(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "NeXT Mathematica Notebook Front End Version 2.2"; NeXTStandardFontEncoding; fontset = title, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e8, 24, "Times"; ; fontset = subtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e6, 18, "Times"; ; fontset = subsubtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, L1, e6, 14, "Times"; ; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, L1, a20, 18, "Times"; ; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, L1, a15, 14, "Times"; ; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, L1, a12, 12, "Times"; ; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 10, "Times"; ; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L1, 12, "Courier"; ; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; ; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, L1, 12, "Courier"; ; fontset = name, inactive, noPageBreakInGroup, nohscroll, preserveAspect, M7, italic, B65535, L1, 10, "Times"; ; fontset = header, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, L1, 12, "Times"; ; fontset = leftheader, L0, 12; fontset = footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, italic, L1, 12, "Times"; ; fontset = leftfooter, L0, 12; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12, "Courier"; ; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; currentKernel; ] :[font = title; inactive; preserveAspect; startGroup] ARCSTAMP :[font = section; inactive; preserveAspect; startGroup] BRIEF ABSTRACT :[font = subsection; inactive; preserveAspect; endGroup] We seek the mathematical description (equations) which describes the cut figure in a flat sheet of metal which when bent around a cylinder gives a prescribed shape. :[font = section; inactive; Cclosed; preserveAspect; startGroup] GENERAL INFORMATION :[font = subsubsection; inactive; preserveAspect; endGroup] FileName: ARCSTAMP Full title: Mathematically describe a flat piece of sheet metal which, when folded about a cylinder of radius 1 unit, will produce a desired curved panel. Last Revision Date: 4 April 1996. Developers: Robert Feller, Richmond High School, Richmond IN 47374 and Brian J. Winkel, Department of Mathematical Sciences, United States Military Academy, West Point NY 10996 USA. Phone: 914-938-3200. Email: ab3646@usma2.usma.edu. FAX: 914-938-2409. Contact: Brian J. Winkel, Department of Mathematical Sciences, United States Military Academy, West Point NY 10996 USA. Phone: 914-938-3200. Email: ab3646@usma2.usma.edu. FAX: 914-938-2409. Aaron D. Klebanoff, Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA. Phone: 812-877-8151. Email: Aaron.Klebanoff@Rose-Hulman.Edu. FAX: 812-877-3198. Support: The production of this material is supported by the National Science Foundation under Division of Undergraduate Education grant DUE-9352849: Development Site for Complex, Technology-Based Problems in Calculus with Applications in Science and Engineering and the Arvin Foundation of Columbus IN. :[font = section; inactive; Cclosed; preserveAspect; startGroup] STATEMENT OF PROBLEM :[font = subsection; inactive; preserveAspect] We consider the problem of manufacturing a curved panel. The panel we wish to make is the front, upper right-hand, quarter of the section of the cylinder x^2 + z^2 = 1 which is cut away by the cylinder y^2 + z^2 = 1. This panel is depicted below in outline on the left to right cylinder and within the open cylinder facing the viewer. :[font = subsection; inactive; preserveAspect] For the purpose of manufacturing sheet metal the sheet metal is flat. We must determine the equation of a function y = f(x) so that when we fold the piece around the left-to right cylinder of radius it will conform exactly to our desired piece depicted above. The following diagram shows the outline of the desired rolled piece PMB and the flat piece which must be cut out (PMC) in order to get the desired rolled piece when the cut piece is folded along the cylinder x^2 + z^2 = 1. :[font = subsection; inactive; preserveAspect] Your job is to find and verify the correctness of the equation y = f(x) which will do the job. :[font = subsection; inactive; preserveAspect; endGroup] You are also to determine separately the precise area of cut out section PMC and the surface area of rolled panel PMB and confirm they are equal. :[font = section; inactive; Cclosed; preserveAspect; startGroup] KEYWORDS :[font = subsection; inactive; preserveAspect; endGroup] Visualization, three dimensional parametric plots, surfaces, trigonometry, integration for surface area in three dimensions and for area in two dimensions . :[font = section; inactive; Cclosed; preserveAspect; startGroup] TEACHER NOTES :[font = subsubsection; inactive; preserveAspect] ISSUES RELATED TO THE PROBLEM :[font = subsection; inactive; preserveAspect; startGroup] Prerequisites :[font = subsubsection; inactive; preserveAspect; endGroup] Three dimensional plotting of parametric plots, equations of surfaces, trigonometry, integration for surface area in three dimensions and for area in two dimensions . :[font = subsection; inactive; preserveAspect; startGroup] Time allotment - time management :[font = subsubsection; inactive; preserveAspect] IF students have to spend all their time constructing the various surfaces mathematically and/or visualizing these by hand then this could take hours and some students might never get it. If reasonable diagrams such as the surface diagram offered in the statement and the line diagram with suggestive lines and curves in the solution are offered then the students' time and frustration can be shortened considerably. :[font = subsubsection; inactive; preserveAspect; endGroup] We would imagine that spending half a class on it with a chance to answer questions, consolidate gains, discuss pictures, etc. at the end of the hour and at the first one quarter of the next hour may be enough for students to tghen go out on their own and do the problem. :[font = subsection; inactive; preserveAspect; startGroup] Expectations :[font = subsubsection; inactive; preserveAspect; endGroup] Students will either see the "flash" solution or will make diagrams (in itself visualization practice) to gain understainding of the surfaces involved. :[font = subsection; inactive; preserveAspect; startGroup] Future payoffs :[font = subsubsection; inactive; preserveAspect; endGroup] Students will be able to visualize in three dimensions and see the shape "rolled out" in two dimensions. :[font = subsection; inactive; preserveAspect; startGroup] Extensions :[font = subsubsection; inactive; preserveAspect; endGroup] One could ask what other figures "drawn" on a cylinder, say a surface of an airliner, would have to look like in their planar version as they are prepared for application to the side of the cylindrical airliner tube. :[font = subsection; inactive; preserveAspect; endGroup] References and Sources :[font = section; inactive; Cclosed; preserveAspect; startGroup] POSSIBLE SOLUTION(S) :[font = subsection; inactive; preserveAspect] This figure will be helpful and we shall refer to it in our solution. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] These cells contain the code to generate this schematic offered above. :[font = input; preserveAspect] p0 = ParametricPlot3D[{u,u,Sqrt[1 - u^2]}, {u,0,1}, AxesLabel->{"x","y","z"}, ViewPoint->{4.000, 0.730, 2.030}, AspectRatio->Automatic] :[font = input; preserveAspect] p1 = ParametricPlot3D[{Cos[t], 0, Sin[t]},{t,0, Pi/2}, AxesLabel->{"x","y","z"}, ViewPoint->{4.000, 0.730, 2.030}, AspectRatio->Automatic] :[font = input; preserveAspect] p2 = ParametricPlot3D[{t+1,Cos[t],0},{t,0, Pi/2}, PlotRange->{{0,3},{0,1},{0,1}}, ViewPoint->{4.000, 0.730, 2.030}, AspectRatio->Automatic] :[font = input; preserveAspect] p3 = Show[Graphics3D[{PointSize[.02], Point[{.8,0,Sqrt[1-.8^2]}], Point[{.8,.8,Sqrt[1-.8^2]}], Point[{.8,.8,0}],Point[{.8,0,0}], Line[{{0,0,0},{1,1,0}}], Line[{{1,0,0},{1,1,0}}], Line[{{.8,0,0},{.8,.8,0}}], Line[{{.8,0,Sqrt[1 - .8^2]}, {.8,.8,Sqrt[1 - .8^2]}}], Line[{{.8,0,0},{.8,0,Sqrt[1 - .8^2]}}], Line[{{.8,.8,0},{.8,.8,Sqrt[1 - .8^2]}}], Line[{{0,0,0},{.8,0,Sqrt[1 - .8^2]}}], Line[{{.8,.8,0},{1,.8,0}}], Line[{{0,0,0},{3,0,0}}], Line[{{0,0,0},{0,0,1.2}}], Line[{{0,0,0},{0,1.2,0}}], {Thickness[.01],Line[{{1,0,0},{1+.7,0,0}}], Line[{{1+.7,0,0},{1+.7, Cos[7],0}}]}, Text["P",{1,-.05,0}], Text["Q",{.8,-.05,Sqrt[1 - .8^2]}], Text["R",{.72,.05,0}], Text["S",{.75,.82,.6}], Text["T",{.75,.82,0}], Text["U",{1.09,.82,0}], Text["V",{1.8,.04,0}], Text["O",{-.02,.03,.03}], Text["N",{1.65,Cos[.7],.03}], Text["M",{.97,1.02,.03}], Text["(x,f(x))",{1.80,Cos[.7] + .08,0}], Text["a",{.25,0,.11}], Text["B",{0,.03,1.02}], Text["C",{1 + Pi/2+.02,.03,0}], Text["w",{.9,.025,0}], Text["w",{.9,.78,0}], Text["w",{1.05,.90,0}]}], PlotRange->{{0,3},{0,1},{0,1}}, AspectRatio->Automatic] :[font = input; preserveAspect; endGroup] solutionPlot = Show[p0,p1,p2,p3] :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] These cells contain the code to generate one of the schematics offered in the statement of the problem. :[font = input; preserveAspect] p0 = ParametricPlot3D[{u,u,Sqrt[1 - u^2]}, {u,0,1}, AxesLabel->{"x","y","z"}, ViewPoint->{4.000, 0.730, 2.030}, AspectRatio->Automatic] :[font = input; preserveAspect] p1 = ParametricPlot3D[{Cos[t], 0, Sin[t]},{t,0, Pi/2}, AxesLabel->{"x","y","z"}, ViewPoint->{4.000, 0.730, 2.030}, AspectRatio->Automatic] :[font = input; preserveAspect] p2 = ParametricPlot3D[{t+1,Cos[t],0},{t,0, Pi/2}, PlotRange->{{0,3},{0,1},{0,1}}, ViewPoint->{4.000, 0.730, 2.030}, AspectRatio->Automatic] :[font = input; preserveAspect] p4 = Show[Graphics3D[{PointSize[.02], Line[{{0,0,0},{1,1,0}}], Line[{{1,0,0},{1,1,0}}], Line[{{0,0,0},{3,0,0}}], Line[{{0,0,0},{0,0,1.2}}], Line[{{0,0,0},{0,1.2,0}}], {Thickness[.01],Line[{{1,0,0},{1+.7,0,0}}], Line[{{1+.7,0,0},{1+.7, Cos[7],0}}]}, Text["P",{1,-.05,0}], Text["V",{1.8,.04,0}], Text["O",{-.02,.03,.03}], Text["M",{.97,1.02,.03}], Text["N",{1.65,Cos[.7],.03}], Text["(x,f(x))",{1.80,Cos[.7] + .08,0}], Text["B",{0,.03,1.02}], Text["C",{1 + Pi/2+.02,.03,0}]}, PlotRange->{{0,3},{0,1},{0,1}}], DisplayFunction->Identity] :[font = input; preserveAspect; endGroup] statementPlot = Show[p0,p1,p2,p4] :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] These cells contain the code to generate one of the graphic images offered in the statement of the problem. :[font = input; preserveAspect] p0 = ParametricPlot3D[{u,u,Sqrt[1 - u^2]}, {u,0,1}, AxesLabel->{"x","y","z"}, ViewPoint->{4.000, 0.730, 2.030}, AspectRatio->Automatic] :[font = input; preserveAspect] q1 = ParametricPlot3D[{Cos[u],v,Sin[u]},{u,0,2 Pi}, {v,-1,1}, AxesLabel->{"x","y","z"}, ViewPoint->{4.000, 0.730, 2.030}, AspectRatio->Automatic] :[font = input; preserveAspect] q2 = ParametricPlot3D[{v,Cos[u],Sin[u]},{u,0,2 Pi}, {v,-1,1}, AxesLabel->{"x","y","z"}, ViewPoint->{4.000, 0.730, 2.030}, AspectRatio->Automatic] :[font = input; preserveAspect] q4 = ParametricPlot3D[{Cos[t], 0, Sin[t]}, {t,0, Pi/2}, AxesLabel->{"x","y","z"}, ViewPoint->{4.000, 0.730, 2.030}, AspectRatio->Automatic] :[font = input; preserveAspect] q5 = Graphics3D[{Thickness[.005], Line[{{1,0,0},{1,1,0}}]}] :[font = input; preserveAspect] q6 = Graphics3D[Text[FontForm["PANEL",{"Courier-Bold",16}], {1,.5,.4}]] :[font = input; preserveAspect; endGroup] shapePlot = Show[p0,q1,q2,q4,q5,q6] :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] We offer a slick, quick, elegant geometric solution as to why f(x) = cos(x). Our colleague Kurt Bryan, Department of Mathematics, Rose-Hulman Institute of Technology, saw this solution in a flash! :[font = subsubsection; inactive; preserveAspect; endGroup] Note from the helpful figure at the start of the solution and below that OQ has length 1 and this means that OR has length cos(a). But OR and RT are the same length as the line QM is at 45 degrees with the x-axis. This makes RW also cos(a). And so if we roll out PQ - length a (arclength a as we are in a unit circle and angle QOP is a radians) until it is exactly PV (distance a) then the length of VN is the same as the length of RT, i.e. cos(a). :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] We present a more tedious analytic solution. :[font = subsubsection; inactive; preserveAspect; startGroup] We consider a portion of the surface area :[font = input; preserveAspect] f[x_] = Sqrt[1 - x^2]; :[font = input; preserveAspect; startGroup] s[w_] = Integrate[Sqrt[1 + f'[x]^2],{x,1-w,1}] :[font = output; output; inactive; preserveAspect; endGroup] Pi/2 - ArcTan[(1 - w)*((2*w - w^2)^(-1))^(1/2)] ;[o] Pi 1 -- - ArcTan[(1 - w) Sqrt[--------]] 2 2 2 w - w :[font = input; preserveAspect; startGroup] curve = ParametricPlot[{s[w],1-w},{w,.0000001,1}, AspectRatio->Automatic] :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; startGroup] s[1-u] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Pi/2 - ArcTan[((2*(1 - u) - (1 - u)^2)^(-1))^(1/2)*u] ;[o] Pi 1 -- - ArcTan[Sqrt[--------------------] u] 2 2 2 (1 - u) - (1 - u) :[font = subsubsection; inactive; preserveAspect; startGroup] If we collect terms we get. :[font = input; preserveAspect; startGroup] express = Pi/2 - ArcTan[u Sqrt[1/Expand[2(1-u) - (1 - u)^2]]] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Pi/2 - ArcTan[u*((1 - u^2)^(-1))^(1/2)] ;[o] Pi 1 -- - ArcTan[u Sqrt[------]] 2 2 1 - u :[font = subsubsection; inactive; preserveAspect] And if we draw a triangle with hypotenuse 1 and sides u and Sqrt[1 - u^2] then it can be shown that express = ArcCos[u]. :[font = subsubsection; inactive; preserveAspect; startGroup] But a plot will "prove" that s[1-u] IS ArcCos[u]. :[font = input; preserveAspect; startGroup] border = Plot[s[1-u],{u,0.0000001,.99999999}, AspectRatio->Automatic] :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; startGroup] arcCos = Plot[ArcCos[u],{u,0,1}, AspectRatio->Automatic] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsubsection; inactive; preserveAspect; startGroup] We see they match perfectly! :[font = input; preserveAspect; startGroup] Show[border,arcCos] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsubsection; inactive; preserveAspect; startGroup] We compute the area of the flat cut out sheet. :[font = input; preserveAspect; startGroup] Integrate[Cos[x],{x,0, Pi/2}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 1 ;[o] 1 :[font = subsubsection; inactive; preserveAspect; startGroup] We compute the desired surface area using the element of surface area in a double integral. :[font = input; preserveAspect; startGroup] g[x_,y_] = Sqrt[1 - x^2] :[font = output; output; inactive; preserveAspect; endGroup] (1 - x^2)^(1/2) ;[o] 2 Sqrt[1 - x ] :[font = input; preserveAspect; startGroup] SA[x_,y_] = Sqrt[D[g[x,y],x]^2 + D[g[x,y],y]^2 + 1] :[font = output; output; inactive; preserveAspect; endGroup] (1 + x^2/(1 - x^2))^(1/2) ;[o] 2 x Sqrt[1 + ------] 2 1 - x :[font = input; preserveAspect; startGroup] inner[x_,y_] = Integrate[SA[x,y],{x,y,1}] :[font = output; output; inactive; preserveAspect; endGroup] Pi/2 - ArcTan[y*((1 - y^2)^(-1))^(1/2)] ;[o] Pi 1 -- - ArcTan[y Sqrt[------]] 2 2 1 - y :[font = input; preserveAspect; startGroup] NIntegrate[inner[x,y],{y,0,1}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 1.0000000000013 ;[o] 1. :[font = subsubsection; inactive; preserveAspect; endGroup; endGroup] We see that they are the same. :[font = section; inactive; Cclosed; preserveAspect; startGroup] ISSUES IN SOLUTION :[font = subsection; inactive; preserveAspect; endGroup; endGroup] The issue in this problem is a good diagram. The issue is visualizing the figure and coming up with an appropriate diagram. Students will spend a good bit of time drawing sketches and labeling points. This is GOOD! It will help their visualization. Even after a solution is known students will grapple with the diagram and the actual significance of rolling out region PBM on the flat x-y-plane. ^*)