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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 17 "Standard Surfaces" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 74 "We study t
he standard surfaces - cylinders, paraboloids, cones, ellipsoids" }}
{PARA 268 "" 0 "" {TEXT -1 4 "From" }{TEXT 266 8 " Options" }{TEXT -1 
8 " choose " }{TEXT 267 12 "Plot Display" }{TEXT -1 17 " and then choo
se " }{TEXT 268 7 "Window." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Cyl
inders - equations with one variable missing" }}{EXCHG {PARA 0 "" 0 "
" {TEXT 256 22 "Consider the equation " }{XPPEDIT 18 0 "z=x^2/4" "/%\"
zG*&%\"xG\"\"#\"\"%!\"\"" }{TEXT 263 162 "   .  Note the y variable is
 missing so this is a cylinder with axis parallel to the y axis. Since
 it is of the form  z=.. we may plot it using the plot3d command" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "restart;\nplot3d( x^2/4, x=-2..2, \+
y=-2..2, orientation=[40,65], \n        axes=boxed, style = patch, tit
le = `x^2 = 4z` ) ;" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "Parabolo
ids " }}{EXCHG {PARA 257 "" 0 "" {TEXT -1 15 "One example is " }
{XPPEDIT 18 0 "z = x^2/4 + y^2/9 " "/%\"zG,&*&%\"xG\"\"#\"\"%!\"\"\"\"
\"*&%\"yG\"\"#\"\"*F)F*" }{TEXT -1 1 " " }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 156 "plot3d( x^2/4 + y^2/9, x=-3..3, y=-3..3, axes=boxed, view=0..
1,\n        title = `z=x^2/4 + y^2/9`, style=patchcontour,            \+
    orientation=[45,75] );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
258 "" 0 "" {TEXT -1 196 "Intersections of this surface with any verti
cal plane through the origin is a parabola as seen by rotating this pi
cture so that phi=90 (outer boundary is a parabola and is the intersec
tion of the " }}{PARA 269 "" 0 "" {TEXT -1 52 "paraboloid with a verti
cal plane through the z axis." }}{PARA 270 "" 0 "" {TEXT -1 237 "\nThe
 intersection of this surface with any horizontal plane is an ellipse \+
as seen by rotating picture so that phi =0 (the contour lines are elli
pses which are the curves of intersection).\n\nSo this surface is name
d an elliptic paraboloid." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Cone
s." }}{EXCHG {PARA 259 "" 0 "" {TEXT -1 25 "One example of a cone is \+
" }{XPPEDIT 18 0 "z^2 =x^2/4 + y^2/9" "/*$%\"zG\"\"#,&*&%\"xG\"\"#\"\"
%!\"\"\"\"\"*&%\"yG\"\"#\"\"*F+F," }{TEXT -1 24 " . We plot it using t
he " }{TEXT 258 15 "implicitplot3d " }{TEXT -1 7 "command" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 187 "with(plots):\nimplicitplot3d( z^2 = x^2/4 +
 y^2/9, x=-2..2, y=-3..3, z=-2..2,\n     style=patchnogrid, axes=boxed
, grid=[10,10,10], \n     title=`z^2 = x^2/4 + y^2/9`, orientation=[60
,75] );" }}}{PARA 263 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT 
-1 44 "We obtain a poor picture because of the way " }{TEXT 259 14 "im
plicitplot3d" }{TEXT -1 98 " works. To improve the picture one would h
ave to take a finer grid. Read about the options later. " }}{PARA 261 
"" 0 "" {TEXT -1 22 "Instead we will use a " }{TEXT 260 26 "parametric
 representation " }{TEXT -1 37 "of this cone to obtain a better plot.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 262 "" 0 "" {TEXT -1 
55 "Using something similar to polar coordinates, the cone " }
{XPPEDIT 18 0 "z^2 = x^2/4 + y^2/9" "/*$%\"zG\"\"#,&*&%\"xG\"\"#\"\"%!
\"\"\"\"\"*&%\"yG\"\"#\"\"*F+F," }{TEXT -1 90 "  has parametric repres
entation    x = 2r cos(t),  y = 3r sin(t), z=+ - r,           r=0.." }
{XPPEDIT 18 0 "\\infinity" "I)infinityG6\"" }{TEXT -1 11 ",    t=0..2
" }{XPPEDIT 18 0 "pi" "I#piG6\"" }{TEXT -1 1 " " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 166 "plot3d( [2*abs(r)*cos(t), 3*abs(r)*sin(t), r], t=0..
2*Pi, r=-1..1,\n        title=`z^2 = x^2/4 + y^2/9`, style=patchcontou
r,\n        axes=boxed, orientation=[60,75] );" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT -1 107 "To obtain a realistic \+
view of the cone choose the CONSTRAINED option from the Projection men
u of the plot. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 0 "" 
{TEXT -1 119 "If we flatten this picture so that phi=0 then we see tha
t sections of this surface with horizontal planes are ellipses." }}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Ellipsoids" }}{PARA 266 "" 0 "" 
{TEXT -1 31 "One example of an ellipsoid is " }{XPPEDIT 18 0 "x^2/4 + \+
y^2/9 + z^2/4 =1 " "/,(*&%\"xG\"\"#\"\"%!\"\"\"\"\"*&%\"yG\"\"#\"\"*F(
F)*&%\"zG\"\"#\"\"%F(F)\"\"\"" }{TEXT -1 124 ". Ellipsoids are spheres
 which have been deformed through stretching in some directions as wil
l be seen from the plot below\n" }}{EXCHG {PARA 267 "" 0 "" {TEXT -1 
20 "Plot obtained using " }{TEXT 261 14 "implicitplot3d" }{TEXT -1 
155 " is not very good. Instead, we plot the surface using its paramet
ric representation. Motivated by spherical coordinates, the parametric
 representation for " }{XPPEDIT 18 0 "x^2/4 + y^2/9 + z^2/4 =1" "/,(*&
%\"xG\"\"#\"\"%!\"\"\"\"\"*&%\"yG\"\"#\"\"*F(F)*&%\"zG\"\"#\"\"%F(F)\"
\"\"" }{TEXT -1 4 " is\n" }{TEXT 269 81 "x=2 sin(p) cos(t) ,    y= 3 s
in(p) sin(t),     z=2 cos(p),                 p =0.." }{XPPEDIT 270 0 
"pi" "I#piG6\"" }{TEXT 271 9 ",  t=0..2" }{XPPEDIT 272 0 "pi" "I#piG6
\"" }{TEXT 273 1 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "plot3d( [2*
cos(t)*sin(p), 3*sin(t)*sin(p), 2*cos(p)], t=0..2*Pi,\n        p = 0..
Pi, axes=boxed, style=patch, orientation=[150,64],\n        title=`x^2
/4 + y^2/9 + z^2/4 = 1` );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 144 
"The plot is misleading - it looks like a sphere. Actually, some direc
tions are stretched more than others. To obtain a realistic picture ch
oose " }{TEXT 274 11 "Constrained" }{TEXT 275 10 " from the " }{TEXT 
276 11 "Projection " }{TEXT 277 140 "menu of the plot window. Also, if
 the picture is flattened so that phi=0 then one sees that horizontal \+
sections of this surface are ellipses" }{TEXT -1 1 "." }}}}}{MARK "2 5
" 0 }{VIEWOPTS 1 1 0 1 1 1803 }