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{SECT 0 {PARA 18 "" 0 "" {TEXT -1 25 "5. Graphics and Animation" }}
{PARA 19 "" 0 "" {TEXT -1 27 "Dr. Saccone, Revised 2/1/01" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 36 "Graphs of Functions of \+
Two Variables" }}{PARA 0 "" 0 "" {TEXT -1 56 "The command for graphing
 a function of two variables is " }{TEXT 265 7 "plot3d." }{TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot3d(x^2+y^2,x=-1..1,y
=-1..1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
56 "If we want to add axes we supply the optional parameter " }{TEXT 
266 11 "axes=normal" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "plot3d(x^2+y^2,x=-1..1,y=-1..1,axes=normal);" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 
267 12 "Axis Labels." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 180 "Note that the axes in the above diagram seem to labele
d in such a way that the coordinate system is not right-handed. Howeve
r, the axis closest to the label \"y\" in this diagram is " }{TEXT 
268 4 "not " }{TEXT -1 83 "the y-axis, it is the x-axis. It seems that
 the way to determine which axis in the " }{XPPEDIT 18 0 "xy;" "6#%#xy
G" }{TEXT -1 235 "-plane Maple is labeling is as follows. Locate a lab
el and draw a line from that label to each axis so that it meets the a
xis at a right angle. The axis that meets this line closest to the ori
gin is the axis this label corresponds to. " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "This method doesn't seem to wor
k with the " }{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 21 "-axis. Note t
hat the " }{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 113 "-axis doesn't a
ppear in the graph, this labeling is optional. You can specify your ow
n labels with the parameter " }{TEXT 269 8 "labels. " }{TEXT -1 32 "Th
is parameter is list of three " }{TEXT 270 9 "strings, " }{TEXT -1 86 
"names appearing withing quotes. Lists are object separated by commas \+
withing brackets." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 61 "plot3d(x+y,x=-1..1,y=-1..1,axes=normal,labels
=[\"x\",\"y\",\"z\"]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 362 "To explore the various options for graphing functio
ns of two variables you can click on the graph and experiment with the
 options on the context bar. You can also option-click (right-click fo
r Windows) the graph for contextual pop-up menus. All of the things yo
u can do from the context bar and contextual pop-up menus can be done \+
by supplying parameters to the " }{TEXT 272 7 "plot3d " }{TEXT -1 67 "
command. See the help on this command for the names of the options." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 139 "If you \+
click on the graph and hold the mouse button down you can rotate the g
raph is two directions. As you do this note the two variables " }
{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 5 " and " }{XPPEDIT 18 
0 "psi;" "6#%$psiG" }{TEXT -1 75 " that appear in the context bar on t
he left. These represent the angles of " }{TEXT 271 12 "orientation " 
}{TEXT -1 71 "and these angles can be supplies as a parameter in the p
lot3d command. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 64 "plot3d(x^2-y^2,x=-1..1,y=-1..1,axes=boxed,orie
ntation=[145,62]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "G
raphs of Equations: " }{TEXT 273 13 "implicitplot " }{TEXT -1 4 "and \+
" }{TEXT 280 14 "implicitplot3d" }}{PARA 0 "" 0 "" {TEXT -1 19 "Within
 the package " }{TEXT 274 6 "plots " }{TEXT -1 18 "are the functions \+
" }{TEXT 275 13 "implicitplot " }{TEXT -1 4 "and " }{TEXT 276 16 "impl
icitplot3d. " }{TEXT -1 186 "These commands are used to plot the graph
s of equations. These algorithms used to produce these plots are numer
ical so the graphs may not look as pretty as the the graphs obtained u
sing " }{TEXT 277 4 "plot" }{TEXT -1 5 " and " }{TEXT 278 6 "plot3d" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "implicitplot(x^2+y^2=1,x=-1..1,y=-1..1);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "implicitplot3d(x^2+y^2+z^2=1,x=-1..
1,y=-1..1,z=-1..1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 167 "This is how we would plot a hemisphere as the \+
graph of a function. Note that the graph gets cut off at places where \+
the function is not defined (where the argument to " }{TEXT 279 5 "sqr
t " }{TEXT -1 13 "is negative)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "
plot3d(sqrt(1-x^2-y^2),x=-1..1,y=-1..1,axes=normal);" }}}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 16 "Quad
ric Surfaces" }}{PARA 0 "" 0 "" {TEXT -1 53 "These are some basic exam
ples of surfaces in 3-space." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 63 "i3dpars := x=-5..5,y=-5..5,z=-5..5,axes=normal,grid=[
15,15,15];" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "The Ellipsoid" }}
{PARA 256 "" 0 "" {XPPEDIT 18 0 "x^2/4+y^2/9+z^2/16 = 1;" "6#/,(*&%\"x
G\"\"#\"\"%!\"\"\"\"\"*&%\"yGF'\"\"*F)F**&%\"zGF'\"#;F)F*F*" }{TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "implicitplot3d(x^2
/4+y^2/9+z^2/16=1,x=-5..5,y=-5..5,z=-5..5,axes=normal,grid=[15,15,15])
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "implicitplot3d(x^2/4+
y^2/9+z^2/16=1,x=-5..5,y=-5..5,z=-5..5,axes=normal,grid=[15,15,15],sty
le=patchcontour);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 75 "plot3d([cos(s)*sin(t),sin(s)*sin(t),cos(t)],
s=0..2*Pi,t=0..Pi,axes=normal);" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 
23 "The Elliptic Paraboloid" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "z = x^
2+y^2;" "6#/%\"zG,&*$%\"xG\"\"#\"\"\"*$%\"yGF(F)" }{TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "plot3d(x^2+y^2,x=-1..1,y=-ab
s(sqrt(1-x^2))..abs(sqrt(1-x^2)),axes=normal);" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "plot3d(x^2+
y^2,x=-1..1,y=-abs(sqrt(1-x^2))..abs(sqrt(1-x^2)),axes=normal,style=pa
tchcontour,orientation=[-4,71]);" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 
-1 25 "The Hyperbolic Paraboloid" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "z
 = x^2-y^2;" "6#/%\"zG,&*$%\"xG\"\"#\"\"\"*$%\"yGF(!\"\"" }{TEXT -1 0 
"" }}{EXCHG {PARA 4 "> " 0 "" {MPLTEXT 1 0 65 "plot3d(x^2-y^2,x=-4..4,
y=-4..4,axes=normal,orientation=[114,57]);" }}}{PARA 4 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 4 "> " 0 "" {MPLTEXT 1 0 84 "plot3d(x^2-y^2,x=-
4..4,y=-4..4,axes=normal,orientation=[114,57],style=patchcontour);" }}
}{EXCHG {PARA 4 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 17 "The Elliptic Cone" }}{PARA 4 "" 0 "" {XPPEDIT 18 0 "x^2
+y^2 = z^2;" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(*$%\"zGF'" }}{EXCHG 
{PARA 4 "> " 0 "" {MPLTEXT 1 0 119 "implicitplot3d(x^2+y^2=z^2,x=-5..5
,y=-5..5,z=-5..5,axes=normal,grid=[15,15,15],style=patchcontour,orient
ation=[54,76]);" }}}{EXCHG {PARA 4 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "The Hyberboloid of One Sheet" }}
{PARA 4 "" 0 "" {XPPEDIT 18 0 "x^2+y^2-z^2 = 1;" "6#/,(*$%\"xG\"\"#\"
\"\"*$%\"yGF'F(*$%\"zGF'!\"\"F(" }}{EXCHG {PARA 4 "> " 0 "" {MPLTEXT 
1 0 77 "implicitplot3d(x^2+y^2-z^2=1,i3dpars,style=patchcontour,orient
ation=[36,78]);" }}}{EXCHG {PARA 4 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "The Hyperboloid of Two Sheets" }}
{PARA 4 "" 0 "" {XPPEDIT 18 0 "z^2-x^2-y^2 = 1;" "6#/,(*$%\"zG\"\"#\"
\"\"*$%\"xGF'!\"\"*$%\"yGF'F+F(" }}{EXCHG {PARA 4 "> " 0 "" {MPLTEXT 
1 0 77 "implicitplot3d(z^2-x^2-y^2=1,i3dpars,style=patchcontour,orient
ation=[45,79]);" }}}{EXCHG {PARA 4 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "The Parabolic Cylinder" }}{PARA 
4 "" 0 "" {XPPEDIT 18 0 "z = y^2;" "6#/%\"zG*$%\"yG\"\"#" }{TEXT -1 0 
"" }}{EXCHG {PARA 4 "> " 0 "" {MPLTEXT 1 0 60 "plot3d(y^2,x=-3..3,y=-3
..3,axes=normal,orientation=[49,64]);" }}}{EXCHG {PARA 4 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 
"" 0 "" {TEXT -1 23 "The Hyperbolic Cylinder" }}{PARA 4 "" 0 "" 
{XPPEDIT 18 0 "y^2-z^2 = 1;" "6#/,&*$%\"yG\"\"#\"\"\"*$%\"zGF'!\"\"F(
" }{TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 4 "> \+
" 0 "" {MPLTEXT 1 0 73 "implicitplot3d(y^2-z^2=1,i3dpars,style=patchco
ntour,orientation=[24,53]);" }}}{EXCHG {PARA 4 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "The Elliptic Cylinder" }
}{PARA 256 "" 0 "" {XPPEDIT 18 0 "x^2/4+z^2/9 = 1;" "6#/,&*&%\"xG\"\"#
\"\"%!\"\"\"\"\"*&%\"zGF'\"\"*F)F*F*" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 87 "implicitplot3d(x^2/4+z^2/9 = 1,x=-3..3,y=-3..3,z=-3..
3,axes=boxed,orientation=[65,62]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 
3 "" 0 "" {TEXT -1 9 "Animation" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 
26 "Functions of Two Variables" }}{PARA 0 "" 0 "" {TEXT -1 143 "Given \+
a function of two variables we can hold one of the variables fixed and
 allow the other to vary. The result is a function of one variable." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 
256 9 "Example. " }{TEXT -1 23 "We define the function " }{XPPEDIT 18 
0 "f(x,t) = (x-t)^2;" "6#/-%\"fG6$%\"xG%\"tG*$,&F'\"\"\"F(!\"\"\"\"#" 
}{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f := (x,t)
 -> (x-t)^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }{TEXT -1 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 56 "Note the syntax f
or defining functions of two variables." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 11 "If we hold " }{XPPEDIT 18 0 "t;" "6#
%\"tG" }{TEXT -1 49 " fixed the result is parabola with vertex on the \+
" }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 10 "-axis. As " }{XPPEDIT 
18 0 "t;" "6#%\"tG" }{TEXT -1 65 " varies the parabolas are shifted to
 the right. For example, let " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT 
-1 7 "=0 and " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 39 "=1 and look
 at the resulting graphs of " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 
18 " as a function of " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 1 "." 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "t:=0;" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 21 "plot(f(x,t),x=-5..5);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 5 "t:=1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 21 "plot(f(x,t),x=-5..5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "W
e can put several of these graphs together in an animated sequence usi
ng the " }{TEXT 258 8 "animate " }{TEXT -1 51 "command. This command i
s part of the Maple package " }{TEXT 259 5 "plots" }{TEXT -1 1 "." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "t:='t'; " }{TEXT -1 22 "Unassign the
 variable " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "animate(f(x,t),x=-10..10,t=0..5);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 3 "To " }{TEXT 260 5 "play " }{TEXT -1 58 "this animation, pl
ace your cursor on the graph and select " }{TEXT 261 5 "Play " }{TEXT 
-1 9 "from the " }{TEXT 262 10 "Animation " }{TEXT -1 66 "menu. You ma
y also experiment with the other options in this menu." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 118 "The default number \+
frames Maple uses when creating these animations is 16. This number ca
n be modified by setting the " }{TEXT 263 7 "frames " }{TEXT -1 106 "p
arameter in the animate command. This will produces a smoother animati
on but will take more time to play." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "animate(f(x,t),x=-10..10,t=0..5,frames=50);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 264 10 "Exercise. " }
{TEXT -1 63 "Suppose the \"wave\" in a sports arena is modeled by the \+
function" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "h(x,t) = 5+cos(.5*x-t);" 
"6#/-%\"hG6$%\"xG%\"tG,&\"\"&\"\"\"-%$cosG6#,&*&$F*!\"\"F+F'F+F+F(F2F+
" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 
"x;" "6#%\"xG" }{TEXT -1 36 " is the seat number in a fixed row, " }
{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 24 " is time in seconds and " }
{XPPEDIT 18 0 "h;" "6#%\"hG" }{TEXT -1 242 " is the height of the pers
on's head (which is moving up and down due to the wave). Create an ani
mation of this wave for seats 0 through 20 during the first four secon
ds. Can you modify the time length so that continuous animation is \"s
mooth\"?" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "Functions of Three V
ariables" }}{PARA 0 "" 0 "" {TEXT -1 28 "Here we can use the command \+
" }{TEXT 281 10 "animate3d " }{TEXT -1 30 "to create an animation of t
he " }{TEXT 282 7 "slices " }{TEXT -1 14 "of a function " }{XPPEDIT 
18 0 "f(x,y,z);" "6#-%\"fG6%%\"xG%\"yG%\"zG" }{TEXT -1 46 " where the \+
third variable is held fixed. When " }{XPPEDIT 18 0 "z;" "6#%\"zG" }
{TEXT -1 164 " is held fixed we obtain a function of two variables whi
ch can be graphed in 3-space. As we vary z the graph will vary and we \+
can animate this variation in 3-space." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 283 9 "Example. " }{TEXT -1 
17 "The bobbing hump." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 26 "Here we study the function" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 39 "f := (x,y,t) -> sin(t)*exp(-(x^2+y^2));" }}}{PARA 
0 "" 0 "" {TEXT -1 24 "Thus we used the symbol " }{XPPEDIT 18 0 "t;" "
6#%\"tG" }{TEXT -1 24 " for the third variable." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "animate3d(f(x,y,t),x=-5
..5,y=-5..5,t=0..Pi,frames=30);" }}}{PARA 0 "" 0 "" {TEXT -1 53 "To pl
ay the animation, click on the graph and select " }{TEXT 284 5 "Play \+
" }{TEXT -1 116 "from the Animation menu (note: the Animation menu wil
l not appear until you click on graphics than can be animated)." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 
287 9 "Warning: " }{TEXT -1 202 "graphics such as animations may use l
ots of memory. If these animations cause your computer to crash you ca
n decrease the number of frames in the animation or you can try lettin
g Maple have more memory." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "Other types of animation" }
}{PARA 0 "" 0 "" {TEXT -1 195 "There may be times when you have create
d your own sequence of graphical objects which is not a sequence of gr
aphs of functions. For example, suppose we examine the level surfaces \+
of the function " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 21 " in the \+
last example." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 39 "f := (x,y,t) -> sin(t)*exp(-(x^2+y^2));" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "We can define a variable
 containing the parameters we like so we do not have to type them repe
atedly. The parameter " }{TEXT 297 5 "grid " }{TEXT -1 63 "can be used
 to control how precisely the implicit plot is done." }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 90 "ourParameters := x=-5..5,y=-5..5,t=0..Pi,
axes=normal,labels=[\"x\",\"y\",\"t\"],grid=[15,15,15];" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 41 "implicitplot3d(f(x,y,t)=0,ourParameters);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "implicitplot3d(f(x,y,t)=1/2,
ourParameters);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "implicit
plot3d(f(x,y,t)=1,ourParameters);" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 60 "The last graph consists of only one point, x=0, y=0, t=Pi
/2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Su
ppose we want to make an animation of these level surfaces as the valu
e of " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 43 " changes from 0 to \+
1. We do this using the " }{TEXT 285 8 "display " }{TEXT -1 17 "comman
d from the " }{TEXT 286 6 "plots " }{TEXT -1 8 "package." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "In the following c
ommand I store some graphics in the variable " }{TEXT 288 12 "myGraphi
cs. " }{TEXT 289 0 "" }{TEXT 290 36 "It is important to use a colon he
re." }{TEXT 291 1 " " }{TEXT 292 0 "" }{TEXT -1 132 "When assigning gr
aphics to variables Maple outputs its internal representation of the g
raphics, and this representation can be long." }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 57 "myGraphics := implicitplot3d(f(x,y,t)=1/2,ourPar
ameters):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "display(myGrap
hics);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
43 "We can display two graphics simultaneously." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "myGraphics1 \+
:= implicitplot3d(f(x,y,t)=0,ourParameters):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 58 "myGraphics2 := implicitplot3d(f(x,y,t)=1/2,ourPa
rameters):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "display(myGra
phics1,myGraphics2);" }}{PARA 13 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "We can animate these
 graphics by using the " }{TEXT 293 11 "insequence " }{TEXT -1 93 "par
ameter. This creates a set of graphics objects that can be played from
 the Animation menu." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "disp
lay([myGraphics1,myGraphics2],insequence=true);" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "Suppose we want to inclu
de more frames. We do not want to type in a long list of graphics vari
ables by hand. Here is where " }{TEXT 294 6 "lists " }{TEXT -1 91 "com
e in handy. We will not go into detail here, you can refer to Maple li
terature for that." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 29 "Here is an example of a list." }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 20 "myList := [1,2,3,4];" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "We can modify a list using a funct
ion." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "g := x -> x^2;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "g(myList);" }}}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "That is not what we wa
nted. To make a new list with the squares of those numbers we use the \+
" }{TEXT 295 4 "map " }{TEXT -1 90 "command, which applies a procedure
 (a function in this case) to every element in the list." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "listOfSquares := map(g,myList);" }}
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "Now let'
s make a list of graphics objects. We will look at the level surfaces \+
of the function " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 67 " above. \+
More precisely we will look at the graphs of the equations " }
{XPPEDIT 18 0 "f = C;" "6#/%\"fG%\"CG" }{TEXT -1 5 " for " }{XPPEDIT 
18 0 "C;" "6#%\"CG" }{TEXT -1 64 " going from 0 to 2 in increments of \+
0.2. Here is how we do this." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "levelList := [0.2*s $ s=0..1
0];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "graphicsMap := s -> \+
implicitplot3d(f(x,y,t)=s,ourParameters);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 44 "graphicsList := map(graphicsMap,levelList): " }
{TEXT -1 0 "" }{TEXT 296 17 "Must use a colon!" }{TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "display(graphicsList,inseque
nce=true);" }}}{PARA 0 "" 0 "" {TEXT -1 95 "Now play the animation. Th
e result of such an animation is on my Maple web page under the link \+
" }{TEXT 298 24 "Level Surface Animation." }{TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "0 \+
0" 3 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }