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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 9 "Math 4530" }}{PARA 257 "
" 0 "" {TEXT -1 18 "Wednesday April 11" }}{PARA 258 "" 0 "" {TEXT -1 
41 "Isothermal coordinates and conformal maps" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 339 "We look at pictures of c
onformal mappings.  Maple maps coordinate curves to coordinate curves,
 so if you start with a grid of (small) squares, the image map will al
so be carved up into approximate squares, when your mapping is a confo
rmal mapping of part of the plane.  The conformal factor tells you how
 much the squares expand or shrink." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 469 "1) Illustrate the fact that stereograph
ic projection is conformal (and its inverse is isothermal/conformal). \+
 We computed the conformal factor rho with  Maple, on Monday.  For the
 inverse of stereographic projection, it was 2/(u^2+v^2+1).  This equa
ls 1-z, where [x,y,z] is the image point on the sphere.  Thus stereogr
aphic projection has conformal factor which is the reciprocal, namely \+
1/(1-z).  These numbers are the factors by which infinitesimal length \+
is scaled." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "restart:\nwit
h(plots):\nwith(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the \+
name changecoords has been redefined\n" }}{PARA 7 "" 1 "" {TEXT -1 80 
"Warning, the protected names norm and trace have been redefined and u
nprotected\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 232 "sphere:=pl
ot3d([2*u/(u^2+v^2+1),2*v/(u^2+v^2+1),(u^2+v^2-1)/(u^2+v^2+1)],\nu=-2.
.2,v=-2..2,grid=[40,40]):\nplane:=plot3d([u,v,0],u=-1..1,v=-1..1,grid=
[20,20]):\ndisplay(\{sphere,plane\}, title = \"Stereographic projectio
n is isothermal\");" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "2) Import \+
procedures 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 0 125 "#Matrix of first fundamenta
l form:\ngij:=proc(X)\nlocal g11,g12,g22,Y;\nY:=evalm(DXq(X));\nsimpli
fy(evalm(transpose(Y)&*Y));\nend:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 116 "#unit normal:\nN:=proc(X)\nlocal Y,Z,s;\nY:=DXq(X);
\nZ:=xp(col(Y,1),col(Y,2));\ns:=nrm(Z);\nsimplify(evalm((1/s)*Z));\nen
d:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 215 "3)  Illustrate the minimal sur
face diagram in which we discuss the composition of isothermal paramet
erization of minimal surface, Gauss map, and stereographic projection;
 the fact that such composition is conformal." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 213 "Cat:=(u,v)->[cos(v)*cosh(u),sin(v)*cosh(u),u]:
\n   #catenoid parameterization\nSt:=(x,y,z)->[x/(1-z),y/(1-z),0]:\n  \+
 #stereographic projection\ngij(Cat(u,v));\n  #verify conformal factor
\nN(Cat(u,v));\n  #unit normal map" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
-%'matrixG6#7$7$*$)-%%coshG6#%\"uG\"\"#\"\"\"\"\"!7$F0F(" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%,$*&*&-%%csgnG6#*$)-%%coshG6#%\"uG
\"\"#\"\"\"F4-%$cosG6#%\"vGF4F4F/!\"\"F9,$*&*&F*F4-%$sinGF7F4F4F/F9F9*
&*&-%%sinhGF1F4F*F4F4F/F9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
136 "plot3d([u,v,0],u=-Pi/2..Pi/2,v=0..2*Pi, grid=[20,40],\ntitle=\"Pa
ramterization domain, with square grid\");\n  #open set U, with square
 grid" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "plot3d(Cat(u,v),u
=-Pi/2..Pi/2,v=0..2*Pi,grid=[20,40],\ntitle=\"Catenoid, with isotherma
l parameterization\");\n  #catenoid piece, square grid\n" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 
"plot3d(N(Cat(u,v)),u=-Pi/2..Pi/2,v=0..2*Pi,grid=[20,40],\ntitle=\"ima
ge of Gauss map, for Catenoid\");" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 154 "plot3d(St(N(Cat(u,v))[1],N(Cat(u,v))[2],N(Cat(u,v))[
3]),\nu=-Pi/2..Pi/2, v=0..2*Pi,grid=[20,40],\ntitle=\"triple compositi
on (St)o(N)oX := g is conformal!\");" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 59 "simplify(St(N(Cat(u,v))[1],N(Cat(u,v))[2],N(Cat(u,v))
[3]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,$*&*&-%%csgnG6#*$)-%%cosh
G6#%\"uG\"\"#\"\"\"F1-%$cosG6#%\"vGF1F1,&F,F1*&-%%sinhGF.F1F'F1!\"\"F:
F:,$*&*&F'F1-%$sinGF4F1F1F6F:F:\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 303 "4)  Complex analytic maps are conformal.  In fact, conformal m
aps from the plane to the plane, when written in complex form f(z) eit
her satisfy f(z) is analytic, or f(z bar) is analytic. More precisely,
 analytic maps are conformal maps which are orientation preserving.  H
ere are 3 interesting examples:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 58 "plot3d([u^2-v^2,2*u*v,0],u=0..1,v=0..1,\ntitle=\"f(z)=z^2\");
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 109 "plot3d([r^2*cos(2*theta),r^2*sin(2*theta),0],r=0..Pi
/2,\ntheta=0..Pi/2,title=\"f(z)=z^2 in polar coordinates\");" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "plot3d([u/(u^2+v^2),-v/(u^2+
v^2),0],u=(0.2)..2,v=(.2)..2,\ngrid=[40,40],\ntitle=\"f(z)=1/z\");" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 125 "plot3d([exp(u)*cos(v),exp(u)*sin(v),0],u=-Pi/2..Pi/2
,\nv=0..2*Pi,grid=[20,40],title=\"f(z)=exp(z) - does this look\nfamili
ar?\");" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "21 0 0" 125 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }