Arc Length of a 3D Curve

Carl Runco & Daniel Charles

253 Honors Spring 2009

restart:

with(Maplets[Elements]):with(Maplets[Tools]):with(plots):

StartEngine();

randomize():

lightorange:="#FFB300":

lightcyan:="#DDFFFF":

lightblue:="#CFCFFF":

lightgreen:="#CCFFCC":

darkgreen:="#00C000":

ColorBkgd:=wheat:

ColorFooter:=lightcyan:

ColorNewButton:=magenta: # or Make, Save, Enter Problem

ColorReturnButton:=magenta:

ColorCloseButton:=pink: # or Quit, Clear, Reset, Cancel

ColorHintButton:=lightorange: # or Syntax

ColorPrevButton:=turquoise: # or Update

ColorEnterButton:=turquoise:

ColorSimplifyButton:=lightorange:

ColorCheckButton:=green:

ColorShowButton:=lightorange:

ColorPlotButton:=tan: ColorPlotButtonFrgd:=black:

ColorTableBkgd:=lightgreen:

ColorCalculatorBkgd:=gold:

ColorMessage:=white: ColorMessageFrgd:=black:

ColorInput:=turquoise: ColorInputFrgd:=black:

ColorCorrect:=green: ColorCorrectFrgd:=black:

ColorIncorrect:=red: ColorIncorrectFrgd:=white:

ColorWarning:=yellow: ColorWarningFrgd:=black:

ColorShow:=lightorange: ColorShowFrgd:=black:

ColorHint:=lightorange: ColorHintFrgd:=black: # or Info, Syntax

ColorDisable:=wheat: ColorDisableFrgd:=black:

FontFooter:=Font("helvetica",10):

FontSymbol:=Font("symbol",12):

FontBig:=Font("times",24):

First:=true:

formCURVE:=[ [a, b, r, r*cos(t), r*sin(t), t, [a,[i*Pi $i=-2..1]], [b,[i*Pi $i=-1..2]], [r, [$1..9]]], #[a, b, r, ln(t), r*t, sqrt(8*r*t), [a,[$1..9]], [b,[$1..9]], [r, [$1..9]]],

NULL]:

randform:=rand(1..nops(formCURVE)):

start:="":

finish:="":

crv:="":

emptyplot:=plot3d(0, x = -3..3, y=-3..3, view=-3..3, axes=normal, labels=["","",""], grid=[2,2], color=white, scaling=constrained):

RandomProblem:=proc()

global formCURVE, randform, start, finish, crv, C;

local RandCURVE, i, pname, pvals;

RandCURVE:=formCURVE[randform()];

C:=RandCURVE[1..8];

for i from 7 to nops(RandCURVE) do

pname:=RandCURVE[i,1];

pvals:=RandCURVE[i,2];

C:=eval( C, pname=pvals[rand(1..nops(pvals))()] );

end do;

if C[1]<>C[2] then

start,finish:=sort(C[1..2], (a,b)->evalf(a)<evalf(b))[];

crv:=[C[4], C[5], C[6]];

return;

end if;

end do;

end proc:

TEXTprob1:=

" Find the arc length of the curve r(t) = ( %a , %a , %a ) from t = %a to t = %a.":

NewProblem:=proc()

global First, start, finish, crv, form, arclen, template, velocity, valueR;

local integrand;

Set(reply="Please wait while the graph is drawn.",

reply(background)=ColorWarning, reply(foreground)=ColorWarningFrgd);

Set(APlot=emptyplot);

if First then

First:=false;

Set(TFvelocity(enabled)=true, TFvelocity(background)=ColorInput);

Set(Bvelck(enabled)=true, Bvelck(background)=ColorCheckButton);

Set(Bvelsh(enabled)=true, Bvelsh(background)=ColorHintButton);

end if;

Set(Bvalck(enabled)=false, Bvalck(background)=ColorDisable);

Set(Bvalsh(enabled)=false, Bvalsh(background)=ColorDisable);

Set(Bformck(enabled)=false, Bformck(background)=ColorDisable);

Set(Bformsh(enabled)=false, Bformsh(background)=ColorDisable);

Set(Barclenck(enabled)=false, Barclenck(background)=ColorDisable);

Set(Barclensh(enabled)=false, Barclensh(background)=ColorDisable); Set(TFvalue(enabled)=true, TFvalue(background)=ColorDisable); Set(TFvelocity(enabled)=true, TFvelocity(background)=ColorInput);

Set(problem=sprintf(TEXTprob1, crv[1], crv[2], crv[3], start, finish)); Set(TFform=" Int(|r'(t)|, t=a..b) "); Set(TFvelocity="[ dx, dy, dz]"); Set(TFvalue="");

Set(TFval="", TFval(background)=ColorMessage, TFval(foreground)=ColorMessageFrgd);

Set(TFform(enabled)=false, TFform(background)=ColorDisable);

Set(formck="",

formck(background)=ColorMessage, formck(foreground)=ColorMessageFrgd):

Set(TFarclen="",

TFarclen(enabled)=false, TFarclen(background)=ColorDisable);

Set(arclenck="",

arclenck(background)=ColorMessage, arclenck(foreground)=ColorMessageFrgd):

Set(MMLform=MathML[Export](L = Int("|r'(t)|",t=a..b))); Set(TFvel="", TFvel(enabled)=true,

TFvel(background)=ColorMessage, TFvel(foreground)=ColorMessageFrgd);

integrand:=sqrt((cat(d,x)/cat(d,t))^2+(cat(d,y)/cat(d,t))^2+(cat(d,z)/cat(d,t))^2);

template:=Int(integrand,var=a..b);

integrand:=simplify(sqrt(diff(crv[1],t)^2+diff(crv[2],t)^2+diff(crv[3],t)^2));

form:=Int( integrand, t=start..finish );

arclen:=value(form); velocity:=[diff(crv[1],t),diff(crv[2],t),diff(crv[3],t)]; valueR:=simplify(sqrt(diff(crv[1],t)^2+diff(crv[2],t)^2+diff(crv[3],t)^2));

Set(APlot=spacecurve([crv[1],crv[2],crv[3]], t=start..finish, axes=normal, scaling=constrained));

Set(reply="Finished",

reply(background)=ColorHint, reply(foreground)=ColorHintFrgd);

end proc:

Check_vel:=proc()

global velocity;

local user_vel, message; message:=""; if Get(TFvelocity)="" then Set(reply=" You must enter an expression to be checked.", reply(background)=ColorWarning, reply(foreground)=ColorWarningFrgd);

return;

end if;

user_vel:=Get(TFvelocity::list, corrections=true, update=true):

if velocity <> user_vel then

Set(TFvel="Incorrect",

TFvel(background)=ColorIncorrect, TFvel(foreground)=ColorIncorrectFrgd): message:="Sorry.":

Set(reply=cat(message, " Try again."),

reply(background)=ColorIncorrect, reply(foreground)=ColorIncorrectFrgd):

else

Set(TFvel="Correct.",

TFvel(background)=ColorCorrect, TFvel(foreground)=ColorCorrectFrgd):

Set(reply="Awesome! You got it right!",

reply(background)=ColorCorrect, reply(foreground)=ColorCorrectFrgd):

end if;

if message<>"" then

Set(TFvalue="",

TFvalue(enabled)=false, TFvalue(background)=ColorDisable);

Set(Bvalck(enabled)=false, Bvalck(background)=ColorDisable);

Set(TFval="",

TFval(enabled)=false, TFval(background)=ColorDisable);

Set(Bvalsh(enabled)=false, Bvalsh(background)=ColorDisable);

Set(TFvel="Incorrect",

TFvel(background)=ColorIncorrect, TFvel(foreground)=ColorIncorrectFrgd):

Set(reply=cat(message, "Try again."),

reply(background)=ColorIncorrect, reply(foreground)=ColorIncorrectFrgd):

else

Set(TFvalue(enabled)=true, TFvalue(background)=ColorInput);

Set(Bvalck(enabled)=true, Bvalck(background)=ColorCheckButton);

Set(TFval="", TFval(enabled)=true,

TFval(background)=ColorMessage, TFval(foreground)=ColorMessageFrgd);

Set(Bvalsh(enabled)=true, Bvalsh(background)=ColorShowButton);

Set(TFvel="Correct",

TFvel(background)=ColorCorrect, TFvel(foreground)=ColorCorrectFrgd):

Set(reply="Great! Now finish it up!",

reply(background)=ColorCorrect, reply(foreground)=ColorCorrectFrgd):

end if;

end proc:

Show_vel:=proc()

global velocity, lightorange;

Set(TFvelocity=velocity):

Set(TFvel="Shown",

TFvel(background)=ColorShow, TFvel(foreground)=ColorShowFrgd):

Set(reply="Try the next steps on your own.",

reply(background)=ColorShow, reply(foreground)=ColorShowFrgd):

Set(TFvalue(enabled)=true, TFvalue(background)=ColorInput);

Set(Bvalck(enabled)=true, Bvalck(background)=ColorCheckButton);

Set(TFval="", TFval(enabled)=true,

TFval(background)=ColorMessage, TFval(foreground)=ColorMessageFrgd);

Set(Bvalsh(enabled)=true, Bvalsh(background)=ColorShowButton);

end proc:

Check_val:=proc()

global valueR;

local user_val, message; message:=""; if Get(TFvalue)="" then Set(reply=" You must enter an expression to be checked.", reply(background)=ColorWarning, reply(foreground)=ColorWarningFrgd);

return;

end if;

user_val:=Get(TFvalue::realcons, corrections=true, update=true):

if valueR <> user_val then

Set(TFval="Incorrect",

TFval(background)=ColorIncorrect, TFval(foreground)=ColorIncorrectFrgd): message:="Sorry.":

Set(reply=cat(message, " Try again."),

reply(background)=ColorIncorrect, reply(foreground)=ColorIncorrectFrgd):

else

Set(TFval="Correct.",

TFval(background)=ColorCorrect, TFval(foreground)=ColorCorrectFrgd):

Set(reply="Awesome! You got it right!",

reply(background)=ColorCorrect, reply(foreground)=ColorCorrectFrgd):

end if;

if message<>"" then

Set(TFform(enabled)=false, TFform(background)=ColorDisable);

Set(Bformck(enabled)=false, Bformck(background)=ColorDisable);

Set(formck="",

formck(enabled)=false, formck(background)=ColorDisable);

Set(Bformsh(enabled)=false, Bformsh(background)=ColorDisable);

Set(TFval="Incorrect",

TFval(background)=ColorIncorrect, TFval(foreground)=ColorIncorrectFrgd):

Set(reply=cat(message, "Try again."),

reply(background)=ColorIncorrect, reply(foreground)=ColorIncorrectFrgd):

else

Set(TFform(enabled)=true, TFvalue(background)=ColorInput);

Set(Bformck(enabled)=true, Bformck(background)=ColorCheckButton);

Set(formck="", formck(enabled)=true,

formck(background)=ColorMessage, formck(foreground)=ColorMessageFrgd);

Set(Bformsh(enabled)=true, Bformsh(background)=ColorShowButton);

Set(TFval="Correct",

TFval(background)=ColorCorrect, TFval(foreground)=ColorCorrectFrgd):

Set(reply="Way to go! Keep it up!",

reply(background)=ColorCorrect, reply(foreground)=ColorCorrectFrgd):

end if;

end proc:

Show_val:=proc()

global valueR, lightorange;

Set(TFvalue=valueR):

Set(TFval="Shown",

TFval(background)=ColorShow, TFval(foreground)=ColorShowFrgd):

Set(reply="I know you can do the rest.",

reply(background)=ColorShow, reply(foreground)=ColorShowFrgd):

Set(TFform(enabled)=true, TFform(background)=ColorInput);

Set(Bformck(enabled)=true, Bformck(background)=ColorCheckButton);

Set(formck="", formck(enabled)=true,

formck(background)=ColorMessage, formck(foreground)=ColorMessageFrgd);

Set(Bformsh(enabled)=true, Bformsh(background)=ColorShowButton);

end proc:

Check_form:=proc()

global form;

local dom, f, user_dom, user_f, user_form, message;

if Get(TFform)="" then

Set(reply="Nothing to Check",

reply(background)=ColorWarning, reply(foreground)=ColorWarningFrgd);

return;

end if;

user_form:=Get(TFform::anything, corrections=true, update=true):

Set(MMLform=MathML[Export](L=user_form)):

message:="";

if not member(op(0,user_form),{int,Int}) then

message:=cat(message, " This response must be a definite integral. (You can use either Int or int.)");

else

f,dom:=op(form);

user_f,user_dom:=op(user_form);

if lhs(dom)<>lhs(user_dom) then

message:=cat(message, " The arc length integral has the wrong integration variable.")

end if;

if rhs(dom)<>rhs(user_dom) then

message:=cat(message, " The limits of integration on the arc length integral are not correct.")

end if;

if simplify(normal(f-user_f))<>0 assuming lhs(dom)>op(1,rhs(dom)), lhs(dom)<op(2,rhs(dom)) then

message:=cat(message, " The integrand of the arc length integral is incorrect.")

end if;

if message<>"" then

Set(TFarclen="",

TFarclen(enabled)=false, TFarclen(background)=ColorDisable);

Set(Barclenck(enabled)=false, Barclenck(background)=ColorDisable);

Set(arclenck="",

arclenck(enabled)=false, arclenck(background)=ColorDisable);

Set(Barclensh(enabled)=false, Barclensh(background)=ColorDisable);

Set(formck="Incorrect",

formck(background)=ColorIncorrect, formck(foreground)=ColorIncorrectFrgd):

Set(reply=cat(message, "Try again."),

reply(background)=ColorIncorrect, reply(foreground)=ColorIncorrectFrgd):

else

Set(TFarclen(enabled)=true, TFarclen(background)=ColorInput);

Set(Barclenck(enabled)=true, Barclenck(background)=ColorCheckButton);

Set(arclenck="", arclenck(enabled)=true,

arclenck(background)=ColorMessage, arclenck(foreground)=ColorMessageFrgd);

Set(Barclensh(enabled)=true, Barclensh(background)=ColorShowButton);

Set(formck="Correct",

formck(background)=ColorCorrect, formck(foreground)=ColorCorrectFrgd):

Set(reply="Great! Now finish it up!",

reply(background)=ColorCorrect, reply(foreground)=ColorCorrectFrgd):

end if;

end proc:

Show_form:=proc()

global form, lightorange;

Set(TFform=form):

Set(MMLform=MathML[Export](L=form)):

Set(TFarclen(enabled)=true, TFarclen(background)=ColorInput);

Set(Barclenck(enabled)=true, Barclenck(background)=ColorCheckButton);

Set(Barclensh(enabled)=true, Barclensh(background)=ColorShowButton);

Set(formck="Shown",

formck(background)=ColorShow, formck(foreground)=ColorShowFrgd):

Set(reply="Next time try it yourself.",

reply(background)=ColorShow, reply(foreground)=ColorShowFrgd):

end proc:

Check_arclen:=proc()

global arclen;

local user_arclen; if Get(TFarclen)="" then Set(reply=" You must enter an expression to be checked.", reply(background)=ColorWarning, reply(foreground)=ColorWarningFrgd);

return;

end if;

user_arclen:=Get(TFarclen::realcons, corrections=true, update=true):

if simplify(normal(arclen - user_arclen))<> 0 then

Set(arclenck="Incorrect",

arclenck(background)=ColorIncorrect, arclenck(foreground)=ColorIncorrectFrgd):

Set(reply=cat("Sorry.", " Try again."),

reply(background)=ColorIncorrect, reply(foreground)=ColorIncorrectFrgd):

else

Set(arclenck="Correct.",

arclenck(background)=ColorCorrect, arclenck(foreground)=ColorCorrectFrgd):

Set(reply="Awesome! You got it right!",

reply(background)=ColorCorrect, reply(foreground)=ColorCorrectFrgd):

end if;

end proc:

Show_arclen:=proc()

global arclen, lightorange;

Set(TFarclen=arclen):

Set(arclenck="Shown",

arclenck(background)=ColorShow, arclenck(foreground)=ColorShowFrgd):

Set(reply=" So that was practice. Try doing it yourself.",

reply(background)=ColorShow, reply(foreground)=ColorShowFrgd):

end proc:

Arc_Length:=Maplet(onstartup=RunWindow(MAIN), Font[helv10]("helvetica",10),

Window[MAIN]( title="3D Arc Length", defaultbutton=Bnew,

[ halign=none, inset=0, spacing=0, background=ColorBkgd,

[ background=ColorBkgd,

Button[Bnew]("New Problem", onclick=ANewProblem, background=ColorNewButton), HorizontalGlue(),

Button("Quit", Shutdown(), background=ColorCloseButton)

[ background=ColorBkgd,

TextBox[problem]("Please click New Problem to begin.",

width=100, height=2, editable=false, background=ColorMessage)

[ background=ColorBkgd,

[ border=true, inset=0, spacing=0, background=ColorBkgd,

caption="Graph",

[ background=ColorBkgd

Plotter[APlot](value=emptyplot,

width=350, height=350)

[ halign=none, background=ColorBkgd, inset=0, spacing=0,

BoxColumn( background=ColorBkgd, inset=0, spacing=0, border=true,

caption="Step 1: Find r'(t), or the velocity of the function.",

[ background=ColorBkgd,

"r'(t) =",

TextField[TFvelocity](width=20,

enabled=false, background=ColorDisable),

HorizontalGlue(),

Button[Bvelck]("Check", Evaluate(function="Check_vel"),

enabled=false, background=ColorDisable),

TextField[TFvel](width=7,

editable=false, background=ColorMessage),

Button[Bvelsh]("Show", Evaluate(function="Show_vel"),

enabled=false, background=ColorDisable),

HorizontalGlue()

]

BoxColumn( background=ColorBkgd, inset=0, spacing=0, border=true,

caption="Step 2: Find the length of r'(t).",

[ background=ColorBkgd,

"|r'(t)| =",

TextField[TFvalue](width=20,

enabled=false, background=ColorDisable),

HorizontalGlue(),

Button[Bvalck]("Check", Evaluate(function="Check_val"),

enabled=false, background=ColorDisable),

TextField[TFval](width=7,

editable=false, background=ColorMessage),

Button[Bvalsh]("Show", Evaluate(function="Show_val"),

enabled=false, background=ColorDisable),

HorizontalGlue()

]

BoxColumn(background=ColorBkgd, inset=0, spacing=0, border=true,

caption="Step 3: Setup the integral for the arc length.",

[ background=ColorBkgd,

"L =",

TextField[TFform]("Int( |r'(t)| ,t=a..b)", width=30,

enabled=false, background=ColorDisable)

[ background=ColorBkgd,

MathMLViewer[MMLform](value=MathML[Export](L = Int("|r'(t)|",t=a..b)), width=300, height=100)

[ background=ColorBkgd,

HorizontalGlue(),

Button[Bformck]("Check", Evaluate(function="Check_form"),

enabled=false, background=ColorDisable),

TextField[formck](width=6,

editable=false, background=ColorMessage),

Button[Bformsh]("Show", Evaluate(function="Show_form"),

enabled=false, background=ColorDisable),

HorizontalGlue()

]

BoxColumn( background=ColorBkgd, inset=0, spacing=0, border=true,

caption="Step 4: Evaluate the integral to find the arc length.",

[ background=ColorBkgd,

"L =",

TextField[TFarclen](width=20,

enabled=false, background=ColorDisable),

HorizontalGlue(),

Button[Barclenck]("Check", Evaluate(function="Check_arclen"),

enabled=false, background=ColorDisable),

TextField[arclenck](width=6,

editable=false, background=ColorMessage),

Button[Barclensh]("Show", Evaluate(function="Show_arclen"),

enabled=false, background=ColorDisable),

HorizontalGlue()

]

)

]

[ background=ColorBkgd,

TextBox[reply](width=40, height=3, editable=false, background=ColorMessage),

Label(Image(Logo)),

Label(Image(LIBLogo))

], [ background="#DDFFFF", inset=0, spacing=0, Label("Programmers: Carl Runco & Daniel Charles", font=helv10), HorizontalGlue(), Label("Copyright P. Yasskin 2009", font=helv10) ]

]),

Action[ANewProblem](

Evaluate(function="RandomProblem"),

Evaluate(function="NewProblem")

)

First:=true:

Maplets[Display]( Arc_Length );