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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 41 "Isothermal coordinates a
nd conformal maps" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 178 "We compute the triple compositions (St)o(U)o(X) for isot
hermal parameterizations of some minimal surfaces, and try to identify
 whether the map is indeed analytic. (It should be!)" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "restart:
\nwith(plots):\nwith(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning,
 new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new \+
definition for trace" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "1) Some p
rocedures to help compute the Gauss map:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 62 "#dot product\ndp:=proc(X,Y)\nX[1]*Y[1]+X[2]*Y[2]+X[3]
*Y[3];\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "#2-norm\nnr
m:=proc(X)\nsqrt(dp(X,X));\nend:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 128 "#cross product:\nxp:=proc(X,Y)\nlocal a,b,c;\na:=X[2
]*Y[3]-X[3]*Y[2];\nb:=X[3]*Y[1]-X[1]*Y[3];\nc:=X[1]*Y[2]-X[2]*Y[1];\n[
a,b,c];\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 206 "#Derivati
ve matrix for mapping X:\nDXq:=proc(X)\nlocal Xu,Xv;\nXu:=matrix(3,1,[
diff(X[1],u),diff(X[2],u),diff(X[3],u)]);\nXv:=matrix(3,1,[diff(X[1],v
),diff(X[2],v),diff(X[3],v)]);\nsimplify(augment(Xu,Xv));\nend:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "#unit normal:\nN:=proc(X)\n
local Y,Z,s;\nY:=DXq(X);\nZ:=xp(col(Y,1),col(Y,2));\ns:=nrm(Z);\nsimpl
ify(evalm((1/s)*Z));\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "2) \+
 Add a procedure for stereographic projection:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "St:=proc(X)
\nsimplify([X[1]/(1-X[3]),X[2]/(1-X[3]),0]);\nend:" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 73 "3)  Check the triple composition for the Catenoid
 and Enneper's Surface !" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "assume(u,real):\nassume(v,real):" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "Cat:=(u,v)->[cos(v)*cosh(
u),sin(v)*cosh(u),u]:\n   #catenoid parameterization\nSt(N(Cat(u,v)));
\n   #can you recognize this?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 52 "Enne:=(u,v)->[u-u^3/3+u*v^2,-v+v^3/3-v*u^2,u^2-v^2]:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "St(N(Enne(u,v)));\n   #how about th
is? " }}}}{MARK "1 0 0" 9 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
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