<< Graphics`ParametricPlot3D`
<< Graphics`Colors`
<< Graphics`Graphics3D`
<< Graphics`Arrow`

def

crtanje

kugla = 
      ParametricPlot3D[{0.04  Cos[u] Sin[v] + 0.5, 
          0.04  Sin[u] Sin[v] + 0.5, 0.04  Cos[v] + 0.25, 
          FaceForm[Black]}, {u, 0, Pi, Pi/6}, {v, 0, 2  Pi, 
          Pi/6}, 
DisplayFunction -> 
          Identity]

kugla1 = 
      ParametricPlot3D[{0.04  Cos[u] Sin[v] + 0.5, 
          0.04  Sin[u] Sin[v] + 0.5, 0.04  Cos[v] + 0.25, 
          SurfaceColor[Black]}, {u, 0, Pi, Pi/6}, {v, 0, 2  Pi, 
          Pi/6}, 
DisplayFunction -> 
          Identity]

tocka = {1/2, 1/2, 1/4}

izvodnica1 = {t, 1/2, 1/2  t}

izvodnica2 = {1/2, t, 1/2  t}

vektor1 = tocka - izvodnica1 /. t -> 1

vektor1/Sqrt[vektor1 . vektor1]

vektor2 = tocka - izvodnica2 /. t -> 1

vektor2/Sqrt[vektor2 . vektor2]

TanRav = tocka + u*vektor1 + v*vektor2

Cross[vektor1, vektor2]

Clear[normala]

normala = tocka - {1/8  t, 1/8  t, -1/4  t}

Simplify[
      Cos[i]*{-(2/(Sqrt[5])), 0, -(1/(Sqrt[5]))} + 
        Sin[i]*{0, -(2/(Sqrt[5])), -(1/(Sqrt[5]))}]

Simplify[tocka + t*%]

tang[i_] := {(1/2) - ((2 t Cos[i])/(Sqrt[5])), 
        (1/2) - ((2 t Sin[i])/(Sqrt[5])), 
        (1/4) - ((t (Cos[i] + Sin[i]))/(Sqrt[5]))}

tang[Pi/4]

tang[3  Pi/4]

tangenta = tocka - tang[i] /. t -> 1

Simplify[
      tocka + u*{(1/8), (1/8), -(1/4)} + 
        v*{((2 Cos[i])/(Sqrt[5])), ((2 Sin[i])/(Sqrt[5])), ((Cos[i] + Sin[i])/(Sqrt[5]))}]

NormalRav[
        i_] := {((4 + u)/8) + ((2 v Cos[i])/(Sqrt[5])), ((4 + u)/8) + ((2 v Sin[i])/(Sqrt[5])), (1/4) - (u/4) + ((v (Cos[i] + Sin[i]))/(Sqrt[5]))}

Simplify[
      Solve[{((4 + u)/8) + ((2 v Cos[i])/(Sqrt[5])) == 
            x, ((4 + u)/8) + ((2 v Sin[i])/(Sqrt[5])) == y, 
          (1/4) - (u/4) + ((v (Cos[i] + Sin[i]))/(Sqrt[5])) == x*y}, {u, 
          v, x}]]

krivulja[
        i_] := {(((5 - 10 y) Cos[i] - (5 + 2 y) Sin[i])/((-2 + 4 y) Cos[i] - 2 (5 + 2 y) Sin[i])), y, 
        y*(((5 - 10 y) Cos[i] - (5 + 2 y) Sin[i])/((-2 + 4 y) Cos[i] - 2 (5 + 2 y) Sin[i]))}

tockice[i_] := 
        If[i == 0, Table[{t, 1/2, 1/2  t}, {t, -1, 1, 0.05}], 
          Table[{(((5 - 10 y) Cos[i] - (5 + 2 y) Sin[i])/((-2 + 4 y) Cos[i] - 2 (5 + 2 y) Sin[i])), y, 
              y*(((5 - 10 y) Cos[i] - (5 + 2 y) Sin[i])/((-2 + 4 y) Cos[i] - 2 (5 + 2 y) Sin[i]))}, {y, -0.95, 0.95, 0.05}]];

krivulja[Pi/4] // Simplify

krivulja[3  Pi/4] // Simplify

kriv = {((5 - 4 y)/(4 + 4 y)), 
        y, -((y (-5 + 4 y))/(4 (1 + y))) + 
          0.01}

zakrivljenost

gcurvature[x_][u_, 
        v_] := 
Simplify[(Det[{D[x[uu, vv], uu, uu], 
                      D[x[uu, vv], uu], 
                      D[x[uu, vv], vv]}]*
Det[{D[x[uu, vv], 
                        vv, vv], D[x[uu, vv], uu], 
                      D[x[uu, vv], vv]}] - 
Det[{D[
                        x[uu, vv], uu, vv], D[x[uu, vv], uu], 
                      D[x[uu, vv], vv]}]^2)/
(D[
                        x[uu, vv], uu] . 
                      D[x[uu, vv], uu]*
D[x[uu, vv], vv] . 
                      D[x[uu, vv], vv] - 
D[x[uu, vv], 
                      uu] . D[x[uu, vv], vv]^2)^2] /. {uu -> u, 
          vv -> v}

meancurvature[x_][u_, v_] := 
      Simplify[(Det[{D[x[uu, vv], uu, uu], D[x[uu, vv], uu], 
                      D[x[uu, vv], vv]}]*
D[x[uu, vv], 
                      vv] . D[x[uu, vv], vv] - 
2 
                   Det[{D[x[uu, vv], uu, vv], D[x[uu, vv], uu], 
                      D[x[uu, vv], vv]}]*
D[x[uu, vv], 
                      uu] . D[x[uu, vv], 
                      vv] + 
Det[{D[x[uu, vv], vv, vv], 
                      D[x[uu, vv], uu], 
                      D[x[uu, vv], vv]}]*
D[x[uu, vv], 
                      uu] . D[x[uu, vv], 
                      uu])/
(2 (D[x[uu, vv], uu] . 
                          D[x[uu, vv], uu]*
D[x[uu, vv], 
                            vv] . D[x[uu, vv], vv] - 
D[
                          x[uu, vv], uu] . D[x[uu, vv], vv]^2)^(3/
                      2))] /. {uu -> u, vv -> v}

H[u_, v_] := {u, v, u*v}

gcurvature[H][1/2, 1/2]

N[%]

meancurvature[H][1/2, 1/2]

N[%]

Solve[{x1*x2 == -(256/529), 
          1/2 (x1 + x2) == -(16/(23 (Sqrt[23])))}, {x1, x2}] // 
      Simplify

N[%]

Clear[K1, K2]

K1 = -((16 (1 + 
                    2 (Sqrt[6])))/(23 (Sqrt[23])));
K2 = ((16 (-1 + 2 (Sqrt[6])))/(23 (Sqrt[23])));

N[%%]

N[%%]

zakrivljenost[phi_] := Cos[phi]^2*K1 + Sin[phi]^2*K2

Table[N[zakrivljenost[i]], {i, 0, Pi, Pi/36}]

N[Solve[Cos[phi]^2*K1 + Sin[phi]^2*K2 == 0, phi, 
          InverseFunctions -> True]] // Simplify

glavne zakrivljenosti i normalna zakrivljenost

drugaGZ[x_][u_, v_] := meancurvature[x][u, v] + 
          Sqrt[meancurvature[x][u, v]^2 - gcurvature[x][u, 
                v]];
prvaGZ[x_][u_, v_] := meancurvature[x][u, v] - 
        Sqrt[meancurvature[x][u, v]^2 - gcurvature[x][u, 
              v]]

normalna[x_][u_, v_][phi_] := 
      Cos[phi]^2*prvaGZ[x][u, v] + 
        Sin[phi]^2*drugaGZ[x][u, v]

Table[
      N[zakrivljenost[i + Pi/4]], {i, -Pi/4, 3  Pi/4 - Pi/44, 
        Pi/44}]

crtezi

Show[
ParametricPlot3D[{x, y, x*y, 
          FaceForm[Hue[0.66, 0.2, 0.9], Hue[0.66, 0.5, 0.9]]}, {x, -1, 1, 
          0.05}, {y, -1, 1, 
          0.05}, 
DisplayFunction -> 
          Identity], 

Graphics3D[{AbsolutePointSize[10], 
          Point[tocka]}], 

Axes -> 
        None, Boxed -> 
        False, 
PlotRange -> {{-1, 1}, {-1, 
            1}, {-1.3, 
            1}}, 
DisplayFunction -> $DisplayFunction, 
      ViewPoint -> {1.1,  -3,  1}, Background -> GrayLevel[0.8], 
      ImageSize -> {400, 400}]

\(Show[
ParametricPlot3D[{x, y, x*y, 
          FaceForm[Hue[0.66, 0.2, 0.9], 
            Hue[0.66, 0.5, 0.9]]}, {x, \(-1\), \(-0.95\), 0.05}, {y, \(-1\), 
          1, 0.05}, 
DisplayFunction -> 
          Identity], 
ParametricPlot3D[{x, y, x*y, 
          FaceForm[Hue[0.66, 0.2, 0.9], Hue[0.66, 0.5, 0.9]]}, {x, 0.95, 1, 
          0.05}, {y, \(-1\), 1, 
          0.05}, 
DisplayFunction -> 
          Identity], 
ParametricPlot3D[{x, y, x*y, 
          FaceForm[Hue[0.66, 0.2, 0.9], Hue[0.66, 0.5, 0.9]]}, {x, \(-1\), 1, 
          0.05}, {y, \(-1\), \(-0.95\), 
          0.05}, 
DisplayFunction -> 
          Identity], 
ParametricPlot3D[{x, y, x*y, 
          FaceForm[Hue[0.66, 0.2, 0.9], Hue[0.66, 0.5, 0.9]]}, {x, \(-1\), 1, 
          0.05}, {y, 0.95, 1, 
          0.05}, 
DisplayFunction -> 
          Identity], 

Table[
        ParametricPlot3D[{i, y, i*y}, {y, \(-0.95\), 0.95}, 
          DisplayFunction -> Identity], {i, \(-0.95\), 0.95, 
          0.05}], 
Table[
        ParametricPlot3D[{x, i, i*x}, {x, \(-0.95\), 0.95}, 
          DisplayFunction -> Identity], {i, \(-0.95\), 0.95, 
          0.05}], 

ParametricPlot3D[
        Append[TanRav, {Hue[0.14, 0.5, 1]}] // Evaluate, {u, \(-6\), 4, 
          8}, {v, \(-6\), 4, 8}, 
        DisplayFunction -> 
          Identity], 

ParametricPlot3D[
        Append[normala, {AbsoluteThickness[3], Hue[0.7]}] // 
          Evaluate, {t, \(-4\), 4}, 
        DisplayFunction -> 
          Identity], 

ParametricPlot3D[
        Append[izvodnica1, {AbsoluteThickness[2]}] // Evaluate, {t, \(-4\), 
          4}, DisplayFunction -> 
          Identity], 
ParametricPlot3D[
        Append[izvodnica2, {AbsoluteThickness[2]}] // Evaluate, {t, \(-4\), 
          4}, DisplayFunction -> 
          Identity], 

Table[
        ParametricPlot3D[
          Append[tang[i], {AbsoluteThickness[1], Hue[1]}] // 
            Evaluate, {t, \(-1\)/2, 1/2}, 
          DisplayFunction -> Identity], {i, 0, Pi, 
          Pi/17}], 

kugla, 

Graphics3D[
        Text[FontForm["n", {"Ariel-Bold", 14}], {0, 0, 1}, {\(-1\), 
            0}]], 
Graphics3D[
        Text[FontForm["T", {"Ariel-Bold", 14}], {0.7, 0.2, 
            0.25}]], 
Graphics3D[
        Text[FontForm["τ", {"Symbol-Bold", 
              16}], {\(-0.7\), \(-1\), \(-1\)}]], 

Axes -> None, 
      Boxed -> 
        False, 
PlotRange -> {{\(-1\), 1}, {\(-1\), 
            1}, {\(-1.3\), 
            1}}, 
DisplayFunction -> $DisplayFunction, 
      ViewPoint -> {1.1, \ \(-3\), \ 1}, Background -> GrayLevel[0.8], 
      Lighting -> False, ImageSize -> {400, 400}]\)

\(Show[
      ParametricPlot3D[{x, y, x*y, 
          FaceForm[Hue[0.66, 0.2, 0.9], 
            Hue[0.66, 0.5, 0.9]]}, {x, \(-1\), \(-0.95\), 0.05}, {y, \(-1\), 
          1, 0.05}, 
DisplayFunction -> 
          Identity], 
ParametricPlot3D[{x, y, x*y, 
          FaceForm[Hue[0.66, 0.2, 0.9], Hue[0.66, 0.5, 0.9]]}, {x, 0.95, 1, 
          0.05}, {y, \(-1\), 1, 
          0.05}, 
DisplayFunction -> 
          Identity], 
ParametricPlot3D[{x, y, x*y, 
          FaceForm[Hue[0.66, 0.2, 0.9], Hue[0.66, 0.5, 0.9]]}, {x, \(-1\), 1, 
          0.05}, {y, \(-1\), \(-0.95\), 
          0.05}, 
DisplayFunction -> 
          Identity], 
ParametricPlot3D[{x, y, x*y, 
          FaceForm[Hue[0.66, 0.2, 0.9], Hue[0.66, 0.5, 0.9]]}, {x, \(-1\), 1, 
          0.05}, {y, 0.95, 1, 
          0.05}, 
DisplayFunction -> 
          Identity], 

Table[
        ParametricPlot3D[{i, y, i*y}, {y, \(-1\), 1}, 
          DisplayFunction -> Identity], {i, \(-1\), 1, 
          0.05}], 
Table[
        ParametricPlot3D[{x, i, i*x}, {x, \(-1\), 1}, 
          DisplayFunction -> Identity], {i, \(-1\), 1, 
          0.05}], 

ParametricPlot3D[
        Append[izvodnica1, {AbsoluteThickness[4], Hue[0.5]}] // 
          Evaluate, {t, \(-4\), 4}, 
        DisplayFunction -> 
          Identity], 
ParametricPlot3D[
        Append[izvodnica2, {AbsoluteThickness[4], Hue[0.5]}] // 
          Evaluate, {t, \(-4\), 4}, 
        DisplayFunction -> 
          Identity], 

ParametricPlot3D[
        Append[TanRav, {Hue[0.14, 0.5, 1]}] // Evaluate, {u, \(-6\), 4, 
          8}, {v, \(-6\), 4, 8}, 
        DisplayFunction -> 
          Identity], 

ParametricPlot3D[
        Append[normala, {AbsoluteThickness[3], Hue[0.7]}] // 
          Evaluate, {t, \(-4\), 4}, 
        DisplayFunction -> 
          Identity], 

Graphics3D[
        Text[FontForm["n", {"Ariel-Bold", 14}], {0, 0, 1}, {\(-1\), 
            0}]], 
Graphics3D[
        Text[FontForm["T", {"Ariel-Bold", 14}], {0.7, 0.2, 
            0.25}]], 
Graphics3D[
        Text[FontForm["τ", {"Symbol-Bold", 
              16}], {\(-0.7\), \(-1\), \(-1\)}]], 
Graphics3D[
        Text[FontForm[\*""(Subscript[a, 1])"", {"Ariel-Bold", 
              14}], {\(-0.8\),  .3, \(-0.5\)}]], 
Graphics3D[
        Text[FontForm[\*""(Subscript[a, 2])"", {"Ariel-Bold", 
              14}], {0.55, \(-0.7\), \(-0.4\)}, {\(-1\), 
            0}]], 

kugla, 


Axes -> None, 
      Boxed -> 
        False, 
PlotRange -> {{\(-1\), 1}, {\(-1\), 
            1}, {\(-1.3\), 
            1}}, 
DisplayFunction -> $DisplayFunction, 
      ViewPoint -> {1.1, \ \(-3\), \ 1}, Lighting -> False, 
      Background -> GrayLevel[0.8], 
      ImageSize -> {400, 400}]\)

\(Show[
ParametricPlot3D[{x, y, x*y, 
          FaceForm[Hue[0.66, 0.2, 0.9], 
            Hue[0.66, 0.5, 0.9]]}, {x, \(-1\), \(-0.95\), 0.05}, {y, \(-1\), 
          1, 0.05}, 
DisplayFunction -> 
          Identity], 
ParametricPlot3D[{x, y, x*y, 
          FaceForm[Hue[0.66, 0.2, 0.9], Hue[0.66, 0.5, 0.9]]}, {x, 0.95, 1, 
          0.05}, {y, \(-1\), 1, 
          0.05}, 
DisplayFunction -> 
          Identity], 
ParametricPlot3D[{x, y, x*y, 
          FaceForm[Hue[0.66, 0.2, 0.9], Hue[0.66, 0.5, 0.9]]}, {x, \(-1\), 1, 
          0.05}, {y, \(-1\), \(-0.95\), 
          0.05}, 
DisplayFunction -> 
          Identity], 
ParametricPlot3D[{x, y, x*y, 
          FaceForm[Hue[0.66, 0.2, 0.9], Hue[0.66, 0.5, 0.9]]}, {x, \(-1\), 1, 
          0.05}, {y, 0.95, 1, 
          0.05}, 
DisplayFunction -> 
          Identity], 

Table[
        ParametricPlot3D[{i, y, i*y}, {y, \(-0.95\), 0.95}, 
          DisplayFunction -> Identity], {i, \(-0.95\), 0.95, 
          0.05}], 
Table[
        ParametricPlot3D[{x, i, i*x}, {x, \(-0.95\), 0.95}, 
          DisplayFunction -> Identity], {i, \(-0.95\), 0.95, 
          0.05}], 


ParametricPlot3D[
        Append[normala, {AbsoluteThickness[3], Hue[0.7]}] // 
          Evaluate, {t, \(-4\), 4}, 
        DisplayFunction -> 
          Identity], 

Table[
        ParametricPlot3D[
          Append[NormalRav[i], White] // Evaluate, {u, \(-1\), 1, 
            1}, {v, \(-0.4\), 0.4, 
            0.4}, 
DisplayFunction -> Identity], {i, 
          0, Pi/2, 
          Pi/2}], 



ParametricPlot3D[
        Append[izvodnica1, {AbsoluteThickness[4], Hue[0.5]}] // 
          Evaluate, {t, \(-4\), 4}, 
        DisplayFunction -> 
          Identity], 
ParametricPlot3D[
        Append[izvodnica2, {AbsoluteThickness[4], Hue[0.5]}] // 
          Evaluate, {t, \(-4\), 4}, 
        DisplayFunction -> 
          Identity], 

kugla, 

Graphics3D[
        Text[FontForm["n", {"Ariel-Bold", 14}], {0, 0, 1}, {\(-1\), 
            0}]], 


Axes -> None, 
      Boxed -> 
        False, 
PlotRange -> {{\(-1\), 1. }, {\(-1\), 
            1}, {\(-1.3\), 
            1}}, 
DisplayFunction -> $DisplayFunction, 
      ViewPoint -> {1.1, \ \(-3\), \ 1}, Lighting -> False, 
      Background -> GrayLevel[0.8], 
      ImageSize -> {400, 400}]\)

\(Show[
      ParametricPlot3D[{x, y, x*y, 
          FaceForm[Hue[0.66, 0.2, 0.9], 
            Hue[0.66, 0.5, 0.9]]}, {x, \(-1\), \(-0.95\), 0.05}, {y, \(-1\), 
          1, 0.05}, 
DisplayFunction -> 
          Identity], 
ParametricPlot3D[{x, y, x*y, 
          FaceForm[Hue[0.66, 0.2, 0.9], Hue[0.66, 0.5, 0.9]]}, {x, 0.95, 1, 
          0.05}, {y, \(-1\), 1, 
          0.05}, 
DisplayFunction -> 
          Identity], 
ParametricPlot3D[{x, y, x*y, 
          FaceForm[Hue[0.66, 0.2, 0.9], Hue[0.66, 0.5, 0.9]]}, {x, \(-1\), 1, 
          0.05}, {y, \(-1\), \(-0.95\), 
          0.05}, 
DisplayFunction -> 
          Identity], 
ParametricPlot3D[{x, y, x*y, 
          FaceForm[Hue[0.66, 0.2, 0.9], Hue[0.66, 0.5, 0.9]]}, {x, \(-1\), 1, 
          0.05}, {y, 0.95, 1, 
          0.05}, 
DisplayFunction -> 
          Identity], 

Table[
        ParametricPlot3D[{i, y, i*y}, {y, \(-1\), 1}, 
          DisplayFunction -> Identity], {i, \(-1\), 1, 
          0.05}], 
Table[
        ParametricPlot3D[{x, i, i*x}, {x, \(-1\), 1}, 
          DisplayFunction -> Identity], {i, \(-1\), 1, 
          0.05}], 

ParametricPlot3D[
        Append[izvodnica1, {AbsoluteThickness[3], Hue[0.5]}] // 
          Evaluate, {t, \(-4\), 4}, 
        DisplayFunction -> 
          Identity], 
ParametricPlot3D[
        Append[izvodnica2, {AbsoluteThickness[3], Hue[0.5]}] // 
          Evaluate, {t, \(-4\), 4}, 
        DisplayFunction -> 
          Identity], 

ParametricPlot3D[
        Append[tang[Pi/4], {AbsoluteThickness[2], Hue[1]}] // 
          Evaluate, {t, \(-0.7\), 2}, 
        DisplayFunction -> 
          Identity], 
ParametricPlot3D[
        Append[tang[3  Pi/4], {AbsoluteThickness[2], Hue[1]}] // 
          Evaluate, {t, \(-0.7\), 0.7}, 
        DisplayFunction -> 
          Identity], 

ParametricPlot3D[
        Append[TanRav, {Hue[0.14, 0.5, 1]}] // Evaluate, {u, \(-6\), 4, 
          8}, {v, \(-6\), 4, 8}, 
        DisplayFunction -> 
          Identity], 

ParametricPlot3D[
        Append[normala, {AbsoluteThickness[3], Hue[0.7]}] // 
          Evaluate, {t, \(-4\), 4}, 
        DisplayFunction -> 
          Identity], 


kugla, 

Graphics3D[
        Text[FontForm["n", {"Ariel-Bold", 14}], {0, 0, 1}, {\(-1\), 
            0}]], 
Graphics3D[
        Text[FontForm["T", {"Ariel-Bold", 14}], {0.5, 0.5, 
            0.4}]], 
Graphics3D[
        Text[FontForm["τ", {"Symbol-Bold", 16}], {\(-0.2\), 0.9, 
            0.1}]], 
Graphics3D[
        Text[FontForm[\*""(Subscript[a, 1])"", {"Ariel-Bold", 
              14}], {\(-0.8\),  .3, \(-0.5\)}]], 
Graphics3D[
        Text[FontForm[\*""(Subscript[a, 2])"", {"Ariel-Bold", 
              14}], {0.55, \(-0.7\), \(-0.4\)}, {\(-1\), 
            0}]], 
Graphics3D[
        Text[FontForm[\*""(Subscript[p, 1])"", {"Ariel-Bold", 
              14}], {0.8,  .1, 0.2}, {\(-1\), 
            0}]], 
Graphics3D[
        Text[FontForm[\*""(Subscript[p, 2])"", {"Ariel-Bold", 
              14}], {\(-0.8\), \(-0.9\), \(-1\)}, {\(-1\), 
            0}]], 



Axes -> None, 
      Boxed -> 
        False, 
PlotRange -> {{\(-1\), 1}, {\(-1\), 
            1}, {\(-1.3\), 
            1}}, 
DisplayFunction -> $DisplayFunction, 
      ViewPoint -> {1.1, \ \(-3\), \ 1}, Lighting -> False, 
      Background -> GrayLevel[0.8], 
      ImageSize -> {400, 400}]\)

\(Show[
ParametricPlot3D[{x, y, x*y, 
          FaceForm[Hue[0.66, 0.2, 0.9], 
            Hue[0.66, 0.5, 0.9]]}, {x, \(-1\), \(-0.95\), 0.05}, {y, \(-1\), 
          1, 0.05}, 
DisplayFunction -> 
          Identity], 
ParametricPlot3D[{x, y, x*y, 
          FaceForm[Hue[0.66, 0.2, 0.9], Hue[0.66, 0.5, 0.9]]}, {x, 0.95, 1, 
          0.05}, {y, \(-1\), 1, 
          0.05}, 
DisplayFunction -> 
          Identity], 
ParametricPlot3D[{x, y, x*y, 
          FaceForm[Hue[0.66, 0.2, 0.9], Hue[0.66, 0.5, 0.9]]}, {x, \(-1\), 1, 
          0.05}, {y, \(-1\), \(-0.95\), 
          0.05}, 
DisplayFunction -> 
          Identity], 
ParametricPlot3D[{x, y, x*y, 
          FaceForm[Hue[0.66, 0.2, 0.9], Hue[0.66, 0.5, 0.9]]}, {x, \(-1\), 1, 
          0.05}, {y, 0.95, 1, 
          0.05}, 
DisplayFunction -> 
          Identity], 

Table[
        ParametricPlot3D[{i, y, i*y}, {y, \(-0.95\), 0.95}, 
          DisplayFunction -> Identity], {i, \(-0.95\), 0.95, 
          0.05}], 
Table[
        ParametricPlot3D[{x, i, i*x}, {x, \(-0.95\), 0.95}, 
          DisplayFunction -> Identity], {i, \(-0.95\), 0.95, 
          0.05}], 


ParametricPlot3D[
        Append[normala, {AbsoluteThickness[3], Hue[0.7]}] // 
          Evaluate, {t, \(-4\), 4}, 
        DisplayFunction -> 
          Identity], 

Table[
        ParametricPlot3D[
          Append[NormalRav[i], White] // Evaluate, {u, \(-1\), 1, 
            1}, {v, \(-0.4\), 0.4, 
            0.4}, 
DisplayFunction -> Identity], {i, 
          Pi/4, 3  Pi/4, Pi/2}], 

Table[
        ParametricPlot3D[
          Append[krivulja[i], {AbsoluteThickness[3], Hue[1]}] // 
            Evaluate, {y, \(-4\), 4}, DisplayFunction -> Identity], {i, 
          Pi/4, 3  Pi/4, 
          Pi/2}], 

kugla, 

Graphics3D[
        Text[FontForm["n", {"Ariel-Bold", 14}], {0, 0, 1}, {\(-1\), 
            0}]], 

Axes -> None, 
      Boxed -> 
        False, 
PlotRange -> {{\(-1\), 1. }, {\(-1\), 
            1}, {\(-1.3\), 
            1}}, 
DisplayFunction -> $DisplayFunction, 
      ViewPoint -> {1.1, \ \(-3\), \ 1}, Lighting -> False, 
      Background -> GrayLevel[0.8], 
      ImageSize -> {400, 400}]\)

\(Show[
ParametricPlot3D[{x, y, x*y, 
          FaceForm[Hue[0.66, 0.2, 0.9], Hue[0.66, 0.5, 0.9]]}, {x, \(-1\), 1, 
          0.05}, {y, \(-1\), 1, 
          0.05}, 
DisplayFunction -> 
          Identity], 

ParametricPlot3D[
        Append[krivulja[Pi/4], {AbsoluteThickness[3], Hue[1]}] // 
          Evaluate, {y, \(-4\), 4}, 
        DisplayFunction -> 
          Identity], 
ParametricPlot3D[
        Append[kriv, {AbsoluteThickness[3], Hue[1]}] // Evaluate, {y, \(-4\), 
          4}, DisplayFunction -> 
          Identity], 

ParametricPlot3D[
        Append[izvodnica1, {AbsoluteThickness[3], Hue[0.5]}] // 
          Evaluate, {t, \(-4\), 4}, 
        DisplayFunction -> 
          Identity], 
ParametricPlot3D[
        Append[izvodnica2, {AbsoluteThickness[3], Hue[0.5]}] // 
          Evaluate, {t, \(-4\), 4}, 
        DisplayFunction -> 
          Identity], 

ParametricPlot3D[
        Append[normala, {AbsoluteThickness[3], Hue[0.7]}] // 
          Evaluate, {t, \(-4\), 4}, 
        DisplayFunction -> 
          Identity], 


Graphics3D[{AbsolutePointSize[12], 
          Point[tocka]}], 
kugla1, 
Graphics3D[
        Text[FontForm["n", {"Ariel-Bold", 14}], {0, 0, 1}, {\(-1\), 
            0}]], 

Axes -> None, 
      Boxed -> 
        False, 
PlotRange -> {{\(-1\), 1}, {\(-1\), 
            1}, {\(-1.3\), 
            1}}, 
DisplayFunction -> $DisplayFunction, 
      ViewPoint -> {1.1, \ \(-3\), \ 1}, Background -> GrayLevel[0.8], 
      ImageSize -> {400, 400}]\)

\(Show[
      Plot[zakrivljenost[x] // Evaluate, {x, 0, Pi}, 
        PlotStyle -> {AbsoluteThickness[2]}, 
        DisplayFunction -> 
          Identity], 

Graphics[{Hue[1], 
          AbsoluteThickness[2], AbsoluteDashing[{2, 5}], 
          Line[{{0, 0}, {0, \(-0.855668\)}}], 
          Line[{{Pi, 0}, {Pi, \(-0.855668\)}}], 
          Line[{{Pi/2, 0}, {Pi/2, 
                0.56556}}]}], 

Graphics[{Hue[1], AbsolutePointSize[6], 
          Point[{0, 0}], Point[{Pi, 0}], 
          Point[{Pi/2, 0}]}], 
Graphics[{AbsolutePointSize[
            6], Point[{0, \(-0.855668\)}], Point[{Pi, \(-0.855668\)}], 
          Point[{Pi/2, 
              0.56556}]}], 

Graphics[{Hue[0.5], 
          AbsolutePointSize[6], Point[{0.8881826, 0}], 
          Point[{2.25341, 
              0}]}], 

Graphics[{{Arrow[{0, 1.1}, {0, 1.2}, HeadWidth -> 0.25, 
              HeadLength -> 0.04]}, 
{Arrow[{3.7, 
                0}, {3.8, 0}, HeadWidth -> 0.25, 
              HeadLength -> 
                0.04]}}], 

Graphics[{Text[
            FontForm["normalna zakrivljenost", {"Arial-Bold", 
                14}], {0.1, 1.1}, {\(-1\), 
              0}]}], 
Graphics[{Text[
            FontForm["φ", {"Symbol-Bold", 14}], {3.6, 
              0.1}, {\(-1\), 
              0}]}], 

Graphics[{Hue[1], 
          Text[FontForm[\*""(Subscript[K, 2])"", {"Arial-Bold", 
                12}], {1.6, 0.3}, {\(-1\), 
              0}]}], 
Graphics[{Hue[1], 
          Text[FontForm[\*""(Subscript[K, 1])"", {"Arial-Bold", 
                12}], {Pi + 0.05, \(-0.5\)}, {\(-1\), 
              0}]}], 
Graphics[{Hue[1], 
          Text[FontForm[\*""(Subscript[K, 1])"", {"Arial-Bold", 
                12}], {\(-0.25\), \(-0.5\)}, {\(-1\), 
              0}]}], 

Graphics[{Hue[
            0.5], AbsolutePointSize[7], 
          Point[{2. ,  .9}]}], 
Graphics[{Hue[1], 
          AbsolutePointSize[7], 
          Point[{2. , 0.75}]}], 
Graphics[{Text[
            FontForm["-  asimptotski smjerovi", {"Arial", 
                10}], {2.1,  .9}, {\(-1\), 
              0}]}], 
Graphics[{Text[
            FontForm["-  glavni smjerovi", {"Arial", 10}], {2.1, 
              0.75}, {\(-1\), 
              0}]}], 

Background -> 
        GrayLevel[
          0.8], 
Ticks -> {{0, Pi/6, 0.888, Pi/2, 
            2.253, 5  Pi/6, Pi}, {\(-1. \), \(-0.855668\), 0, \(-1\), 
            0.56556, 1}}, AspectRatio -> 1, 
      PlotRange -> {{\(-0.5\), 3.8}, {\(-1\), 
            1.2}}, 
DisplayFunction -> $DisplayFunction, ImageSize -> {300, 250}]\)

animacije

\(Table[
      Show[
ParametricPlot3D[{x, y, x*y, 
            FaceForm[Hue[0.66, 0.2, 0.9], 
              Hue[0.66, 0.5, 0.9]]}, {x, \(-1\), \(-0.95\), 
            0.05}, {y, \(-1\), 1, 
            0.05}, 
DisplayFunction -> 
            Identity], 
ParametricPlot3D[{x, y, x*y, 
            FaceForm[Hue[0.66, 0.2, 0.9], Hue[0.66, 0.5, 0.9]]}, {x, 0.95, 1, 
            0.05}, {y, \(-1\), 1, 
            0.05}, 
DisplayFunction -> 
            Identity], 
ParametricPlot3D[{x, y, x*y, 
            FaceForm[Hue[0.66, 0.2, 0.9], Hue[0.66, 0.5, 0.9]]}, {x, \(-1\), 
            1, 0.05}, {y, \(-1\), \(-0.95\), 
            0.05}, 
DisplayFunction -> 
            Identity], 
ParametricPlot3D[{x, y, x*y, 
            FaceForm[Hue[0.66, 0.2, 0.9], Hue[0.66, 0.5, 0.9]]}, {x, \(-1\), 
            1, 0.05}, {y, 0.95, 1, 
            0.05}, 
DisplayFunction -> 
            Identity], 

Table[
          ParametricPlot3D[{i, y, i*y}, {y, \(-0.95\), 0.95}, 
            DisplayFunction -> Identity], {i, \(-0.95\), 0.95, 
            0.05}], 
Table[
          ParametricPlot3D[{x, i, i*x}, {x, \(-0.95\), 0.95}, 
            DisplayFunction -> Identity], {i, \(-0.95\), 0.95, 
            0.05}], 


ParametricPlot3D[
          Append[normala, {AbsoluteThickness[3], Hue[0.7]}] // 
            Evaluate, {t, \(-4\), 4}, 
          DisplayFunction -> 
            Identity], 

ParametricPlot3D[
          Append[NormalRav[i], White] // Evaluate, {u, \(-1\), 1, 
            1}, {v, \(-0.4\), 0.4, 
            0.4}, 
DisplayFunction -> 
            Identity], 

ScatterPlot3D[
          tockice[i] // Evaluate, 
          PlotStyle -> {Hue[1], AbsolutePointSize[5]}, 
          DisplayFunction -> 
            Identity], 

kugla, 

Graphics3D[
          Text[FontForm["n", {"Ariel-Bold", 14}], {0, 0, 1}, {\(-1\), 
              0}]], 

Axes -> None, 
        Boxed -> 
          False, 
PlotRange -> {{\(-1\), 
              1. }, {\(-1\), 1}, {\(-1.3\), 
              1}}, 
DisplayFunction -> $DisplayFunction, ViewPoint -> {1.1, \ \(-3\), \ 1}, Lighting -> False, 
        Background -> GrayLevel[0.8], 
        ImageSize -> {300, 300}], {i, \(-Pi\)/4, 3  Pi/4 - Pi/45, 
        Pi/45}]\)

\(Table[
      Show[Plot[zakrivljenost[x] // Evaluate, {x, \(-Pi\)/45, i + Pi/45}, 
          PlotStyle -> {AbsoluteThickness[2]}, 
          DisplayFunction -> 
            Identity], 

Graphics[{Hue[
              1], AbsoluteThickness[2], AbsoluteDashing[{2, 5}], 
            Line[{{0, 0}, {0, \(-0.855668\)}}], 
            Line[{{Pi, 0}, {Pi, \(-0.855668\)}}], 
            Line[{{Pi/2, 0}, {Pi/2, 
                  0.56556}}]}], 

Graphics[{Hue[1], AbsolutePointSize[6], 
            Point[{0, 0}], Point[{Pi, 0}], 
            Point[{Pi/2, 
                0}]}], 
Graphics[{AbsolutePointSize[6], 
            Point[{0, \(-0.855668\)}], Point[{Pi, \(-0.855668\)}], 
            Point[{Pi/2, 
                0.56556}]}], 

Graphics[{Hue[0.5], 
            AbsolutePointSize[6], Point[{0.8881826, 0}], 
            Point[{2.25341, 
                0}]}], 

Graphics[{{Arrow[{0, 1.1}, {0, 1.2}, HeadWidth -> 0.25, 
                HeadLength -> 0.04]}, 
{Arrow[{3.7, 
                  0}, {3.8, 0}, HeadWidth -> 0.25, 
                HeadLength -> 
                  0.04]}}], 

Graphics[{Text[
              FontForm["normal curvature", {"Arial-Bold", 14}], {0.1, 
                1.1}, {\(-1\), 0}]}], 
Graphics[{Text[
              FontForm["φ", {"Symbol-Bold", 14}], {3.6, 
                0.1}, {\(-1\), 
                0}]}], 

Graphics[{Hue[
              1], Text[
              FontForm[\*""(Subscript[K, 2])"", {"Arial-Bold", 12}], {1.6, 
                0.3}, {\(-1\), 0}]}], 
Graphics[{Hue[1], 
            Text[FontForm[\*""(Subscript[K, 1])"", {"Arial-Bold", 
                  12}], {Pi + 0.05, \(-0.5\)}, {\(-1\), 
                0}]}], 
Graphics[{Hue[1], 
            Text[FontForm[\*""(Subscript[K, 1])"", {"Arial-Bold", 
                  12}], {\(-0.25\), \(-0.5\)}, {\(-1\), 
                0}]}], 

Graphics[{Hue[
              0.5], AbsolutePointSize[7], 
            Point[{1. , \(-0.75\)}]}], 
Graphics[{Hue[1], 
            AbsolutePointSize[7], 
            Point[{1. , \(-0.9\)}]}], 
Graphics[{Text[
              FontForm["-  asymptotic  directions", {"Arial", 
                  10}], {1.1, \(- .75\)}, {\(-1\), 
                0}]}], 
Graphics[{Text[
              FontForm["-  principal  direction", {"Arial", 
                  10}], {1.1, \(-0.9\)}, {\(-1\), 
                0}]}], 

Background -> GrayLevel[
            0.8], 
Ticks -> {{0, Pi/6, 0.888, Pi/2, 
              2.253, 5  Pi/6, Pi}, {\(-1. \), \(-0.855668\), 0, \(-1\), 
              0.56556, 1}}, AspectRatio -> 1, 
        PlotRange -> {{\(-0.5\), 3.8}, {\(-1\), 
              1.2}}, 
DisplayFunction -> $DisplayFunction, ImageSize -> {300, 300}], {i, 0, Pi - Pi/45, 
        Pi/45}]\)

crtezi

ParametricPlot3D[Append[H[u, v], Hue[gcurvature[H][u, v]]] // Evaluate,
	{u, -2, 2, 0.1},{v, -2, 2, 0.1},
Axes -> None,Boxed -> False,
BoxRatios -> {1, 1, 1},
DisplayFunction -> $DisplayFunction,ViewPoint -> {0.9,  -3, 1},Background -> GrayLevel[0.8],ImageSize -> {400, 400},Lighting -> False]

Plot3D[gcurvature[H][u, v] /. {u -> x, v -> y} // 
          Evaluate,{x, -2, 2},{y, -2, 2},
PlotPoints -> 100,AxesLabel -> {x, y, gcurvature},AxesEdge -> {{-1, -1}, {1, -1}, {-1, -1}},
BoxRatios -> {1, 1, 1},PlotRange -> {{-2, 2}, {-2, 2}, {-1, 
              0}},Mesh -> False,ViewPoint -> {0.9,  -3, 1},Ticks -> {{-1, 0, 1, 2}, {-1, 0, 1, 
              2}, {-0.6, -0.5, -0.4, -0.3, -0.2, -0.1, 0}},Background -> GrayLevel[0.8],ImageSize -> {400, 400},AmbientLight -> OrangeRed,FaceGrids -> All]

ParametricPlot3D[Append[H[u, v], Hue[meancurvature[H][u, v]]] // 
          Evaluate,
	{u, -2, 2, 0.1},{v, -2, 2, 0.1},
Axes -> None,Boxed -> False,
BoxRatios -> {1, 1, 1},
DisplayFunction -> $DisplayFunction,ViewPoint -> {1.1,  -3,  1},Background -> GrayLevel[0.8],ImageSize -> {400, 400},Lighting -> False]

Plot3D[meancurvature[H][u, v] /. {u -> x, v -> y} // 
          Evaluate,{x, -2, 2},{y, -2, 2},
PlotPoints -> 100,AxesLabel -> {x, y, meancurvature},AxesEdge -> {{-1, -1}, {1, -1}, {-1, -1}},
BoxRatios -> {1, 1, 1},PlotRange -> {{-2, 2}, {-2, 2}, {-0.25, 
              0.25}},Mesh -> False,ViewPoint -> {0.9,  -3, 1},Ticks -> {{-1, 0, 1, 2}, {-1, 0, 1, 
              2}, {-0.2, -0.15, -0.1, -0.05, 0, 0.05, 0.1, 
              0.15, 0.2}},Background -> GrayLevel[0.8],ImageSize -> {400, 400},AmbientLight -> Indigo,FaceGrids -> All]