{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 }