(*^ ::[ 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, 12; fontset = footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, italic, L1, 12, "Times"; ; fontset = leftfooter, 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; paletteColors = 128; automaticGrouping; currentKernel; ] :[font = title; inactive; preserveAspect; startGroup] LOVEWRIT :[font = subtitle; inactive; preserveAspect] Spelling out LOVE with functions :[font = section; inactive; preserveAspect; startGroup] BRIEF ABSTRACT :[font = subsection; inactive; preserveAspect; endGroup] Here's a quick challenge for students learning how to plot parametrically: write a word with parametric curves. :[font = section; inactive; Cclosed; preserveAspect; startGroup] GENERAL INFORMATION :[font = subsection; inactive; preserveAspect] FileName: LOVEWRIT :[font = subsection; inactive; preserveAspect] Full title: Spelling out LOVE with functions :[font = subsection; inactive; preserveAspect] Last Update: 6/5/96 :[font = subsection; inactive; preserveAspect] Developer: Lynn Kiaer, Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA :[font = subsection; inactive; preserveAspect] Contact: Aaron Klebanoff, Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA. Phone: 812-877-8151. Email: Klebanoff@rose-hulman.edu. FAX: 812-877-3198. :[font = subsection; inactive; preserveAspect; endGroup] 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; endGroup] Write a set of parametric equations whose graph is the word LOVE in block capitals, with the bottom left corner of L at (1, 1). :[font = section; inactive; Cclosed; preserveAspect; startGroup] KEYWORDS :[font = subsection; inactive; preserveAspect; endGroup] Functions, parametric plotting, geometry. :[font = section; inactive; Cclosed; preserveAspect; startGroup] TEACHER NOTES :[font = subsection; inactive; preserveAspect; startGroup] ISSUES RELATED TO THE PROBLEM :[font = subsubsection; inactive; preserveAspect; endGroup] It's not necessary to specify the location of the first letter. It was provided here to make the problem more concrete for the students. :[font = subsection; inactive; preserveAspect; startGroup] Prerequisites :[font = subsubsection; inactive; preserveAspect; endGroup] Understanding of lines and circles and their parametric plots. :[font = subsection; inactive; preserveAspect; startGroup] Time allotment - time management :[font = subsubsection; inactive; preserveAspect; endGroup] This can be done in 30 minutes with groups of students - each taking parts of the word L O V E. :[font = subsection; inactive; preserveAspect] Expectations :[font = subsection; inactive; preserveAspect] Future payoffs :[font = subsection; inactive; preserveAspect; startGroup] Extensions :[font = subsubsection; inactive; preserveAspect; endGroup] Students can parameterize their names, other words, and they can even solve the ultimate problem of writing in cursive! :[font = subsection; inactive; preserveAspect; endGroup] References and Sources :[font = section; inactive; Cclosed; preserveAspect; startGroup] POSSIBLE SOLUTION(S) :[font = subsection; inactive; preserveAspect; startGroup] Suppose we decide that the O will be a circle with radius 1. Then the heights of the letters should be 2, and the other letters should probably be roughly of width 1.5. (Other decisions are possible, of course. Making the O an ellipse with small eccentricity is somewhat more visually pleasing.) :[font = subsubsection; inactive; preserveAspect; startGroup] Here is the vertical bar of the L :[font = input; preserveAspect; endGroup] x[t_] := 1 /; t<=1 && t >= 0 y[t_] := 3 - 2t /; t<=1 && t >= 0 :[font = subsubsection; inactive; preserveAspect; startGroup] And here is the horizontal :[font = input; preserveAspect] x[t_] := t /; t>1 && t<2.5 y[t_] := 1 /; t>1 && t<2.5 :[font = input; preserveAspect; startGroup] L = ParametricPlot[{x[t],y[t]},{t,0,2.5}, Axes->False] :[font = message; inactive; preserveAspect] ParametricPlot::pptr: {CompiledFunction[{t}, <<2>>][t], <<1>>} does not evaluate to a pair of real numbers at t = 2.5. :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsubsection; inactive; preserveAspect; startGroup] Now for the O. It will have its center at (3.5, 2). :[font = input; preserveAspect] x[t_] := 3.5 + Cos[t Pi] /; t>2.5 && t<4.5 y[t_] := 2 + Sin[t Pi] /; t>2.5 && t<4.5 :[font = input; preserveAspect; startGroup] Oh = ParametricPlot[{x[t],y[t]},{t,2.50001,4.5}, Axes->False,AspectRatio->Automatic] :[font = message; inactive; preserveAspect] ParametricPlot::pptr: {CompiledFunction[{t}, <<2>>][t], <<1>>} does not evaluate to a pair of real numbers at t = 4.5. :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; startGroup] Show[L, Oh] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsubsection; inactive; preserveAspect; startGroup] Now for the V. We'll put the upper left end at (4.5,3), the vertex at (5.5,1) and the upper right at (6.5,3). :[font = input; preserveAspect] :[font = input; preserveAspect] x[t_] := t /; t>4.5 && t<=6.5 y[t_] := 3 - 2(t-4.5) /; t>4.5 && t<=5.5 y[t_] := 1 + 2(t-5.5) /; t>5.5 && t<=6.5 :[font = input; preserveAspect; startGroup] V = ParametricPlot[{x[t],y[t]},{t,4.50001,6.5}, Axes->False,AspectRatio->Automatic] :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; startGroup] Show[L,Oh,V] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsubsection; inactive; preserveAspect; startGroup] All that's left is the E. But what a letter! The upper left corner is at about (6.6,3), and it starts out a lot like the L, only translated. :[font = input; preserveAspect] x[t_] := 6.6 /; t<=6.6 && t>6.5 y[t_] := 3 - 20(t - 6.5) /; t<=6.6 && t>6.5 :[font = input; preserveAspect] x[t_] := t /; t>6.6 && t<=8.1 y[t_] := 1 /; t>6.6 && t<=8.1 :[font = input; preserveAspect] x[t_] := t-1.5 /; t>8.1 && t<=9.6 y[t_] := 3 /; t>8.1 && t<=9.6 :[font = input; preserveAspect] x[t_] := t-3 /; t>9.6 && t<=10.6 y[t_] := 2 /; t>9.6 && t<=10.6 :[font = input; preserveAspect; startGroup] Ebottom = ParametricPlot[{x[t],y[t]},{t,6.50001,8.1}, Axes->False] :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; startGroup] Etop = ParametricPlot[{x[t],y[t]},{t,8.10001,9.6}, Axes->False] :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; startGroup] Emiddle = ParametricPlot[{x[t],y[t]},{t,9.60001,10.6}, Axes->False] :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; startGroup] EE = Show[Ebottom,Emiddle,Etop] :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; startGroup] Show[L,Oh,V,EE] :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; startGroup] Show[{L,Oh,V,EE},AspectRatio->Automatic] :[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = section; inactive; Cclosed; preserveAspect; startGroup] ISSUES IN SOLUTION :[font = subsection; inactive; preserveAspect; startGroup] One of the frustrations in solving this problem is the fact that Mathematica wants to connect everything. If you ParametricPlot the whole thing, you get ;[s] 3:0,0;65,1;76,2;153,-1; 3:1,12,9,Times,1,14,0,0,0;1,13,10,Times,3,14,0,0,0;1,12,9,Times,1,14,0,0,0; :[font = input; preserveAspect; startGroup] ParametricPlot[{x[t],y[t]},{t,0,10.6},Axes->False] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsection; inactive; preserveAspect; endGroup; endGroup] We can make the E look okay by defining it from the upper right to the lower right, and traversing the center crossbar twice, but I haven't figured out how to get around the other problems. If anyone knows how to make a Mathematica function discontinuous... ;[s] 3:0,0;221,1;232,2;258,-1; 3:1,12,9,Times,1,14,0,0,0;1,13,10,Times,3,14,0,0,0;1,12,9,Times,1,14,0,0,0; ^*)