Lecture 2/19/99; Gaussian and Mean Curvature

1  +  1

Enneper's Surface

x[u_, v_]  := 
       {u  -  u^3/3  +  u v^2,  v  -  v^3/3  +  v u^2,  
        u^2  -  v^2}

ParametricPlot3D[x[u, v], {u, -2, 2}, {v, -2, 2}]

D[x[u, v], u]

xu  =  %

D[x[u, v], v]

xv  =  %

xu  .  xu

Simplify[%]

EE  =  %

xu  .  xv

Simplify[%]

FF  =  %

xv  .  xv

Simplify[%]

GG  =  %

Cross[xu, xv]

% / Sqrt[%  .  %]

Simplify[%]

PowerExpand[%]

U  =  %

U  .  U

Simplify[%]

xu

D[xu, u]

xuu  =  %

D[x[u, v], u, u]

xuv  =  D[xu, v]

xvv  =  D[xv,  v]

U  .  xuu

Simplify[%]

l  =  %

U  .  xuv

m  =  0

U  .  xvv

Simplify[%]

n  = %

(l n  -  m^2)/(EE GG  -  FF^2)

(GG l  +  EE n  -  2 FF m)/
      (2 (EE GG  -  FF^2))

Hyperboloid of Two Sheets

x[u_, v_]  := 
       {a Sinh[u] Cos[v],  b Sinh[u] Sin[v],  c Cosh[u]}

ParametricPlot3D[x[u, v], {u, -1, 1}, {v, 0, 2  Pi}]

\(\(a\  = \ 8
\)\)

b  =  8

c  =  8

Clear[a]

Clear[b]

Clear[c]

x[u, v]

D[x[u, v], u]

xu  =  %

xv  =  Simplify[D[x[u, v], v]]

xu  .  xu

Simplify[%]

EE  =  %

FF  =  Simplify[xu  .  xv]

GG  =  Simplify[xv . xv]

Cross[xu, xv]

Simplify[%]

% / Sqrt[ %  .  %]

Simplify[%]

U  =  %

D[xu, u]

xuu  =  %

D[xu, v]

xuv  =  %

D[xv, v]

xvv  =  %

U  .  xuu

Simplify[%]

l  =  %

m  =  Simplify[ U  .  xuv]

PowerExpand[l]

n  =  Simplify[U  .  xvv]

(l n  -  m^2)/(EE GG  -  FF^2)

Simplify[%]

x[u, v]

x[u, v][[1]]

x[u, v][[2]]

1/(
        a^2 b^2 c^2 
          (x[u, v][[1]]^2/a^4  +  x[u, v][[2]]^2/b^4  + 
                 x[u, v][[3]]^2/c^4)^2)

%  -  %82

Simplify[%]

H  =  
      (GG l  +  EE n  -  2 FF m)/
        (2 (EE GG  -  FF^2))

Simplify[%]

PowerExpand[%]

Surface of Revolution

x[u_, v_]  :=  {g[u],  h[u] Cos[v] ,  h[u] Sin[v]}

xu  =  D[x[u, v], u]

xv  =  D[x[u, v], v]

Cross[xu, xv]

% / Sqrt[%  .  %]

Simplify[%]

U  =  %

PowerExpand[%]

U  =  %

xuu  =  D[xu, u]

xuv  =  D[xu,  v]

xvv  =  D[xv ,  v]

EE  =  xu  .  xu

Simplify[%]

EE  ==  %

EE  =  %105

FF  =  Simplify[xu  .  xv]

GG  =  Simplify[xv  .  xv]

l  =  Simplify[U  .  xuu]

m  =  Simplify[U  .  xuv]

n  =  Simplify[U  .  xvv]

(l n  -  m^2)/(EE GG  -  FF^2)

Simplify[%]

K  =  %

The Torus

g[u_]  :=  r Sin[u]

h[u_]  :=  R  +  r Cos[u]

K

Simplify[%]

r  =  2

R  =  6

ParametricPlot3D[x[u, v], {u, 0,  2 Pi},  {v,  0,  2 Pi}]