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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 0 "" }{TEXT 
257 0 "" }{TEXT 258 39 "PARAMETRIC SURFACES IN THREE DIMENSIONS" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Start wit
h the usual:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 12 "with(plots);" }}{PARA 12 "" 1 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "
Suppose we are given a surface parametrically:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "                     x = \+
x(u, v),    y = y(u, v),    z = z(u, v),     " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 184 "with appropriate bounds \+
on u and v.  Here, u and v represent the independent variables (much l
ike t in our usual parametric equations), and x, y, and z depend on th
e values of u and v." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 79 "We may plot this surface using the 'plot3d' command.  T
he syntax is as follows:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 22 "             plot3d( [" }{TEXT 259 25 "x(u, v), y(
u, v), z(u, v)" }{TEXT -1 7 "], u = " }{TEXT 260 11 "lower limit" }
{TEXT -1 4 " .. " }{TEXT 261 11 "upper limit" }{TEXT -1 6 ", v = " }
{TEXT 262 11 "lower limit" }{TEXT -1 4 " .. " }{TEXT 263 11 "upper lim
it" }{TEXT -1 3 " );" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 295 "Note the use of square brackets around the x, y, and z
 part of the syntax.  (Yes, I know that when we plotted parametric cur
ves in three space, the square bracket went at the end of the upper li
mit for t and that now it goes at the end of the z expression.  That's
 the way Maple is programmed.  " }{TEXT 265 32 "Ours is not to reason \+
why......." }{TEXT -1 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 19 "Let's try graphing:" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 121 "                     x = v (1 - c
os (5u)) cos u,    y = v (1 - cos (5u)) sin u,    z = v^2,     u = 0 t
o 2 Pi, v = 0 to 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 75 "plot3d([v*(1-cos(5*u))*cos(u),v*(1-cos(5*u)
)*sin(u),v^2],u=0..2*Pi,v=0..2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "Interesting, but we have no axes a
nd the picture is \"blocky.\"  Try:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "plot3d([v*(1-cos(5*u))*co
s(u),v*(1-cos(5*u))*sin(u),v^2],u=0..2*Pi,v=0..2,axes=normal,grid=[60,
30]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1 1 0
" 0 }{VIEWOPTS 1 1 0 1 1 1803 }