{VERSION 4 0 "IBM INTEL LINUX22" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Headi ng 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 1 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 256 27 "Curve Operations with Ma ple" }}{PARA 4 "" 0 "" {TEXT -1 27 "Curve construction examples" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "This work sheet gives some examples of operations that can be done with Maple on curves" }}{PARA 0 "" 0 "" {TEXT -1 84 "and vector operations. See \+ Section 2.4 for explanations of some of the words. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT -1 9 "Conchoids" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "The idea of a con choid is to start with some curve " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 42 " rold(t) = ( x(t),y (t) )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 " We think of r(t) as a radius vector from the origin. The new curve is formed" }}{PARA 0 "" 0 "" {TEXT -1 79 "by increasing the len gth of the vector r(t) by a fixed amount, say 1. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "This means that t he new curve has vector equation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 " rnew(t) \+ = rold(t) + rold(t)/| rold(t) | ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 39 "The new curve is called a conchoid. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "Here ar e some examples that show how to build a conchoid efficiently using Ma ple. Note" }}{PARA 0 "" 0 "" {TEXT -1 76 "that all the data can be \+ changed merely by editing the first input line. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "The example below shows how to construct the conchoid based on the curve " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "x := 2+t; y := -2*t; a := -3; b := 3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "radius := sqrt(x^2+y^2);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "plot(\{[x,y,t=a..b], [x*(1+1 /radius),y*(1+1/radius),t=a..b] \},scaling=constrained);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT -1 8 "On speed" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "Here a re some commands that are pertinent to speed. Recall that if a \+ curve is given" }}{PARA 0 "" 0 "" {TEXT -1 16 "by the vector " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 " \+ r(t) = (x(t),y(t)), " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "then the speed is the scalar \+ sqrt( x(t)^2 + y(t)^2 ). It's another function of t." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "To ``see'' the speed on a parametric curve, one way is to plot points at equal time intervals" }}{PARA 0 "" 0 "" {TEXT -1 57 "along the curve. Here's h ow. First we define a curve:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "x := t+co s(2*t); y := 1+sin(2*t); a := 0; b := 2*Pi;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "Now we can plot the curve, but let's do it with 50 \+ points rather than as a continuous curve:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "plot( [x,y,t=a..b], style=po int, numpoints=50,scaling=constrained);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "Why aren't the points equally spaced? Because the curve is not parametrized with constant" }}{PARA 0 "" 0 "" {TEXT -1 6 "spee d." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Let 's try another curve:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "x := cos(t); y := sin(t); a := 0; b := 2*Pi; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "plot( [x,y,t=a..b], sty le=point, numpoints=50,scaling=constrained);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 59 "This time the plot points DO seem to be equally spaced. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "G iven a curve, we can use Maple to find its length. Here are some he lpful commands." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "x := cos(t); y := s in(t); a:=0; b := 2*Pi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " dx := diff(x,t); dy := diff(y,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "speed := sqrt( dx^2+dy^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "int(%,t=a..b);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "13" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }