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{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 }