{VERSION 6 0 "IBM INTEL NT" "6.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 
0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 
0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal
" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 
1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 
-1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 
0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 
2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 
256 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 
0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "T
imes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }
}
{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 42 "Three Dimensional Graphi
ng\nAn Introduction" }}{PARA 257 "" 0 "" {TEXT -1 21 "Math 252, Spring
 2005" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "# last update 1/24
/2005" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 20 "Plotting with plot3d" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 163 "As always, we can use the function or the expres
sion approach to obtain a three dimensional graph of a function.  Let'
s illustrate both methods for the paraboloid " }{XPPEDIT 18 0 "z = x^2
+y^2;" "6#/%\"zG,&*$%\"xG\"\"#\"\"\"*$%\"yGF(F)" }{TEXT -1 23 " or, in
 function form  " }{XPPEDIT 18 0 "f(x,y) = x^2+y^2;" "6#/-%\"fG6$%\"xG
%\"yG,&*$F'\"\"#\"\"\"*$F(F+F," }{TEXT -1 21 "  , and begin in the " }
{TEXT 256 16 "function setting" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 22 "f := (x,y) -> x^2+y^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 22 "plot3d(f,-3..3,-3..3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 
"The same graph in the " }{TEXT 261 18 "expression setting" }{TEXT -1 
15 " is obtained by" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "z := x^2+y^2
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot3d(z,x=-3..3,y=-3.
.3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 251 "Study the differences ca
refully!  In the function setting you indicate the range of the variab
les, regardless of their names, in the expression setting you have to \+
specify that you mean x and y.  In the sequel I will use either method
.  Here are some " }{TEXT 257 7 "options" }{TEXT -1 48 " you may use t
o improve the looks of your graph:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 107 "plot3d(f(x,y),x=-3.5..3.5,y=-3.5..3.5,axes=normal, v
iew=[-3.5..3.5,-3.5..3.5 ,-1..10], title='Paraboloid');" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 244 "Other useful options include\nscaling = \+
constrained (for 1:1 appearence), style=patchcontour (to show curve of
 equal \"elevation\", numpoints = 1000 (or higher, to improve the reso
lution of your graph), color = ...(set the color of your surface). " }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 14 "Several Graphs" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 177 "
Several surfaces can be displayed in a common graph with  plot3d as we
ll, just enclose the respective equations with brackets.  Example (the
 definition of f is still in effect):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 103 "plot3d([f(x,y), 4 + x - y/2],x=-3.5..3.5, y=-3.5..3.
5, axes=normal, view=[-3.5..3.5,-3.5..3.5,-1..10]);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 224 "A second method to show several surfaces in a \+
common graph is to apply the display command.  Save each graph as G1, \+
G2, etc. and then generate the combinded graph by display(G1,G2,G3).  \+
A \"with(plots)\" is required.  Example:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 214 "with(plots):\nPara := plot3d( f(x,y),x=-3.5..3.5,y=-
3.5..3.5,axes=normal, view=[-3.5..3.5,-3.5..3.5 ,-1..10], title='Parab
oloid'):\nplane := plot3d( 4+x-y/2, x=-3.5..3.5,y=-3.5..3.5, color=blu
e):\ndisplay(Para,plane);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Countours" }}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 95 "Curves of equal function values can be displaye
d with the style=patchcontopur option.  Example:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "f := (x,y) -> (x+y)/(x^2+y^2+1);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 57 "plot3d(f,-10..10,-10..10,style=patchcontour,
axes=normal);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 460 "Be encopuraged \+
to twist and turn the surface so that the xy-plane is horizontal and t
he z-axis is vertical.  On my version of maple the different levels of
 function values appear in increments of 0.1 in the z-directiopn.  If \+
you turn the graph such that you view the surface from above, you will
 see a family of circles in the xy-plane.  These are the level curves.
  Level curves can be obtained directly with the contourplot command (
\"with(plots):\" required)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with
(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "contourplot(f,-
10..10,-10..10);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Some options \+
for contourplot are included below." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
91 "contourplot(f,-10..10,-10..10,contours=12,coloring=[blue,red], fil
led=true,numpoints=1000);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 
"" 0 "" {TEXT -1 37 "Cylindrical and Spherical Coordinates" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "resta
rt;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "The syntax for " }{TEXT 
258 22 "cylindrical coordinate" }{TEXT -1 191 " plots is   plot3d( r ,
 theta range, z range,  .... coords = cylindrical).  Note that coords=
cylindrical is used as a plotting option.  Conversion to cylindrical c
oordinates for our example  " }{XPPEDIT 18 0 "z = x^2+y^2;" "6#/%\"zG,
&*$%\"xG\"\"#\"\"\"*$%\"yGF(F)" }{TEXT -1 10 "  yields  " }{XPPEDIT 
18 0 "z = r^2;" "6#/%\"zG*$%\"rG\"\"#" }{TEXT -1 39 " .  The plot is a
ccomplished as follows" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "p
lot3d(sqrt(z),t = 0..2*Pi,z=-1..10,coords=cylindrical,axes=normal); " 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 129 "For spherical coordinates the \+
syntax is  plot3d( rho , theta range, phi range, .. coords=spherical .
.).  In our example we have  " }{XPPEDIT 18 0 "rho = cos(phi)/(sin(phi
)^2);" "6#/%$rhoG*&-%$cosG6#%$phiG\"\"\"*$-%$sinG6#F)\"\"#!\"\"" }
{TEXT -1 24 "  and the graph becomes " }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "R := cos(s)/(sin(s)^2);\nplot3d(R,t
=0..2*Pi,s=0..Pi/2,coords=spherical);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 25 "In the example the angle " }{XPPEDIT 18 0 "theta;" "6#%&t
hetaG" }{TEXT -1 134 "  never occurred explicitly due to the rotationa
l symmetry.  Plots in spherical or cylindrical coordinates can also be
 obtained using " }{TEXT 259 10 "sphereplot" }{TEXT -1 4 " or " }
{TEXT 260 12 "cylinderplot" }{TEXT -1 53 ".  There require a \"with(pl
ots)\". An example follows." }{MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 60 "with(plots):\nsphereplot( 1.2^t*sin(s), t=-Pi.
.2*Pi,s=0..Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 14 "Implicit Plots" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 342 "Suppose that a \+
surface is given by an equation and that it is difficult or impossible
 to solve the equation for z explicitly.  In this case the can attempt
 to use the implicitplot3d command.   Implicitplot will require ranges
 for x, y and z, and it will attempt to mark all points within that cu
be, which satisfy the given equation.  Example:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 21 "restart; with(plots):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 34 "S := z^2 = x^2 - 2*x + 4*y^2 + 2;\n" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 138 "Solving the equation above for z would r
equire + or - a square root, therefore we prefer to proceed with plott
ing the equation implicitly." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "imp
licitplot3d(S,x=-2..4,y=-2..2,z = -3..3, numpoints=5000, \naxes=normal
, scaling=constrained); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 17 "Curves in 3-space" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 208 "The met
hod to plot curves in space is straightforward, use the spacecurve com
mand, provide formulas for x,y,and z, and give a range for the paramet
er..  We proceed with an example:  Graph the curve given by  " }
{XPPEDIT 18 0 "x = (cos*t)^2;" "6#/%\"xG*$*&%$cosG\"\"\"%\"tGF(\"\"#" 
}{TEXT -1 3 " , " }{XPPEDIT 18 0 "y = (sin*t)^2;" "6#/%\"yG*$*&%$sinG
\"\"\"%\"tGF(\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "z = sin(8*t);
" "6#/%\"zG-%$sinG6#*&\"\")\"\"\"%\"tGF*" }{TEXT -1 23 " for   t betwe
en 0 and " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 1 "." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart; with(plots):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "spacecurve([cos(t)^2, sin(t)^2, si
n(8*t)], t=0..Pi, numpoints=200, thickness=2, axes=normal, scaling=con
strained);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Another example:  A
 line segment which connects two given points on a graph." }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 277 "f := (x,y) -> x*y;\na := 1; b := 2; c := -1
 ; d := 2;\nP := [a, b, f(a,b)]; \nQ := [c ,d, f(c,d)];\nv := Q-P;\nX \+
:= P[1] + t*v[1];\nY := P[2] + t*v[2];\nZ := P[3] + t*v[3];\nL := spac
ecurve([X,Y,Z],t=0..1,thickness=3, color=blue, axes=normal):\nS := plo
t3d(f,-2..2,-1..3):\ndisplay(S,L);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}
{MARK "2" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 
1 1 }