Exercise 2.5

<<  curvature.m

Helcat

\(\(x[u_, v_]\  := 
      \ {Cos[t]\ Sinh[v]\ Sin[u]\  + \ Sin[t]\ Cosh[v]\ Cos[u], 
        \(-\ Cos[t]\)\ Sinh[v]\ Cos[u]\  + \ Sin[t]\ Cosh[v]\ Sin[u], \ 
        u\ Cos[t]\  + \ v\ Sin[t]}
\)\)

KK[x, u, v]

HH[x, u, v]

t  =  0

ParametricPlot3D[x[u, v], {u, 0, 8 Pi}, {v, -2, 2}]

t  =  Pi/2

ParametricPlot3D[x[u, v], {u, 0, 8 Pi}, {v, -2, 2}]

Henneberg's Surface

x[u_, v_]  := 
       {2 Sinh[u] Cos[v]  -  2/3 Sinh[3 u] Cos[3 v],  
        2 Sinh[u] Sin[v]  +  2/3 Sinh[3 u] Sin[3 v],  
        2 Cosh[2  u] Cos[2  v]}

KK[x, u, v]

HH[x, u, v]

ParametricPlot3D[x[u, v], {u, -1.5, 1.5}, {v, 0,  2 Pi}]

Catalan's Surface

x[u_, v_]  := 
       {u  -  Sin[u] Cosh[v] ,  1  -  Cos[u] Cosh[v],  
        4 Sin[u/2] Sinh[v/2]}

KK[x, u, v]

HH[x, u, v]

ParametricPlot3D[x[u, v], {u, -4 Pi,  4 Pi}, {v, -2, 2}]

, "Output"]
}, Operface (Please see tutorial)", "Subsubsection"]

Scherk's fifth surface

x[u_, v_]  :=  {ArcSinh[u],  ArcSinh[v],  ArcSin[u v]}

KK[x, u, v]

HH[x, u, v]

ParametricPlot3D[x[u, v], {u, -1, 1}, {v, -1, 1}]

Planar lines of curvature surface

x[u_, v_]  := 
       {(c u  +  a Sin[u] Cosh[v])/Sqrt[1 - c^2],  
        (v  +  a c Cos[u] Sinh[v])/Sqrt[1 - c^2],  
        a Cos[u] Cosh[v]}

a  =  1

x[u, v]

KK[x, u, v]

HH[x, u, v]

c  =  0;

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

c =  .5;

ParametricPlot3D[x[u, v], {u, 0, 6  Pi}, {v, -2, 2}]

c  =   .9

ParametricPlot3D[x[u, v], {u, 0, 6 Pi}, {v, -2, 2}]

Clear[c]

a  =  -1

x[u, v]

KK[x, u, v]

HH[x, u, v]