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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 13 "Lecture 8 (a)" }}}{SECT 
1 {PARA 3 "" 0 "" {TEXT -1 29 "Basic three-dimensional plots" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "First we consider the ploting of \+
" }{TEXT 282 24 "an expression or formula" }{TEXT -1 12 " of the form
" }{TEXT 256 11 " z = f(x,y)" }{TEXT -1 58 " for x and y in a given ra
nge.  First we plot the surface " }{TEXT 257 12 "z = cos(x*y)" }{TEXT 
-1 139 " with x between -3 and 3 and y between -3 and 3. Of course, th
ere is nothing special about x, y and z. Any variablew could be used i
nstead." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "plot3d(cos(x*y),
x=-3..3,y=-3..3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 226 "Note that i
f you click on the graph you can rotate it with your cursor. Also a ro
w of options, different types of spheres, etc. appear on the menu. By \+
clicking on these you can see the effect on the graph. Try out the com
mand " }{TEXT 258 15 "?plot3d[options" }{TEXT -1 75 " ]to see what som
e of the many additional features you have to choose from." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "?plot3d[options]" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 34 "As for ordinary plots you may use " }{TEXT 283 
10 "procedures" }{TEXT -1 12 " instead of " }{TEXT 284 21 "algebraic e
xpressions" }{TEXT -1 88 " as in the above example.To illustrate we co
nvert the above expression into a procedure:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 25 "f:=unapply(cos(x*y),x,y);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 22 "plot3d(f,-3..3,-3..3);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Changes made with th
e mouse can also be accomplished by adding various options to the " }
{TEXT 285 6 "plot3d" }{TEXT -1 36 " command itself: Here is an example
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "plot3d(f,-3..3,-3..3,g
rid=[50,50],axes=box, style = patchcontour);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 25 "Space \+
curves and surfaces" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Execute the
 commands " }{TEXT 259 11 "with(plots)" }{TEXT -1 5 " and " }{TEXT 
260 15 "with(plottools)" }{TEXT -1 114 " to see some of the many other
 ploting procedures in Maple. I will illustrate a few more below: Firs
t note that a " }{TEXT 263 5 "curve" }{TEXT -1 36 " in three space may
 be specified by " }{TEXT 268 20 "parametric equations" }{TEXT -1 2 ":
 " }}{PARA 0 "" 0 "" {TEXT -1 30 "\n                             " }
{TEXT 261 22 "x=f(t), y=g(t), z=h(t)" }{TEXT -1 13 "  \nor simply " }}
{PARA 0 "" 0 "" {TEXT -1 7 "       " }}{PARA 0 "" 0 "" {TEXT 262 44 " \+
                           [f(t),g(t),h(t)]" }{TEXT -1 1 " " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "where t is in s
ome interval: Here's a simple example:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 77 "spacecurve([cos(t),sin(t),t],t=0..6*Pi,color=black, thickness=3,
axes=normal);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "A " }{TEXT 264 7 
"surface" }{TEXT -1 66 " in three space may be defined by parametric e
quations of the form" }}{PARA 0 "" 0 "" {TEXT -1 26 " \n              \+
          " }{TEXT 265 30 "x = f(s,t), y=g(s,t), z=h(s,t)" }{TEXT -1 
2 " \n" }}{PARA 0 "" 0 "" {TEXT -1 74 "where s and t lies in some regi
on in the plane: Here we map the rectangle " }{TEXT 266 13 "[0,Pi]x[-1
,1]" }{TEXT -1 23 " to half of a cylinder:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 79 "plot3d([cos(s),sin(s),t], s=0..2*Pi,t=-1..1,\nscaling
=constrained,\naxes=normal);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "H
ere's a helicoid:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "plot3d
([r*cos(s),r*sin(s),s], r=0..1,s=0..6*Pi,grid=[15,45],style=patch,shad
ing=zhue, axes=box); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 49 "Surfaces in cylindrical and sp
herical coordinates" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "One can als
o use " }{TEXT 269 8 "spherica" }{TEXT -1 5 "l or " }{TEXT 270 23 "cyl
indrical coordinates" }{TEXT -1 10 " with the " }{TEXT 271 6 "plot3d" 
}{TEXT -1 46 " procedure: Here is an example of each:   See " }{TEXT 
290 7 "?coords" }{TEXT -1 6 "  and " }{TEXT 291 15 "?plot3d[coords]" }
{TEXT -1 128 " for more details on each of the coordinate systems. The
re are over 30 different types of coordinate systems supported by Mapl
e." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Rec
all that in " }{TEXT 288 21 "spherical coordinates" }{TEXT -1 47 " the
re are three parameters usually denoted by " }{XPPEDIT 18 0 "rho;" "6#
%$rhoG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT 
-1 6 ", and " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 98 ".  If you
 have forgotten look it up in your calculus book. Most commonly surfac
es are of the form " }{XPPEDIT 18 0 "rho = f(theta,phi);" "6#/%$rhoG-%
\"fG6$%&thetaG%$phiG" }{TEXT -1 17 ". In our example " }{XPPEDIT 18 0 
"rho = 1;" "6#/%$rhoG\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "plot3d(1, theta=
0..2*Pi, phi=0..Pi,style=patch,coords=spherical, scaling = constrained
, title=\"Sphere of Radius 1\");\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 4 "For " }{TEXT 289 23 "cylindrical coordinates" }{TEXT -1 21 " the
 variable are r, " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 44 "
, and z. Usually the surface is give by r = " }{XPPEDIT 18 0 "f(theta,
z);" "6#-%\"fG6$%&thetaG%\"zG" }{TEXT -1 23 ".In our example r = 2. " 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "plot3d(2,theta=0..2*Pi, \+
z=-1..1,style=patch, coords=cylindrical, scaling=constrained, title=\"
Cylinder of Radius 2\");" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 40 "We can put these in the same plot using \+
" }{TEXT 267 7 "display" }{TEXT -1 18 ": Remember to use " }{TEXT 286 
7 "display" }{TEXT -1 14 " you need the " }{TEXT 287 5 "plots" }{TEXT 
-1 39 " package which we already loaded above." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 260 "P1:=plot3d
(1, theta=0..2*Pi, phi=0..Pi,style=patch,coords=spherical, scaling = c
onstrained, color = red):\n\nP2:=plot3d(2,theta=0..2*Pi, z=-1..1,style
=patch, coords=cylindrical, scaling=constrained, color = green):\n\nP3
:=plot3d(0,x=-3..3,y=-3..3, color = white):\n" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 67 "display([P1,P2,P3], title=\"Sphere Inside Cylind
er Cut By A Plane\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 8 "Tubeplot" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 292 9 "Tubeplots" }{TEXT 
-1 93 " are plots of tubes about one or more space curves: First let's
 do a circle in the x-y plane:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Curve:=[cos(theta),sin(theta
),0],theta=0..2*Pi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "spac
ecurve(Curve, axes = normal, color = black);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 52 "Now we put a tube around the circle to make a torus:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "tubeplot(Curve,radius=1/4
, numpoints=20, tubepoints=10,scaling=constrained, style=patch, axes =
 normal);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 91 "The tube radius can be made to depend on the curve
's parameter as in the following example:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 144 "tubeplot([cos(theta),sin(theta),0],theta=0..(2.5)*
Pi,radius=theta/8, numpoints=40, tubepoints=20,scaling=constrained, st
yle=patch, axes = none);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 14 "Space station " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 848 "rest
art:\nwith(plots):\n\naxel:=tubeplot([t,0,0],t=-1..11, radius=sin(t)^2
*1/2,scaling=constrained, color=yellow):\n\np1:=tubeplot([0,4*cos(t),4
*sin(t)],t=0..2*Pi,radius=1/2,scaling=constrained,color=red):\n\nq:=tu
beplot([0,cos(t)/2,sin(t)/2],t=0..2*Pi,radius=1/8,scaling=constrained,
 color=red):\n\nfor i from 0 to 9 do\np[i]:=tubeplot([0,t*cos(i*2*Pi/1
0),t*sin(i*2*Pi/10)],t=.5..4,radius=1/8,scaling=constrained,color=gree
n):\nod:\n\np12:=tubeplot([10,4*cos(t),4*sin(t)],t=0..2*Pi,radius=1/2,
scaling=constrained,color=red):\n\nq2:=tubeplot([10,cos(t)/2,sin(t)/2]
,t=0..2*Pi,radius=1/8,scaling=constrained, color=red):\n\nfor i from 0
 to 9 do\np3[i]:=tubeplot([10,t*cos(i*2*Pi/10),t*sin(i*2*Pi/10)],t=.5.
.4,radius=1/8,scaling=constrained,color=green):\nod:\n\ndisplay(\{axel
,q,p1,seq(p[i],i=0..9), q2,p12,seq(p3[i],i=0..9)\},scaling=constrained
, title = \"Space Station\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 32 "Three dimensional impl
icit plots" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 87 "This refers to the ploting of surfaces that are solution \+
sets of equations of the form " }}{PARA 0 "" 0 "" {TEXT -1 23 "\n     \+
                 " }{TEXT 272 11 "f(x,y,z)=0 " }{TEXT -1 1 "\n" }}
{PARA 0 "" 0 "" {TEXT -1 127 "that cannot be solved for one variable i
n terms of the other two. For example let's see what the surface \n\n \+
                   " }{TEXT 273 1 " " }{XPPEDIT 274 0 "x^2+y^2-z^2 = 1
;" "6#/,(*$%\"xG\"\"#\"\"\"*$%\"yGF'F(*$%\"zGF'!\"\"F(" }{TEXT -1 14 "
 \n\nlooks like:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart
:\nwith(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "implici
tplot3d(x^2 + y^2 - z^2 = 1  ,x=-3..3,y=-3..3,z=-3..3, grid=[20,20,20]
,\nstyle=patchcontour, contours=20,axes=box);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 37 "Packag
es are tables and filled = true" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 32 "Note that you don't have to use " }{TEXT 275 11 "w
ith(plots)" }{TEXT -1 36 " to be able to use a command in the " }
{TEXT 276 5 "plots" }{TEXT -1 84 " package. You may instead use the fo
llowing method. Note that a package is really a " }{TEXT 281 5 "table
" }{TEXT -1 123 " and the indices of the table consist of the names of
 the procedures in the table. \n\nThis is an interesting feature of th
e " }{TEXT 280 12 "implicitplot" }{TEXT -1 9 " command:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "plots[implicitplot](x^2+y^2=1, x=-2
..2,y=-2..2,\nscaling = constrained,\nfilled=true);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 11 "The option " }{TEXT 278 13 "filled = true" }
{TEXT -1 28 " also works with the simple " }{TEXT 277 4 "plot" }{TEXT 
-1 9 " command:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "plot(x^2
,x=-1..1, filled = true);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Here
's another example of the use of a package procedure without use of " 
}{TEXT 279 5 "with." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "A:=m
atrix([[a,b],[c,d]]);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "d
et(A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "linalg[det](A);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(combinat);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "type(combinat,table);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "combinat[powerset](\{1,2,3\}
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "indices(combinat);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 36 "The package p
lottools -- stop lights" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 115 "The package plottools has a number of us
eful procedures. I will illustrate just one of them, namely, the proce
dure " }{TEXT 293 6 "disk. " }{TEXT -1 12 "The command " }{TEXT 294 
27 "disk([x,y],r, color = red) " }{TEXT -1 142 " creates a two-dimensi
onal plot data object which when displayed, is a disk of radius r cent
ered at [x,y] colored red.  Note that the command " }{TEXT 295 7 "disp
lay" }{TEXT -1 34 " must be used to display the disk." }}}{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 16 "with(plottools);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "DD:=disk([0,0],4, color=green, scal
ing = constrained):\ndisplay(DD, axes = none);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 162 "L:=[green,yellow,red]:  display([seq(disk([1,
i],.45,colour=L[i]),i=1..3)],scaling=constrained, axes = boxed, xtickm
arks=[], ytickmarks=[], title = \"Stop Lights\");" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "5 0 0" 7 }{VIEWOPTS 0 0 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }