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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 27 "K=-1 Surfaces of Revoluti
on" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "This worksheet shows how M
aple may be used to find surfaces of revolution with constant Gauss cu
rvature " }{XPPEDIT 18 0 "K=-1" "/%\"KG,$\"\"\"!\"\"" }{TEXT -1 25 ". \+
We start with the usual" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "
with(plots):with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
14 "EFG := proc(X)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "       local \+
E,F,G,Xu,Xv;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "           Xu := [
diff(X[1],u),diff(X[2],u),diff(X[3],u)];\n           Xv := [diff(X[1],
v),diff(X[2],v),diff(X[3],v)];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " \+
          E := dotprod(Xu,Xu);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " \+
          F := dotprod(Xu,Xv);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " \+
          G := dotprod(Xv,Xv);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 " \+
          simplify([E,F,G]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "   \+
    end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 177 "The following proced
ure gives the unit normal to the input surface X. If you wish to take \+
UN(X), then X must be given in parametrized form in square brackets as
 done down below." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "UN := \+
proc(X)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "      local Xu,Xv,Z,s;
\n          Xu := [diff(X[1],u),diff(X[2],u),diff(X[3],u)];\n         \+
 Xv := [diff(X[1],v),diff(X[2],v),diff(X[3],v)];" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "          Z := crossprod(Xu,Xv);" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 42 "          s := sqrt(Z[1]^2+Z[2]^2+Z[3]^2);" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "          simplify([Z[1]/s,Z[2]/s,Z
[3]/s]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "      end:" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 271 "The formula for Gauss curvature requires
 dotting certain second partial derivatives with the unit normal. This
 is done as follows. See the formulas for K and H given on page 91 and
 Lemma 2.1 on page 92. Now compare the Examples on pages 93-97 with th
e calculations below." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "lm
n := proc(X)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "       local Xu,Xv,
Xuu,Xuv,Xvv,U,l,m,n;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "           \+
Xu := [diff(X[1],u),diff(X[2],u),diff(X[3],u)];" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 58 "           Xv := [diff(X[1],v),diff(X[2],v),diff(X[3]
,v)];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "           Xuu := [diff(Xu
[1],u),diff(Xu[2],u),diff(Xu[3],u)];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 62 "           Xuv := [diff(Xu[1],v),diff(Xu[2],v),diff(Xu[3],v)];" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "           Xvv := [diff(Xv[1],v),
diff(Xv[2],v),diff(Xv[3],v)];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "  \+
         U := UN(X);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "           \+
l := dotprod(U,Xuu);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "           \+
m := dotprod(U,Xuv);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "           \+
n := dotprod(U,Xvv);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "           \+
simplify([l,m,n]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "       end:" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "Finally we can calculate Gauss \+
curvature K as follows." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "
GK := proc(X)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "      local E,F,G,
l,m,n,S,T;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "          S := EFG(X)
;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "          T := lmn(X);" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "          E := S[1];" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 20 "          F := S[2];" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "          G := S[3];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 20 "          l := T[1];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "     \+
     m := T[2];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "          n := T
[3];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "          simplify((l*n-m^2
)/(E*G-F^2));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "      end:" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Mean curvature is given by" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "MK := proc(X)" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 28 "      local E,F,G,l,m,n,S,T;" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 22 "          S := EFG(X);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "          T := lmn(X);" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 20 "          E := S[1];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "   \+
       F := S[2];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "          G :=
 S[3];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "          l := T[1];" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "          m := T[2];" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 20 "          n := T[3];" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "          simplify((G*l+E*n-2*F*m)/(2*E*G-2*F^2));" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "      end:" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 110 "Now we can enter a function h which will be the pro
file curve of a surface of revolution with parametrization " }
{XPPEDIT 18 0 "x(u,v)=[h(u)*cos(v),h(u)*sin(v),g(u)]" "/-%\"xG6$%\"uG%
\"vG7%*&-%\"hG6#F&\"\"\"-%$cosG6#F'F-*&-F+6#F&F--%$sinG6#F'F--%\"gG6#F
&" }{TEXT -1 32 " and the parametrization itself." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 22 "h:=t->h(t);g:=t->g(t);" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 40 "surfrev:=[h(u)*cos(v),h(u)*sin(v),g(u)];" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "We can take the Gauss curvature us
ing the GK procedure above." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
29 "gauss:=simplify(GK(surfrev));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
42 "Of course we may assume the profile curve " }{XPPEDIT 18 0 "alpha=
[h(u),0,g(u)]" "/%&alphaG7%-%\"hG6#%\"uG\"\"!-%\"gG6#F(" }{TEXT -1 32 
" is unit speed. This means that " }{XPPEDIT 18 0 "diff(g(u),u)^2+diff
(h(u),u)^2=1" "/,&*$-%%diffG6$-%\"gG6#%\"uGF+\"\"#\"\"\"*$-F&6$-%\"hG6
#F+F+\"\"#F-\"\"\"" }{TEXT -1 59 ". If we differentiate this expressio
n on both sides, we get" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "
diff(diff(g(u),u)^2+diff(h(u),u)^2,u)=diff(1,u);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 135 "We can now simplify the Gauss curvature of the su
rface of revolution using the two relations we have just found. The wa
y to do this is " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "gaussK
:=simplify(gauss,\{diff(g(u),u)*diff(diff(g(u),u),u)=-diff(h(u),u)*dif
f(diff(h(u),u),u), diff(g(u),u)^2+diff(h(u),u)^2=1\});" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 103 "So now we have an expression for the Gau
ss curvature assuming the profile curve is unit speed. We want " }
{XPPEDIT 18 0 "K=1" "/%\"KG\"\"\"" }{TEXT -1 88 " , so we solve the di
fferential equation obtained when we set the curvature equal to 1. " }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "sol:=dsolve(gaussK=-1,h(u)
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "h_ans:=expand(solve(s
ol[2],h(u)));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Let's put in som
e initial conditions to make h_ans explicit." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 40 "hh:=eval(subs(\{_C1=0,_C2=ln(2)\},h_ans));" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Now that we have " }{XPPEDIT 18 0 
"h(u)=exp(u)" "/-%\"hG6#%\"uG-%$expG6#F&" }{TEXT -1 48 ", we get g(u) \+
by using the unit speed condition " }{XPPEDIT 18 0 "diff(g(u),u)^2+dif
f(h(u),u)^2=1" "/,&*$-%%diffG6$-%\"gG6#%\"uGF+\"\"#\"\"\"*$-F&6$-%\"hG
6#F+F+\"\"#F-\"\"\"" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 68 "g_ans:=rhs(simplify(dsolve(diff(g(u),u)^2+diff(hh,u)^
2=1,g(u))[1]));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "int(sqrt
(1-exp(2*u)),u);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "gg:=sub
s(_C1=0,g_ans);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "We can now pl
ot the surface of revolution by substituting our functions into the su
rfrev parametrization." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "
plot3d(subs(\{g(u)=gg,h(u)=hh\},surfrev),u=-2..0,v=0..2*Pi,style=patch
,scaling=constrained,shading=zhue,lightmodel=light3,orientation=[36,80
]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Let's try another set of i
nitial conditions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "hh2:=
eval(subs(\{_C1=-1,_C2=ln(2)\},h_ans));" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 17 "Now that we have " }{XPPEDIT 18 0 "h(u)=exp(u)+1/4/exp(u)
" "/-%\"hG6#%\"uG,&-%$expG6#F&\"\"\"*(\"\"\"F+\"\"%!\"\"-F)6#F&F/F+" }
{TEXT -1 48 ", we get g(u) by using the unit speed condition " }
{XPPEDIT 18 0 "diff(g(u),u)^2+diff(h(u),u)^2=1" "/,&*$-%%diffG6$-%\"gG
6#%\"uGF+\"\"#\"\"\"*$-F&6$-%\"hG6#F+F+\"\"#F-\"\"\"" }{TEXT -1 2 ". \+
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "g_ans2:=rhs(simplify(ds
olve(diff(g(u),u)^2+diff(hh2,u)^2=1,g(u))[1]));" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 24 "gg2:=subs(_C1=0,g_ans2);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 179 "We can now plot the surface of revolution by subs
tituting our functions into the surfrev parametrization. First we chec
k that the square roots don't have negative numbers in them." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "evalf(solve(24*exp(-2*u)-16-
exp(-4*u)=0));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot(24*e
xp(-2*u)-16-exp(-4*u),u=0..2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 78 "gg3:=-1/4*int(exp(-s)*((24*exp(-2*s)-16-exp(-4*s))*exp(4*s))^(
1/2),s=-1.5..u);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "plot3d
(subs(\{g(u)=gg3,h(u)=hh2\},surfrev),u=-1.5..0.18,v=0..2*Pi,style=patc
h,scaling=constrained,shading=zhue,lightmodel=light3,orientation=[36,8
0]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 247 "Here is a general proced
ure for creating surfaces of revolution with constant negative curvatu
re. Below, you will see what you have to do if you want the pseudosphe
re. Note that we need something special for it to avoid negatives in s
quare roots. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "plotnegcon
stcurv:=proc(c,d)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 289 "local X,desys
,gu,s,h,kk,init;\nif d=0 then h:=c*exp(u);\ns:=-2.5..ln(1/c) fi;\nif c
>0 and d>0 then h:=2*sqrt(c*d)*cosh(u);\ns:=arcsinh(-1/(2*sqrt(c*d))).
.arcsinh(1/(2*sqrt(c*d))) fi;\nif c>=0 and d<0 then h:=2*sqrt(-c*d)*si
nh(u);\nkk:=c*d;\nif kk>=-1/4 then s:=0..arccosh(1/(2*sqrt(-c*d)))fi; \+
fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "X:=[h*cos(v),h*sin(v),g]; " 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "desys:=dsolve(\{diff(g(u),u)=sqrt
(1-diff(h,u)^2),g(0)=0\},g(u),\ntype=numeric,output=listprocedure):" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "gu:=subs(desys,g(u)); " }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 99 "plot3d(subs(g='gu'(u),X),u=s,v=0..2*Pi,scal
ing=constrained,style=patch,\nshading=zhue,grid=[15,40]);" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 22 "plotnegconstcurv(1,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 24 "plotnegconstcurv(0.5,1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 
"Let A=1, B=0 --- the pseudosphere." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "plotnegconstcurv(1,0);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "48 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }