -1 10 "Parameters" }} {PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 23 22 "r - name or procedur e" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 23 5 "t0 - " }{TEXT -1 26 "algebraic expression\n " }{TEXT 23 5 "a - " }{TEXT -1 19 "rea l constant\n " }{TEXT 23 5 "b - " }{TEXT -1 14 "real constant\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "info" {TEXT -1 11 "Description" }}{PARA 15 "" 0 "" {TEXT 498 26 "frenetFrame(r, t0 , a .. b)" }{TEXT -1 127 " plots the Frenet Frame (i.e., the unit tang ent vector, the principal unit normal vector, and the binormal vector) at a point " }{TEXT 502 5 "r(t0)" }{TEXT -1 21 " on a space curve \+ " }{TEXT 500 4 "s ->" }{TEXT -1 1 " " }{TEXT 499 4 "r(s)" }{TEXT -1 26 ". Included is the plot of " }{TEXT 503 4 "r(s)" }{TEXT -1 5 " for \+ " }{TEXT 504 2 "s " }{TEXT -1 1 "=" }{TEXT 505 7 " a .. b" }{TEXT -1 2 ".\n" }}}{SECT 1 {PARA 4 "" 0 "examples" {TEXT -1 8 "Examples" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "r := t -> [t, t^3/3, (1-t^2) /sqrt(2)]:\nfrenetFrame(r,1,0..2);" }}{PARA 13 "" 1 "" {GLPLOT3D 365 365 365 {PLOTDATA 3 "6'-%'CURVESG6%7T7%$\"\"!F)F($\"+7y1rq!#57%$\"+`Ej \"3%!#6$\"+n#HmE#!#9$\"+2wGfqF,7%$\"+1`Ej\")F0$\"+9MI8=!#8$\"+$*p%R-(F ,7%$\"+'z*[C7F,$\"+B!**)>hF;$\"+qf/lpF,7%$\"+hIlK;F,$\"+IFk]9!#7$\"+QX e#)oF,7%$\"+Ej\"3/#F,$\"+#e'GLGFJ$\"+(pilx'F,7%$\"+\"fz*[CF,$\"+7#>f*[ FJ$\"+Z/)pk'F,7%$\"+cG9dGF,$\"+x$QXx(FJ$\"+(yPQ\\'F,7%$\"+@hIlKF,$\"+$ =90;\"F0$\"+=Z8iF0$\"+8x9iZF,7%$\"+w*[C7'F,$\"+kP()\\wF0$\"+!y@0 U%F,7%$\"+TAhIlF,$\"+iM6%G*F0$\"+QaLbSF,7%$\"+1bxQpF,$\"+d\\f86F,$\"+' o)emOF,7%$\"+r(QpM(F,$\"+'=)*=K\"F,$\"+E:GaKF,7%$\"+O?5bxF,$\"+75oa:F, $\"+cRT=GF,7%$\"+,`Ej\")F,$\"+5MI8=F,$\"+xf)*eBF,7%$\"+m&G9d)F,$\"+g`7 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\"normalPlane expects its second argume nt to be an algebraic expression.\";\nelif not type(args[3],range) the n\n error \"normalPlane expects its third argument to be a range in \+ the form a .. b.\";\nend if:\n \nr := args[1]:\nt0 := args[2]:\na := o p(1,args[3]):\nb := op(2,args[3]):\n\nif not type(a,realcons) and type (b,realcons) then\n error \"normalPlane expects its third argument to be a..b where a and b are real constants.\"\nelif not type(evalf(b- a),positive) then\n error \"normalPlane expects its third argument to be a..b where a and b are real constants with a < b.\"\nend if;\n\n if nargs = 3 then\n xHalfRange := (b-a)/2:\n yHalfRange := (b-a)/2: \+ \n zHalfRange := (b-a)/2:\nelse\n if not type(args[4], list) then \n error \"normalPlane expects its fourth argument to be of the fo rm [h1,h2,h3].\"\n elif nops(args[4]) <> 3 then\n error \"norma lPlane expects its fourth argument to be of the form [h1,h2,h3].\"\n \+ end if:\nxHalfRange := args[4][1]:\nyHalfRange := args[4][2]:\nzHalfR ange := args[4][3]:\n if not type(xHalfRange, positive) or not type (yHalfRange, positive)\n or not type(xHalfRange, positive) then\n \+ error \"normalPlane expects its fourth argument to be of the form [h1, h2,h3] with h1, h2, h3 positive.\"\n end if:\nend if:\n\nif not type( r(s), list) then\n error \"normalPlane expects its first argument to be a vector-valued function.\";\nend if;\n\nN := nops(r(s)):\n\nif N \+ <> 3 then \n error \"normalPlane expects its first argument to be a \+ vector-valued function with three components.\" \nend if;\n\nvelocity \+ := []:\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 if subs(s=t0,speed) = 0 \n then error \"singular point encountered.\";\n end if:\n\n\nunit TangentVector := map(z->z/speed,velocity):\nunitTangentVector := subs( s=t0, unitTangentVector):\n\n\n\nsc := plots[spacecurve](r(t),t=a..b,c olor=GRAY,thickness=3,tickmarks=[0,0,0]):\n\n\nnormalP := plots[implic itplot3d](unitTangentVector[1]*(x-r(t0)[1])+unitTangentVector[2]*(y-r( t0)[2])+unitTangentVector[3]*(z-r(t0)[3])=0,x=r(t0)[1]-xHalfRange..r(t 0)[1]+xHalfRange,y=r(t0)[2]-yHalfRange..r(t0)[2]+yHalfRange,z=r(t0)[3] -zHalfRange..r(t0)[3]+zHalfRange);\nplots[display](sc, normalP);\nend \+ proc;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%,normalPlaneGf*6\"60%\"rG%# t0G%\"aG%\"bG%#scG%(normalPG%\"jG%\"NG%&speedG%)velocityG%2unitTangent VectorG%+xHalfRangeG%+yHalfRangeG%+zHalfRangeGF&F&C6@*529#\"\"$2\"\"%F :YQMnormalPlane~expects~three~or~four~arguments.F&4-%%typeG6$&9\"6#\" \"\"<$%*procedureG%%nameGYQ\\onormalPlane~expects~its~first~argument~t o~be~a~name~or~procedure.F&4-FB6$&FE6#\"\"#%*algebraicGYQaonormalPlane ~expects~its~second~argument~to~be~an~algebraic~expression.F&4-FB6$&FE 6#F;%&rangeGYQconormalPlane~expects~its~third~argument~to~be~a~range~i n~the~form~a~..~b.F&>8$FD>8%FP>8&-%#opG6$FGFY>8'-F_o6$FRFY@&34-FB6$F]o %)realconsG-FB6$FboFjoYQ`pnormalPlane~expects~its~third~argument~to~be ~a..b~where~a~and~~b~~are~real~constants.F&4-FB6$-%&evalfG6#,&FboFGF]o !\"\"%)positiveGYQ[qnormalPlane~expects~its~third~argument~to~be~a..b~ where~a~and~~b~~are~real~constants~with~a~<~b.F&@%/F:F;C%>8/,&*&#FGFRF GFboFGFG*&#FGFRFGF]oFGFfp>80F_q>81F_qC'@&4-FB6$&FE6#F=%%listGYQ`onorma lPlane~expects~its~fourth~argument~to~be~of~the~form~[h1,h2,h3].F&0-%% nopsG6#F]rF;YFar>F^q&F]rFF>Feq&F]rFQ>Fgq&F]rFZ@$433-FB6$F^qFgp-FB6$Feq FgpFasYQipnormalPlane~expects~its~fourth~argument~to~be~of~the~form~[h 1,h2,h3]~with~h1,~h2,~h3~positive.F&@$4-FB6$-Fin6#%\"sGF_rYQaonormalPl ane~expects~its~first~argument~to~be~a~vector-valued~function.F&>8+-Fd r6#F[t@$0FatF;YQgpnormalPlane~expects~its~first~argument~to~be~a~vecto r-valued~function~with~three~components.F&>8-7\">8,\"\"!?(8*FGFGFat%%t rueGC$>F\\u,&F\\uFG*$)-%%diffG6$&F[t6#F_uF]tFRFGFG>Fit7$-F_o6#FitFfu>F \\u-%)simplifyG6#-%%sqrtG6#F\\u@$/-%%subsG6$/F]tF[oF\\uF]uYQ8.-%$mapG6$f*6#%\"zGF&6$%)operatorG%&arrowGF&*&9$ FGT#FfpF&F&6$F0F\\uFit>F_w-Fiv6$F[wF_w>8(-&%&plotsG6#%+spacecurveG6'-F in6#%\"tG/Fjx;F]oFbo/%&colorG%%GRAYG/%*thicknessGF;/%*tickmarksG7%F]uF ]uF]u>8)-&Fdx6#%/implicitplot3dG6&/,(*&&F_wFFFG,&%\"xGFG&-Fin6#F[oFFFf pFGFG*&&F_wFQFG,&%\"yGFG&FczFQFfpFGFG*&&F_wFZFG,&FewFG&FczFZFfpFGFGF]u /Faz;,&FbzFGF^qFfp,&FbzFGF^qFG/Fhz;,&FizFGFeqFfp,&FizFGFeqFG/Few;,&F][ lFGFgqFfp,&F][lFGFgqFG-&Fdx6#%(displayG6$FaxFfyF&F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 17 " Description of " }{TEXT 468 11 "normalPlane" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 11 "normalPlane" }{TEXT 30 34 " - normal plane to a space c urve" }}{PARA 4 "" 0 "usage" {TEXT -1 16 "Calling Sequence" }}{PARA 0 "" 0 "" {TEXT -1 17 " normalPlane(" }{TEXT 35 1 "r" }{TEXT -1 3 " \+ , " }{TEXT 35 2 "t0" }{TEXT -1 3 " , " }{TEXT 473 7 "a .. b " }{TEXT -1 19 ")\n normalPlane(" }{TEXT 35 1 "r" }{TEXT -1 3 " , " }{TEXT 35 2 "t0" }{TEXT -1 3 " , " }{TEXT 479 7 "a .. b " }{TEXT 480 1 "," } {TEXT 481 12 " [h1,h2,h3] " }{TEXT -1 2 ")\n" }}{PARA 4 "" 0 "" {TEXT -1 10 "Parameters" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 23 22 " r - name or procedure" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 23 5 "t0 - " }{TEXT -1 26 "algebraic expression\n " }{TEXT 23 5 "a - " }{TEXT -1 19 "real constant\n " }{TEXT 23 5 "b - " }{TEXT -1 19 "real constant\n " }{TEXT 23 5 "h1 - " }{TEXT -1 21 "positiv e number\n " }{TEXT 23 5 "h2 - " }{TEXT -1 21 "positive number\n \+ " }{TEXT 23 5 "h3 - " }{TEXT -1 17 "positive number\n\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "info" {TEXT -1 11 "Descr iption" }}{PARA 15 "" 0 "" {TEXT 469 26 "normalPlane(r, t0, a .. b)" } {TEXT -1 46 " plots the plane that is (1) through a point " }{TEXT 474 5 "r(t0)" }{TEXT -1 21 " on a space curve " }{TEXT 472 4 "s -> " }{TEXT -1 1 " " }{TEXT 470 4 "r(s)" }{TEXT -1 41 " and (2) has the \+ unit tangent vector of " }{TEXT 475 1 "r" }{TEXT -1 15 " at the point \+ " }{TEXT 471 5 "r(t0)" }{TEXT -1 47 " as its normal vector. Included \+ is the plot of " }{TEXT 476 4 "r(s)" }{TEXT -1 5 " for " }{TEXT 477 2 "s " }{TEXT -1 1 "=" }{TEXT 478 7 " a .. b" }{TEXT -1 2 ".\n" }}{PARA 15 "" 0 "" {TEXT -1 12 "The numbers " }{TEXT 484 1 "a" }{TEXT -1 7 " \+ and " }{TEXT 485 1 "b" }{TEXT -1 15 " in the call " }{TEXT 482 26 " normalPlane(r, t0, a .. b)" }{TEXT -1 16 " must satisfy " }{TEXT 483 5 "a < b" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 9 "Suppose \+ " }{TEXT 486 20 "r(t0) = [x0, y0, z0]" }{TEXT -1 101 ". In a three arg ument call the normal plane is plotted for\n \+ \n " }{TEXT 488 16 "x = x0-h .. x0+h" }{TEXT -1 19 ", \+ \n " }{TEXT 489 16 "y = y0-h .. y0+h" }{TEXT -1 24 ", \+ and \n " }{TEXT 490 16 "z = z0-h .. z0+h" }{TEXT -1 12 " \n\n where " }{TEXT 487 11 "h = (b-a)/2" }{TEXT -1 2 ". " }}{PARA 15 "" 0 "" {TEXT -1 9 "Suppose " }{TEXT 491 20 "r(t0) = [x0, y0, z0] " }{TEXT -1 29 ". In the four argument call " }{TEXT 495 40 "normalPl ane(r, t0, a .. b, [h1, h2, h3])" }{TEXT -1 76 " the normal plane is \+ plotted for\n \n " }{TEXT 492 18 "x = x0-h1 .. x0+h1" }{TEXT -1 19 ", \n " }{TEXT 493 18 "y = y0-h2 .. y0+h2" }{TEXT -1 24 ", and \n " } {TEXT 494 18 "z = z0-h3 .. z0+h3" }{TEXT -1 0 "" }}}{SECT 1 {PARA 4 " " 0 "examples" {TEXT -1 8 "Examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "r := u -> [cos(u), sin(u), u]: \nnormalPlane(r, Pi/4 , 0..2);\n\n\n" }}{PARA 13 "" 1 "" {GLPLOT3D 639 263 263 {PLOTDATA 3 " 6&-%'CURVESG6%7T7%$\"\"\"\"\"!$F*F*F+7%$\"+%Hr;***!#5$\"+L%*\\!3%!#6$ \"+`Ej\"3%F27%$\"+\\!*pm**F/$\"+4=?a\")F2$\"+1`Ej\")F27%$\"+i[7D**F/$ \"+S@V@7F/$\"+'z*[C7F/7%$\"+')z,n)*F/$\"+3&4ai\"F/$\"+hIlK;F/7%$\"+6_Z #z*F/$\"+l$zm-#F/$\"+Ej\"3/#F/7%$\"+12i,(*F/$\"+)HtXU#F/$\"+\"fz*[CF/7 %$\"+6eg%f*F/$\"+?&G%=GF/$\"+cG9dGF/7%$\"+'y3;Z*F/$\"+p*)e2KF/$\"+@hIl KF/7%$\"+7X$GL*F/$\"+0kS\"f$F/$\"+'QpMn$F/7%$\"+^T^y\"*F/$\"+([T#pRF/$ 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args[1]:\nt0 := args[2]:\na := \+ op(1,args[3]):\nb := op(2,args[3]):\n\nif not type(a,realcons) and typ e(b,realcons) then\n error \"osculatingPlane expects its third argume nt to be a..b where a and b are real constants.\"\nelif not type(eva lf(b-a),positive) then\n error \"osculatingPlane expects its third ar gument to be a..b where a and b are real constants with a < b.\"\nen d if;\n\nif nargs = 3 then\n xHalfRange := (b-a)/2:\n yHalfRange := \+ (b-a)/2: \n zHalfRange := (b-a)/2:\nelse\n if not type(args[4], li st) then\n error \"osculatingPlane expects its fourth argument to \+ be of the form [h1,h2,h3].\"\n elif nops(args[4]) <> 3 then\n e rror \"osculatingPlane expects its fourth argument to be of the form [ h1,h2,h3].\"\n end if:\nxHalfRange := args[4][1]:\nyHalfRange := arg s[4][2]:\nzHalfRange := args[4][3]:\n if not type(xHalfRange, posit ive) or not type(yHalfRange, positive)\n or not type(xHalfRange, po sitive) then\n error \"normalPlane expects its fourth argument to be \+ of the form [h1,h2,h3] with h1, h2, h3 positive.\"\n end if:\nend if; \n\nif not type(r(s), list) then\n error \"osculatingPlane expects i ts first argument to be a vector-valued function.\";\nend if;\n\nN := \+ nops(r(s)):\n\nif N <> 3 then \n error \"osculatingPlane expects its first argument to be a vector-valued function with three components. \" \nend if;\n\nvelocity := []:\n\nspeed := 0:\n for j from 1 to N d o \n speed := speed + diff(r(s)[j],s)^2:\n velocity := [op(veloci ty),diff(r(s)[j],s)]:\n end do:\nspeed := simplify(sqrt(speed)):\n \+ if subs(s=t0,speed) = 0\n then error \"singular point encountered. \";\n end if:\n\n\nunitTangentVector := map(z->z/speed,velocity):\nn ormalLength := 0:\nnormalVector := map(z->diff(z,s),unitTangentVector) :\n\nunitTangentVector := subs(s=t0, unitTangentVector):\nnormalVector := subs(s=t0, normalVector):\n\n\nfor j from 1 to N do\nnormalLength \+ := normalLength + (normalVector[j])^2:\nend do:\nnormalLength := sqrt( normalLength);\n\nprincipalUnitnormalVector := map(z->evalf(z/normalLe ngth),normalVector):\n\n\n\nbinormalVector := convert(linalg[crossprod ](convert(unitTangentVector,vector),convert(principalUnitnormalVector, vector)),list);\nbinormalVector := map(evalf,binormalVector);\n\n\nsc \+ := plots[spacecurve](r(t),t=a..b,color=COLOR(RGB, 0.2500, 0.4000, 0.50 00),thickness=3,tickmarks=[0,0,0],numpoints=600):\n\n\nosculatingP := \+ plots[implicitplot3d](binormalVector[1]*(x-r(t0)[1])+binormalVector[2] *(y-r(t0)[2])+binormalVector[3]*(z-r(t0)[3])=0,x=r(t0)[1]-xHalfRange.. r(t0)[1]+xHalfRange,y=r(t0)[2]-yHalfRange..r(t0)[2]+yHalfRange,z=r(t0) [3]-zHalfRange..r(t0)[3]+zHalfRange,color=COLOR(RGB, 0.8500, 0.8300, 0 .8000),style = WIREFRAME,scaling=CONSTRAINED);\n\n\nplots[display](sc, osculatingP);\nend proc;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%0osculat ingPlaneGf*6\"64%\"rG%#t0G%\"aG%\"bG%#scG%,osculatingPG%\"jG%\"NG%&spe edG%)velocityG%:principalUnitnormalVectorG%2unitTangentVectorG%-normal VectorG%-normalLengthG%/binormalVectorG%+xHalfRangeG%+yHalfRangeG%+zHa lfRangeGF&F&C>@*529#\"\"$2\"\"%F>YQQosculatingPlane~expects~three~or~f our~arguments.F&4-%%typeG6$&9\"6#\"\"\"<$%*procedureG%%nameGYQ`ooscula tingPlane~expects~its~first~argument~to~be~a~name~or~procedure.F&4-FF6 $&FI6#\"\"#%*algebraicGYQeoosculatingPlane~expects~its~second~argument ~to~be~an~algebraic~expression.F&4-FF6$&FI6#F?%&rangeGYQgoosculatingPl ane~expects~its~third~argument~to~be~a~range~in~the~form~a~..~b.F&>8$F H>8%FT>8&-%#opG6$FKFgn>8'-Fco6$FVFgn@&34-FF6$Fao%)realconsG-FF6$FfoF^p YQdposculatingPlane~expects~its~third~argument~to~be~a..b~where~a~and~ ~b~~are~real~constants.F&4-FF6$-%&evalfG6#,&FfoFKFao!\"\"%)positiveGYQ _qosculatingPlane~expects~its~third~argument~to~be~a..b~where~a~and~~b ~~are~real~constants~with~a~<~b.F&@%/F>F?C%>83,&*&#FKFVFKFfoFKFK*&#FKF VFKFaoFKFjp>84Fcq>85FcqC'@&4-FF6$&FI6#FA%%listGYQdoosculatingPlane~exp ects~its~fourth~argument~to~be~of~the~form~[h1,h2,h3].F&0-%%nopsG6#Far F?YFer>Fbq&FarFJ>Fiq&FarFU>F[r&FarFhn@$433-FF6$FbqF[q-FF6$FiqF[qFesYQi pnormalPlane~expects~its~fourth~argument~to~be~of~the~form~[h1,h2,h3]~ with~h1,~h2,~h3~positive.F&@$4-FF6$-F]o6#%\"sGFcrYQeoosculatingPlane~e xpects~its~first~argument~to~be~a~vector-valued~function.F&>8+-Fhr6#F_ t@$0FetF?YQ[qosculatingPlane~expects~its~first~argument~to~be~a~vector -valued~function~with~three~components.F&>8-7\">8,\"\"!?(8*FKFKFet%%tr ueGC$>F`u,&F`uFK*$)-%%diffG6$&F_t6#FcuFatFVFKFK>F]u7$-Fco6#F]uFju>F`u- %)simplifyG6#-%%sqrtG6#F`u@$/-%%subsG6$/FatF_oF`uFauYQ8/-%$mapG6$f*6#%\"zGF&6$%)operatorG%&arrowGF&*&9$FKT#Fj pF&F&6$F0F`uF]u>81Fau>80-Few6$f*FhwF&FjwF&-F[v6$F^xFatF&F&F&Fcw>Fcw-F] w6$F_wFcw>Fdx-F]w6$F_wFdx?(FcuFKFKFetFdu>Fbx,&FbxFK*$)&FdxF^vFVFKFK>Fb x-Fhv6#Fbx>8.-Few6$f*FhwF&FjwF&-Fgp6#F]xF&F&6$F5FbxFdx>82-%(convertG6$ -&%'linalgG6#%*crossprodG6$-Fdz6$Fcw%'vectorG-Fdz6$FjyF^[lFcr>Fbz-Few6 $FgpFbz>8(-&%&plotsG6#%+spacecurveG6(-F]o6#%\"tG/F^\\l;FaoFfo/%&colorG 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" [h1,h2,h3] " }{TEXT 591 2 ")\n" }} {PARA 4 "" 0 "" {TEXT 592 10 "Parameters" }}{PARA 0 "" 0 "" {TEXT 593 27 " r - name or procedure" }}{PARA 0 "" 0 "" {TEXT 594 158 " \+ t0 - algebraic expression\n a - real constant\n b - real co nstant\n h1 - positive number\n h2 - positive number\n h3 \+ - positive number\n\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "info" {TEXT 595 11 "Description" }}{PARA 15 "" 0 "" {TEXT 508 30 "osculatingPlane(r, t0, a .. b)" }{TEXT 596 46 " plots th e plane that is (1) through a point " }{TEXT 513 5 "r(t0)" }{TEXT 597 21 " on a space curve " }{TEXT 511 4 "s ->" }{TEXT 598 1 " " } {TEXT 509 4 "r(s)" }{TEXT 599 37 " and (2) has the binormal vector of " }{TEXT 514 1 "r" }{TEXT 600 15 " at the point " }{TEXT 510 5 "r(t0 )" }{TEXT 601 47 " as its normal vector. Included is the plot of " } {TEXT 515 4 "r(s)" }{TEXT 602 5 " for " }{TEXT 516 2 "s " }{TEXT 603 1 "=" }{TEXT 517 7 " a .. b" }{TEXT 604 2 ".\n" }}{PARA 15 "" 0 "" {TEXT 605 12 "The numbers " }{TEXT 523 1 "a" }{TEXT 606 7 " and " } {TEXT 524 1 "b" }{TEXT 607 15 " in the call " }{TEXT 521 30 "osculat ingPlane(r, t0, a .. b)" }{TEXT 608 16 " must satisfy " }{TEXT 522 5 "a < b" }{TEXT 609 1 "." }}{PARA 15 "" 0 "" {TEXT 610 9 "Suppose " }{TEXT 525 20 "r(t0) = [x0, y0, z0]" }{TEXT 611 105 ". In a three argu ment call the osculating plane is plotted for\n \+ \n " }{TEXT 527 16 "x = x0-h .. x0+h" }{TEXT 612 19 " , \n " }{TEXT 528 16 "y = y0-h .. y0+h" }{TEXT 613 24 " , and \n " }{TEXT 529 16 "z = z0-h .. z0+h" }{TEXT 614 12 " \n\n where " }{TEXT 526 11 "h = (b-a)/2" }{TEXT 615 2 ". " }}{PARA 15 "" 0 "" {TEXT 616 9 "Suppose " }{TEXT 530 20 "r(t0) = [x0, y0, z0]" }{TEXT 617 29 ". In the four argument call " }{TEXT 534 44 "osculatingPlane(r, t0, a .. b, [h1, h2, h3])" }{TEXT 618 76 " the no rmal plane is plotted for\n \n \+ " }{TEXT 531 18 "x = x0-h1 .. x0+h1" }{TEXT 619 19 ", \n \+ " }{TEXT 532 18 "y = y0-h2 .. y0+h2" }{TEXT 620 24 ", and \n \+ " }{TEXT 533 18 "z = z0-h3 .. z0+h3" }{TEXT 621 0 "" }}} {SECT 1 {PARA 4 "" 0 "examples" {TEXT 622 9 "Examples\n" }}{PARA 0 "" 0 "" {TEXT 623 43 "One view of an osculating plane of a helix:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "r := u -> [cos(u), sin(u), u/5]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 219 "principalNormalVectorPlot := plots[arrow](r(Pi/4),r( Pi/4)+principalUnitNormalVector(r,Pi/4),difference,color=BLUE):\nunitT angentVectorPlot := plots[arrow](r(Pi/4),r(Pi/4)+unitTangentVector(r,P i/4),difference,color=BLUE):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "plots[display](osculatingPlane(r, Pi/4, -7*Pi/4..9*Pi/4,[0.95,0 .95,0.95]),principalNormalVectorPlot, unitTangentVectorPlot);" }} {PARA 13 "" 1 "" {GLPLOT3D 493 458 458 {PLOTDATA 3 "6/-%'CURVESG6%7d`m 7%$\"+7y1rq!#5F($!+Hub*4\"!\"*7%$\"+t$z6#pF*$\"+$GWy@(F*$!+Y;O&4\"F-7% $\"+I\\ConF*$\"+\">W9O(F*$!+ie;\"4\"F-7%$\"+!zJBh'F*$\"+OV!=](F*$!+z+( p3\"F-7%$\"+r&3NX'F*$\"+ZH')QwF*$!+'HuF3\"F-7%$\"+s^%=H'F*$\"+.(fDx(F* $!+8&y&y5F-7%$\"+QFTFhF*$\"+ld$G!zF*$!+IFQu5F-7%$\"+QOGgfF*$\"+*zL'H!) 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