Osculating Circle MapletJoanna Ellis-Monaghan, Colin Kriwox, Laura McLane25 February 2003Click Edit/Execute/Worksheet or the !!! button to initiate the Maplet.

This Maplet plots a parameterized space curve together with its osculating circle (the circle that best approximates the curve at the point of tangency) at a time t.restart; interface( warnlevel=0 ): with(VectorCalculus): with(LinearAlgebra): with(plots):

axesType:=proc(norm,box,fram,none) if norm then NORMAL elif box then BOXED elif fram then FRAMED else NONE end ifend:scaleType:=proc(unc,con) if con then CONSTRAINED else UNCONSTRAINED end ifend proc:

getT:=proc(tl,th,t)local returnVal:returnVal:= evalf(tl + ((t/100) * (th - tl))); end proc:

ocscircle1 := proc (h,tmin,tmax, p, hView, vView, AXES, SCALE)local center, oscirc, curve,R1, p1,h1,h2,cph, Tp, T, T1, Np, check,Tprime, returnVal,centerpoint: h1:=evalf(subs(t=p,diff(h,t)));h2:=evalf(subs(t=p,diff(h, t,t)));cph:=CrossProduct(h1,h2);check:=evalf(Norm(cph,2));R1:=evalf((Norm(h1,2)^3)/check);Tp:=h1/Norm(h1,2);T:=diff(h,t)/Norm(diff(h,t),2) assuming t::real; Tprime:=diff(T,t) assuming t::real;T1:=evalf(subs(t=p,Tprime));Np:=T1/Norm(T1,2);p1:=eval(h, t=p);"center" is the center of the osculating circle.center:=evalm(R1*Np+p1);oscirc:=spacecurve(evalm((R1*cos(s)*Np+R1*sin(s)*Tp)+center), s=0..2*Pi, color=red):curve:=spacecurve([h[1],h[2],h[3]], t=tmin..tmax):display({oscirc, curve}, axes=AXES, scaling=SCALE, orientation=[hView, vView]);end:

DrawSpaceCurve := proc(funcF, funcG, funcH, tLow, tHigh, vView, hView, pnt,AXES,SCALE) local failPlot; try ocscircle1(<funcF, funcG, funcH>, tLow,tHigh,pnt,hView,vView, AXES,SCALE): ##ocscircle(<t*cos(t), 2*t*sin(t), t>, 0,4,3,90,90): catch: failPlot := textplot3d([4,4,3,"Please check the Maple syntax and for singularities."]): end try;end:

with(Maplets[Elements]):b := 'border'=true: s1 := 'inset'=0, 'spacing'=0: c := 'background'="#DDFFFF": dc := 'background'="#CCFFFF": lc := 'background'="#EEFFFF": assignT:=(tlow,newtlow,thigh,newthigh,t,newt)->Action(Evaluate(tlow=newtlow),Evaluate(thigh=newthigh),Evaluate(t='getT(newtlow,newthigh,newt)'),Evaluate(Plot='DrawSpaceCurve(FuncF, FuncG, FuncH, xLow, xHigh, VertView, HorView, pointx, axesType(RB1a,RB2a,RB3a,RB4a),scaleType(RB1s,RB2s))')): OsculatingCircle := Maplet( 'onstartup' = RunWindow('MainWin'), Font['F1']('family'="Comic Sans MS", 'bold'='false', 'italic'='false', size=12), Font['F2']('family'="Comic Sans MS", 'bold'='true', 'italic'='false', size=14), ButtonGroup['RG1'](), # for MenuRadioButtons AXES menu ButtonGroup['RG2'](), # for MenuRadioButtons SCALE Window['MainWin']('title' = "Osculating Circle", 'menubar'='Menu', 'layout' = 'Body'), Window['DescriptionWin']('title' = "Description", 'layout' = 'DescriptionBody', width=400, height=250),Window['SyntaxWin']('title' = "Maple Syntax", 'layout' = 'SyntaxBody', width=400, height=430),Window['NotesWin']('title' = "Notes", 'layout' = 'NotesBody', width=400, height=300),Window['PrintingWin']('title' = "Printing", 'layout' = 'PrintingBody', width=400, height=250),Window['AboutWin']('title' = "About", 'layout' = 'AboutBody', width=400, height=300),Window['ExploreWin']('title' = "Osculating Circle", 'layout' = 'ExploreBody', width=400, height=300),ConfirmDialog['continue']('question', "Somecalculations cause this Maplet to stall. If th is Maplet appears to have stalled you may need to restart it. Are you sure you want to continue?", 'onapprove' = 'EvoluteWarning' ), BoxLayout['DescriptionBody']( BoxColumn(b, 'inset'=0, 'spacing'=4, BoxRow(s1, TextBox('wrapped'=true, 'width'=80, 'height'=15, c, 'font'='F1', 'editable'='false', 'value'= ############################ DESCRIPTION TEXT (BELOW) ##################"This Maplet plots a parameterized space curve together with its osculating circle (the circle that best approximates the curve at the point of tangency) at a time t. The osculating circle lies in the plane determined by the tangent and normal to the curve, its radius is 1/curvature, and the line from the center of the circle to the point of tangency to the curve is parallel to the normal vector. The space curve has three coordinate functions of time: F(t), G(t), and H(t). The endpoints of the desired time interval are entered in the text boxes, and the time t when the osculating circle is computed is adjusted with the slider. Sliders may be moved with the mouse or changed incremetally by selecting with the mouse and then using the arrow keys. The textboxes under Time of Tangency are output only. Various buttons on the menu bar change the appearance of the plot." ############################ DESCRIPTION TEXT (ABOVE) ################## ) ), BoxRow(s1, Button("Close", 'font'='F1', CloseWindow('DescriptionWin') ) ) )),BoxLayout['SyntaxBody']( BoxColumn(b, 'inset'=0, 'spacing'=4, BoxRow(s1, TextBox('wrapped'=true, 'width'=80, 'height'=15, c, 'font'='F1', 'editable'='false', 'value'= ############################ SYNTAX TEXT (BELOW) ##################"Maple Syntax: Entries in textboxes must use Maple Syntax. For example: \n Addition: a+b \n Subtraction: a-b \n Multiplication: a*b \n Division: a/b \n Powers: a^b \n Exponents of e: exp(a), use exp(1) for e = 2.718... \n Square root: sqrt(a) \n Log base 10: log10(a) \n Natural log: ln(a) \n Trigonometric functions: sin(a) \n Pi (3.14...): Pi \n Lists: [a,b,c] \n \n Use parentheses to clearly indicate order of operation." ############################ SYNTAX TEXT (ABOVE) ################## ) ), BoxRow(s1, Button("Close", 'font'='F1', CloseWindow('SyntaxWin') ) ) )),BoxLayout['NotesBody']( BoxColumn(b, 'inset'=0, 'spacing'=4, BoxRow(s1, TextBox('wrapped'=true, 'width'=80, 'height'=15, c, 'font'='F1', 'editable'='false', 'value'= ############################ NOTES TEXT (BELOW) ##################"1. Coordinate functions (or their derivatives) with singularities (e.g. division by zero), and times with 0 or infinite curvatures may generate an error dispatch box. Close this box and adjust the time interval away from the problem values. \n 2. If 'constrained' is selected under the 'scale' option on the menu bar, and the axes still appear unequal, it may be necessary to resize the entire Maplet window itself (by clicking on a corner and dragging) until the aspect looks correct.\n\n3. If the osculating circle has a very small or very large radius, either the circle or the space curve may not be easily visible. If one or the other fails to appear, note the scale on the axes. If the circle is too small, try a smaller time interval or increment t slightly. If the circle is too large, either try a larger time interval, or choose a time interval away from the straight-ish part of the curve, or increment t slightly." ############################ NOTES TEXT (ABOVE) ################## ) ), BoxRow(s1, Button("Close", 'font'='F1', CloseWindow('NotesWin') ) ) )),BoxLayout['PrintingBody']( BoxColumn(b, 'inset'=0, 'spacing'=4, BoxRow(s1, TextBox('wrapped'=true, 'width'=80, 'height'=15, c, 'font'='F1', 'editable'='false', 'value'= ############################ PRINTING TEXT (BELOW) ##################"Currently the best way to get a printout of this Maplet (on a PC at least) is to press Alt PrntScrn while the Maplet is the active window. This puts a copy of the window image on the clipboard. You can then paste this image into another application, such as Word, and then print it from there. Please check the Maple website at www.maplesoft.com for future developments!" ############################ PRINTING TEXT (ABOVE) ################## ) ), BoxRow(s1, Button("Close", 'font'='F1', CloseWindow('PrintingWin') ) ) )),BoxLayout['AboutBody']( BoxColumn(b, 'inset'=0, 'spacing'=6, background="#990099", BoxColumn(s1, background="#990099", Label(font=F2, "Saint Michael\'s College", foreground="#FFFF00", background="#990099"), Label(font=F1, "Colchester, VT", foreground="#FFFF00", background="#990099"), Label("blank space",background="#990099", foreground="#990099"), Label(c, font=F1, "Development Team:", foreground="#FFFF00", background="#990099"), Label(c, font=F1, "\t\tJoanna Ellis-Monaghan", foreground="#FFFFFF", background="#990099"), Label(c, font=F1, "\t\tColin Kriwox", foreground="#FFFFFF", background="#990099"), Label(c, font=F1, "\t\tLaura McLane", foreground="#FFFFFF", background="#990099"), Label(c, font=F1, "\t\twith George Ashline and Zsuzsanna Kadas", foreground="#FFFFFF", background="#990099"), Label("blank space",background="#990099", foreground="#990099"), Button("Close", 'font'='F1', CloseWindow('AboutWin')) ) )),BoxLayout['ExploreBody']( BoxColumn(b, 'inset'=0, 'spacing'=4, BoxRow(s1, TextBox('wrapped'=true, 'width'=80, 'height'=15, c, 'font'='F1', 'editable'='false', 'value'= ############################ Explore TEXT (BELOW) ################## "Here are a few suggested activities to begin exploring the concepts illustrated by this Maplet. \n\nTo begin with, consider the following space curves: \n\nS = < t*sin(t), t, sin(t) > S = < t^2, sin(t), t> S = < t^3, t, cos(t) > S = < 2*t, 3*t, t^2 > S = < sin(t), cos(t), t^2 > S = < cos(t), sin(t), t^4 > \n \na.Use the osculating circle at various points of the time interval to help you describe the underlying motion of the space curve. If possible, think of a real-life context in which you may see this space curve in action.\n\nb. At which times(s) does the spacecurve seem to have maximum and minimum curvature? What happens to the osculating circle at those points? Hint: Recall that the radius of the osculating circle is the reciprocal of the curvature.\n\nc. Are there any point(s) where the osculating circle is undefined? If so, what does this tell you about the underlying curvature there?\n\nd. Over which time interval(s) does it appear that the curvature of the spacecurve changes the most? The least? " ############################ Explore TEXT (ABOVE) ################## ) ), BoxRow(s1, Button("Close", 'font'='F1', CloseWindow('ExploreWin') ) ) )), BoxLayout['Body']( BoxRow(s1, c, BoxColumn(s1, c, b, 'caption'="Space Curve and Osculating Circle", BoxRow(s1, c, Plotter['Plot'](lc), Slider['VertView'](lower=0,background="#DDFFFF",filled=true,majorticks=30,minorticks=10,orientation=vertical,snapticks=false,upper=360,value=70,onchange=assignT(tLow,xLow,tHigh,xHigh,pointx,sliderT))), Slider['HorView'](lower=0,background="#DDFFFF",filled=true,majorticks=30,minorticks=10,orientation=horizontal,snapticks=false,upper=360,value=70, onchange=assignT(tLow,xLow,tHigh,xHigh,pointx,sliderT))), BoxColumn(s1, c, 'inset'=1, 'spacing'=1, BoxRow(c,b,'caption'="Spacecurve as a function of t : S = <F(t), G(t), H(t)>", 'inset'=1, 'spacing'=1, BoxColumn(c, 'inset'=1, 'spacing'=1, BoxRow(c, 'inset'=1, 'spacing'=1, Label("F(t) = ", 'font'='F1', c), TextField['FuncF'](16, lc, 'value'=t*sin(t), 'tooltip'="Enter a function") ), BoxRow(c, 'inset'=1, 'spacing'=1, Label("G(t) = ", 'font'='F1', c), TextField['FuncG'](16, lc, 'value'=t*cos(t), 'tooltip'="Enter a function") ), BoxRow(c, 'inset'=1, 'spacing'=1, Label("H(t) = ", 'font'='F1', c), TextField['FuncH'](16, lc, 'value'=t, 'tooltip'="Enter a function") ) ) ), BoxRow(s1, c, b, 'caption'="Time Interval", Label("t", 'font'='F1', c), Label(" = ", 'font'='F1', c), TextField['xLow'](2, lc, 'value'=-5), Label("..", 'font'='F1', c), TextField['xHigh'](2, lc, 'value'=5) ), BoxColumn(s1, c, b, 'caption'="Time of Tangency:", [TextField[tLow](4,lc,value=-5,editable=false, enabled=false), Slider[sliderT](c,lower=0,filled=true,upper=100, lower=0, value=50, showlabels=false,minorticks=1, majorticks=25, onchange=assignT(tLow,xLow,tHigh,xHigh,pointx,sliderT), orientation=horizontal,snapticks=false), TextField[tHigh](4,lc,value=5,editable=false, enabled=false),c], [Label("Frame at t = ", 'font'='F1', c), TextField['pointx'](2, lc, enabled=false),c] ), BoxRow('inset'=0, 'spacing'=0, c, Button("Plot", dc, 'tooltip'="Plot", onclick=assignT(tLow,xLow,tHigh,xHigh,pointx,sliderT)), Button("Close", Shutdown(), dc) ) ) ) ), MenuBar['Menu']( Menu("Explore", MenuItem("Osculating Circle", onclick=RunWindow('ExploreWin')) ), Menu("Axes", RadioButtonMenuItem['RB1a']("Normal",'value'=true,'group'='RG1', onclick=assignT(tLow,xLow,tHigh,xHigh,pointx,sliderT)), RadioButtonMenuItem['RB2a']("Boxed",'group'='RG1', onclick=assignT(tLow,xLow,tHigh,xHigh,pointx,sliderT)), RadioButtonMenuItem['RB3a']("Framed",'group'='RG1', onclick=assignT(tLow,xLow,tHigh,xHigh,pointx,sliderT)), RadioButtonMenuItem['RB4a']("None", 'group'='RG1', onclick=assignT(tLow,xLow,tHigh,xHigh,pointx,sliderT)) ), Menu("Scale", RadioButtonMenuItem['RB1s']("Unconstrained",'value'=true,'group'='RG2', onclick=assignT(tLow,xLow,tHigh,xHigh,pointx,sliderT)), RadioButtonMenuItem['RB2s']("Constrained",'group'='RG2', onclick=assignT(tLow,xLow,tHigh,xHigh,pointx,sliderT)) ), Menu("Help", MenuItem("Description", onclick=RunWindow('DescriptionWin')), MenuItem("Notes", onclick=RunWindow('NotesWin')), MenuItem("Maple Syntax", onclick=RunWindow('SyntaxWin')), MenuItem("Printing", onclick=RunWindow('PrintingWin')), MenuItem("About", onclick=RunWindow('AboutWin')) ) ), Action['EvoluteWarning']( Evaluate('Plot'='DrawSpaceCurve('FuncF', 'FuncG', 'FuncH', 'xLow', 'xHigh', 'pointx', 'VertView', 'HorView', 3)') ) ):Maplets[Display] (OsculatingCircle);