The Paraboloid

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

ParametricPlot3D[Evaluate[x[u, v]], {u, -3, 3}, {v, -3, 3}]

The Helicoid

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

a  =  4

b  =  2

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

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[Evaluate[x[u, v]], {u, -2, 2}, {v, -2, 2}, 
      PlotPoints  ->  {25, 25}]

Scherk's Fifth Surface

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

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

Calculation of Curvatures

<<  curvature.m

x[u, v]

KK[x, u, v]

HH[x, u, v]

The Torus

x[u_, v_]  := 
       { (R  + r Cos[u]) Cos[v],  (R  +  r Cos[u]) Sin[v], 
        r Sin[u]}

R  =  3

r  =  1

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

Clear[r, R]

KK[x, u, v]

HH[x, u, v]

Seeing Curvatures

r = 1

R  =  3

KK[x, u, v]

B[u_, v_]  :=  Cos[u]/(3  +  Cos[u])

ParametricPlot3D[{(R + r Cos[u]) Cos[v], 
        (R + r Cos[u]) Sin[v], r Sin[u], Hue[B[u, v]]}, {u, 0, 2 Pi}, 
       {v, 0,  2 Pi}, Lighting  ->  False, PlotPoints  ->  {25, 25}]

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

KK[x, u, v]

HH[x, u, v]

B[u_, v_]  :=  -4/(1  +  u^2  +  v^2)^4

ParametricPlot3D[{u - u^3/3  +  u v^2,  v  -  v^3/3  +  v u^2, 
         u^2  -  v^2, Hue[2.5  B[u, v]]}, {u, -1.5, 1.5}, 
       {v, -1.5, 1.5}, Lighting  ->  False, PlotPoints  ->  {50, 50}]

Geodesics on the Torus

x[u_, v_]  := 
       {(R  +  r Cos[u]) Cos[v],  (R  +  r Cos[u]) Sin[v], 
         r Sin[u]}

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

torus2  =  Show[Graphics3D[{EdgeForm[], First[torus]}]]

\(SolveThis\  := 
      \ {\((3\  + \ Cos[u[t]])\)*Sin[u[t]]*\(\(Derivative[1]\)[v]\)[t]^2\  + 
            \ \(\(Derivative[2]\)[u]\)[t]\  == \ \ 0, \ 
        \((\(-2\)*Sin[u[t]]*\(\(Derivative[1]\)[u]\)[t]*
                  \(\(Derivative[1]\)[v]\)[t])\)/
\ \ \ \ \ 
              \((3\  + \ Cos[u[t]])\)\  + \ \(\(Derivative[2]\)[v]\)[t]\  == 
          \ 0, \ u[0]\  == \ 0, \ v[0]\  == \ 0, \ 
\ \ 
        \(\(Derivative[1]\)[u]\)[0]\  == \ 1, \ 
        \(\(Derivative[1]\)[v]\)[0]\  == \  .2}\)sol  =  NDSolve[SolveThis,  {u, v}, {t, 30}]\(geodesic\  = \ 
      ParametricPlot3D[Evaluate[x[u[t], v[t]]\  /. \ sol], {t, 0, 30}, 
        PlotRange\  -> \ All]
\)

\(SolveThis\  := 
      \ {\((3\  + \ Cos[u[t]])\)*Sin[u[t]]*\(\(Derivative[1]\)[v]\)[t]^2\  + 
            \ \(\(Derivative[2]\)[u]\)[t]\  == \ \ 0, \ 
        \((\(-2\)*Sin[u[t]]*\(\(Derivative[1]\)[u]\)[t]*
                  \(\(Derivative[1]\)[v]\)[t])\)/
\ \ \ \ \ 
              \((3\  + \ Cos[u[t]])\)\  + \ \(\(Derivative[2]\)[v]\)[t]\  == 
          \ 0, \ u[0]\  == \ 0, \ v[0]\  == \ 0, \ 
\ \ 
        \(\(Derivative[1]\)[u]\)[0]\  == \ 1, \ 
        \(\(Derivative[1]\)[v]\)[0]\  == \ 0}\)sol  =  NDSolve[SolveThis,  {u, v}, {t, 30}]geodesic  =  
      ParametricPlot3D[Evaluate[x[u[t], v[t]]  /.  sol], {t, 0, 30}, 
        PlotRange  ->  All]torus2  =  Show[Graphics3D[{EdgeForm[], First[torus]}]]Show[geodesic, torus2]

\(SolveThis\  := 
      \ {\((3\  + \ Cos[u[t]])\)*Sin[u[t]]*\(\(Derivative[1]\)[v]\)[t]^2\  + 
            \ \(\(Derivative[2]\)[u]\)[t]\  == \ \ 0, \ 
        \((\(-2\)*Sin[u[t]]*\(\(Derivative[1]\)[u]\)[t]*
                  \(\(Derivative[1]\)[v]\)[t])\)/
\ \ \ \ \ 
              \((3\  + \ Cos[u[t]])\)\  + \ \(\(Derivative[2]\)[v]\)[t]\  == 
          \ 0, \ u[0]\  == \ 0, \ v[0]\  == \ 0, \ 
\ \ 
        \(\(Derivative[1]\)[u]\)[0]\  == \ 0, \ 
        \(\(Derivative[1]\)[v]\)[0]\  == \ 1}\)sol  =  NDSolve[SolveThis,  {u, v}, {t, 10}]geodesic  =  
      ParametricPlot3D[Evaluate[x[u[t], v[t]]  /.  sol], {t, 0, 10}, 
        PlotRange  ->  All]torus2  =  Show[Graphics3D[{EdgeForm[], First[torus]}]]Show[geodesic, torus2]

\(SolveThis\  := 
      \ {\((3\  + \ Cos[u[t]])\)*Sin[u[t]]*\(\(Derivative[1]\)[v]\)[t]^2\  + 
            \ \(\(Derivative[2]\)[u]\)[t]\  == \ \ 0, \ 
        \((\(-2\)*Sin[u[t]]*\(\(Derivative[1]\)[u]\)[t]*
                  \(\(Derivative[1]\)[v]\)[t])\)/
\ \ \ \ \ 
              \((3\  + \ Cos[u[t]])\)\  + \ \(\(Derivative[2]\)[v]\)[t]\  == 
          \ 0, \ u[0]\  == \ 0, \ v[0]\  == \ 0, \ 
\ \ 
        \(\(Derivative[1]\)[u]\)[0]\  == \ 1, \ 
        \(\(Derivative[1]\)[v]\)[0]\  == \  .2}\)sol  =  NDSolve[SolveThis,  {u, v}, {t, 30}]geodesic  =  
      ParametricPlot3D[Evaluate[x[u[t], v[t]]  /.  sol], {t, 0, 30}, 
        PlotRange  ->  All]torus2  =  Show[Graphics3D[{EdgeForm[], First[torus]}]]Show[geodesic, torus2, ViewPoint -> {0.223,  -0.246,  3.367}]

\(SolveThis\  := 
      \ {\((3\  + \ Cos[u[t]])\)*Sin[u[t]]*\(\(Derivative[1]\)[v]\)[t]^2\  + 
            \ \(\(Derivative[2]\)[u]\)[t]\  == \ \ 0, \ 
        \((\(-2\)*Sin[u[t]]*\(\(Derivative[1]\)[u]\)[t]*
                  \(\(Derivative[1]\)[v]\)[t])\)/
\ \ \ \ \ 
              \((3\  + \ Cos[u[t]])\)\  + \ \(\(Derivative[2]\)[v]\)[t]\  == 
          \ 0, \ u[0]\  == \ 0, \ v[0]\  == \ 0, \ 
\ \ 
        \(\(Derivative[1]\)[u]\)[0]\  == \ Cos[ .5], \ 
        \(\(Derivative[1]\)[v]\)[0]\  == \ Sin[ .5]}\)sol  =  NDSolve[SolveThis,  {u, v}, {t, 30}]geodesic  =  
      ParametricPlot3D[Evaluate[x[u[t], v[t]]  /.  sol], {t, 0, 30}, 
        PlotRange  ->  All]torus2  =  Show[Graphics3D[{EdgeForm[], First[torus]}]]Show[geodesic, torus2, ViewPoint -> {0.196,  3.037,  1.479}]