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0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 204 "" 0 "" {TEXT 204 38 "Plotting algebraic curv es and surfaces" }{TEXT 204 0 "" }}{PARA 205 "" 0 "" {TEXT 205 0 "" }} {PARA 206 "" 0 "" {TEXT 206 15 "Mike O'Sullivan" }{TEXT 206 0 "" }} {PARA 207 "" 0 "" {TEXT 207 26 "San Diego State University" }{TEXT 207 0 "" }}{PARA 208 "" 0 "" {TEXT 208 12 "January 2005" }{TEXT 208 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 205 "" 0 "" {TEXT 205 99 "The syntax for plot ting curves is not always obvious, so here are some examples to get yo u started." }{TEXT 205 0 "" }}{PARA 205 "" 0 "" {TEXT 205 114 "There a re numerous parameters that can be used to alter the presentation of a plot (color, axes, line width etc.)." }{TEXT 205 0 "" }}{PARA 205 "" 0 "" {TEXT 205 86 "These are also available from the menu: enter the p lot command then click on the plot." }{TEXT 205 0 "" }}{PARA 205 "" 0 "" {TEXT 205 65 "Plots can be exported as in several formats including postscript." }{TEXT 205 0 "" }}}{SECT 1 {PARA 210 "" 0 "" {TEXT 209 49 "Plotting plane curves defined by a function of x." }{TEXT 209 0 "" }}{EXCHG {PARA 205 "" 0 "" {TEXT 205 89 "Here we define a polynomial, substitute 3 for x, and plot the graph in the range [-3..3]." }{TEXT 205 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 15 "f := x^3+x^2+1 ;" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 14 "\nsubs(x=3, f);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 18 "\nplot(f, x=-3..3);" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 1 "\n" }}}{EXCHG {PARA 205 "" 0 "" {TEXT 205 76 "The same graph, but here we define f as a function rather than a polynomial." }{TEXT 205 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 21 "f := x \+ -> x^3+x^2 +1;" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 6 "\nf(3);" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 24 "\nplot(f(x), x = -3.. 3);" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 205 "" 0 "" {TEXT 205 88 "Now introd uce a parameter so that we can see how the curve varies as the paramet er does." }{TEXT 205 0 "" }{TEXT 205 54 "\nChanging t re-enter, and yo u'll get a different curve" }{TEXT 205 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 8 "a := -2;" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 18 "\nf := x^3+a*x^2+1;" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 19 "\nplot(f, x= -3 ..3);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 205 "" 0 "" {TEXT 205 91 "Yo u can animate the family of curves using the procedure animate() in t he package plots[]." }{TEXT 205 0 "" }}{PARA 205 "" 0 "" {TEXT 205 61 "Enter the following then choose Play from the Animation menu." } {TEXT 205 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 15 "f := x^3 +u*x+1;" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 47 "\nplots[animate](plot, [ f, x= -3..3], u= -2..2);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 205 "" 0 "" {TEXT 205 24 "Plotting several curves." }{TEXT 205 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 30 "plot([x^3-1, x^2+2], x=-3..3);" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 210 "" 0 "" {TEXT 209 41 "Plotting plane curves defined implicitly." }{TEXT 209 0 "" }}{EXCHG {PARA 205 "" 0 "" {TEXT 205 105 "This is computation ally much more difficult. For each x it is necessary to find roots of a polynomial to" }{TEXT 205 0 "" }}{PARA 205 "" 0 "" {TEXT 205 32 "fi nd the corresponding y values." }{TEXT 205 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 17 "f := y^2 - x^3-x;" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 205 "" 0 "" {TEXT 205 33 "Again, using the plots[] packag e." }{TEXT 205 0 "" }}{PARA 205 "" 0 "" {TEXT 205 43 "Here ranges on x and y seem to be required." }{TEXT 205 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 41 "plots[implicitplot](f, x=-3..3, y=-3..3);" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 205 "" 0 "" {TEXT 205 80 "You can al so use the algcurves[] package for algebaically defined plane curves. " }{TEXT 205 0 "" }{TEXT 205 51 "\nHere the ranges on x and y seem to \+ not be allowed." }{TEXT 205 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 36 "algcurves[plot_real_curve](f, x, y);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 205 "" 0 "" {TEXT 205 78 "For an interesting expe riment try the two methods on the following polynomial." }{TEXT 205 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 31 "f:=(x^2+y^2)^2 + 3*x ^2*y - y^3;" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 210 "" 0 "" {TEXT 209 45 "Plotting parametrically defined plane curves." } {TEXT 209 0 "" }}{EXCHG {PARA 205 "" 0 "" {TEXT 205 40 "Here is the fo rmat for parametric plots." }{TEXT 205 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 33 "f := t-> t^3-1; g := t -> t^2+2;" }{MPLTEXT 1 0 31 "\nplot([f(u), g(u) , u=-3..3]); " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 205 "" 0 "" {TEXT 205 92 "Two curves can be graphed together. N otice the extra set of brackets around the two curves." }{TEXT 205 0 " " }}{PARA 205 "" 0 "" {TEXT 205 71 "There are lots of options to plot( ). Here I show how to choose colors." }{TEXT 205 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 63 "plot([ [t^3-1, t^2+2, t=-3..3], [ t^2+t, t^3-2*t+3, t=-2..2] ], " }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 23 " \ncolor = [red, black]);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{SECT 1 {PARA 210 "" 0 "" {TEXT 209 24 "Plotting in 3-dimensions" }{TEXT 209 0 "" }}{EXCHG {PARA 205 "" 0 "" {TEXT 205 42 "Try the simi lar thins with the procedures:" }{TEXT 205 0 "" }{TEXT 205 9 "\nplot3d ()" }{TEXT 205 0 "" }{TEXT 205 24 "\nplots[implicitplot3d]()" }{TEXT 205 0 "" }{TEXT 205 1 "\n" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 12 "with(plots);" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 37 "plot3 d(x^2-y^2, x= -2..2, y = -2..2);" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 59 "implicitplot3d(z^2-x^2-y^2, x= -2..2, y = -2..2, z= - 2..2);" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 211 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }