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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 30 "Weierstrass-Enneper procedure:
" }}{PARA 259 "" 0 "" {TEXT -1 44 "construct minimal surfaces from com
plex data" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
399 "The following code will work for local construction of minimal su
rfaces from analytic data, provided the data is simple enough for Mapl
e to evaluate.  For reconstruction over non-simply connected domains, \+
or over Riemann surfaces, or for complicated data, the code would need
 to be customized.  This code is adapted from the textbook by Oprea, \+
\"Differential Geometry and its Applications\", page 249 " }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart:with(plots):" }}{PARA 7 "" 
1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "assume(u,real);\nassume
(v,real);\nassume(t,real);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
306 "#Procedure which uses general gauss map g, adapted from Oprea pag
e 249\nWE1:=proc(f,g)\nlocal Z1,Z2,Z3, X1,X2,X3,X;\nZ1:=int(f*(1-g^2),
z);\nZ2:=int(I*f*(1+g^2),z);\nZ3:=int(2*f*g,z);\nX1:=evalc(Re(subs(z=u
+I*v,Z1)));\nX2:=evalc(Re(subs(z=u+I*v,Z2)));\nX3:=evalc(Re(subs(z=u+I
*v,Z3)));\nX:=simplify([X1,X2,X3]);\nend:" }}}{PARA 0 "" 0 "" {TEXT 
-1 21 "Test it on Enneper.  " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
107 "Enne1:=WE1(1,z);\nplot3d(Enne1,u=-1..1,v=-1..1,color=black,style=
wireframe,\nscaling=constrained,axes=boxed);" }}{PARA 11 "" 1 "" 
{TEXT -1 0 "" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 19 "Here's a variation:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 110 "Enne2:=WE1(1,z^2);\nplot3d(Enne2,u=-1..1,v=-1..1,color=black,
style=wireframe,\nscaling=constrained,axes=boxed);\n" }}{PARA 12 "" 1 
"" {TEXT -1 0 "" }}}{PARA 260 "" 1 "" {TEXT -1 129 "Interesting famili
es of surfaces:  This family interpolates the catenoid to the helicoid
, through a family of isometric surfaces." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "Helcat:=WE1(-exp(I*t)*exp(-z)/2,-exp(z));" }}{PARA 
12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "
animate3d(Helcat,u=-1..1,v=0..2*Pi,t=0..2*Pi,frames=28,\nscaling=const
rained);" }}}{PARA 261 "" 1 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 82 "If the following surface is the catenoid when t=1, what will th
e family look like?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "duh:
=WE1(-t*exp(-z)/2,-exp(z));\nanimate3d(duh,u=-1..1,v=0..2*Pi,t=0..2*Pi
,frames=20,\nscaling=constrained);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }
}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 27 "Someth
ing picked at random:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "pla
y:=WE1(exp(t*z),z);\nanimate3d(play,u=-1..1,v=-1..1,t=1..3,frames=20,
\nscaling=constrained);" }}{PARA 12 "" 1 "" {TEXT -1 0 "" }}{PARA 13 "
" 1 "" {TEXT -1 0 "" }}}}{MARK "17 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 
1 1 }{PAGENUMBERS 0 1 2 33 1 1 }