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

def

ploha, tocka, pravci, ravnine, krivulja u tangencijalnoj ravnini, tangente u tangencijalnoj ravnini

F[u_, z_] := {Cos[u], Sin[u] (1 - z), z}

F[u, 0] := {Cos[u], Sin[u] (1 - z), z}

tocka = {(1/2), (1/2) (Sqrt[3]) , 0}

izvodnica = {(1/2), (1/2) (Sqrt[3]) (1 - t), t}

gltangenta = {(3/4) t + 
          1/2, -(3/4) (Sqrt[3]) (-(2/3) + t), t}

normala = {(1/2) + (t/2), ((Sqrt[3])/2) + (((Sqrt[3]) t)/2), ((3 t)/4)}

{(t/2), (((Sqrt[3]) t)/2), ((3 t)/4)} + {(1/2), ((Sqrt[3])/2), 0}

tangenta1 = {(1/2) + ((3 t)/(2 (Sqrt[13]))), ((Sqrt[3])/2) + (-((3 (Sqrt[(3/13)]))/2) - (Sqrt[(3/7)])) t, ((2/(Sqrt[7])) + 
              (2/(Sqrt[13]))) t}

tangenta2 = {(1/2) + ((3 t)/(2 (Sqrt[13]))), ((Sqrt[3])/2) - (((3 (Sqrt[(3/13)]))/2) - (Sqrt[(3/7)])) t, -((2/(Sqrt[7])) - (2/(Sqrt[13]))) t}

ravTang = {(1/2) - (((Sqrt[3]) u)/2), ((Sqrt[3])/2) + (u/2) - (((Sqrt[3]) v)/2), 
        v}

asimptRav1 = {(1/2) + ((2 u)/5), ((Sqrt[3])/2) + ((2 (Sqrt[3]) u)/5) - (Sqrt[(3/7)]) v, ((3 u)/5) + ((2 v)/(Sqrt[7]))}

asimptRav2 = {(1/2) + ((2 u)/5) + ((3 v)/(2 (Sqrt[13]))), ((Sqrt[3])/2) + ((2 (Sqrt[3]) u)/5) - (3/2) (Sqrt[(3/13)]) v, ((3 u)/5) + ((2 v)/(Sqrt[13]))}

glavRav1 = {(1/2) + ((2 u)/5) + ((3 v)/(2 (Sqrt[13]))), 
        (1/910) (Sqrt[3]) (364 u - 
              5 (-91 + 26 (Sqrt[7]) v + 
                    21 (Sqrt[13]) v)), ((3 u)/5) + ((2 v)/(Sqrt[7])) + ((2 v)/(Sqrt[13]))}

glavRav2 = {(1/2) + ((2 u)/5) - ((3 v)/(2 (Sqrt[13]))), 
        (1/910) (Sqrt[3]) (364 u + 
              5 (91 - 26 (Sqrt[7]) v + 
                    21 (Sqrt[13]) v)), ((3 u)/5) + ((2 v)/(Sqrt[7])) - ((2 v)/(Sqrt[13]))}

Y[x_, z_, i_] := 
        1/2 Sqrt[3] - 
          1/2 Sqrt[3] z + (i - 1) (x - 1/2)/Sqrt[3];

A[i_, 
          z_] := (4 - 8  i + i^2 + 6  i*z - 3  z^2)/(8 - 4  i + 
              2  i^2 - 12  z + 6  z^2);
B[i_, z_] := 
        Evaluate[Y[x, z, i] /. x -> A[i, z]];

    
KRIV[i_][u_, z_] := 
        ParametricPlot3D[{0.035  Cos[u] + A[i, z], 0.035  Sin[u] + B[i, z], 
              z, Red} // Evaluate, {u, 0, 2  Pi, Pi/6}, {z, -1.5, 
            2.5,  .1}, DisplayFunction -> Identity];

    
Kriv1[i_][z_] := {A[i, z], B[i, z], z};

Kugla [i_] := 
      ParametricPlot3D[{0.07  Cos[u] Sin[v] + 0.5, 
          0.07  Sin[u] Sin[v] +  .5  Sqrt[3] (1 - i), 
          0.06  Cos[v] + i + 0.01, Black}, {u, 0, Pi, Pi/6}, {v, 0, 2  Pi, 
          Pi/4}, DisplayFunction -> Identity]

Kugla1 [i_] := 
      ParametricPlot3D[{0.05  Cos[u] Sin[v] + 0.5, 
          0.05  Sin[u] Sin[v] +  .5  Sqrt[3] (1 - i), 
          0.04  Cos[v] + i + 0.01, SurfaceColor[Hue[1]]}, {u, 0, Pi, 
          Pi/6}, {v, 0, 2  Pi, Pi/4}, 
        DisplayFunction -> Identity]

tang[
        i_] := {(1/2) - (1/2) (Sqrt[3]) t Cos[i], ((Sqrt[3])/2) + 
          t (Cos[i]/2 - (Sqrt[(3/7)]) Sin[
                    i]), ((2 t Sin[i])/(Sqrt[7]))}

ravnine normalnog presjeka

ravNormal[i_] := {(1/2), (1/2) (Sqrt[3]) , 0} + 
        u*{(2/5), ((2 (Sqrt[3]))/5), (3/5)} + 
        v*{-(((Sqrt[3]) Cos[
                      i])/(2 (Sqrt[((3 Cos[i]^2)/4) + ((4 Sin[i]^2)/7) + (((Cos[i]/2 - (Sqrt[(3/7)]) Sin[i]))^2)]))), ((Cos[i]/2 - (Sqrt[(3/7)]) Sin[i])/(Sqrt[((3 Cos[i]^2)/4) + ((4 Sin[i]^2)/7) + (((Cos[i]/2 - (Sqrt[(3/7)]) Sin[i]))^2)])), (2 Sin[
                    i])/((Sqrt[7]) √(((3 Cos[i]^2)/4) + ((4 Sin[i]^2)/7) + (((Cos[i]/2 - (Sqrt[(3/7)]) Sin[i]))^2)))}

{(1/2), (1/2) (Sqrt[3]) , 0} + u*{(2/5), ((2 (Sqrt[3]))/5), (3/5)} + 
      v*{-(((Sqrt[3]) Cos[
                    i])/(2 (Sqrt[((3 Cos[i]^2)/4) + ((4 Sin[i]^2)/7) + (((Cos[i]/2 - (Sqrt[(3/7)]) Sin[i]))^2)]))), ((Cos[i]/2 - (Sqrt[(3/7)]) Sin[i])/(Sqrt[((3 Cos[i]^2)/4) + ((4 Sin[i]^2)/7) + (((Cos[i]/2 - (Sqrt[(3/7)]) Sin[i]))^2)])), (2 Sin[
                  i])/((Sqrt[7]) √(((3 Cos[i]^2)/4) + ((4 Sin[i]^2)/7) + (((Cos[i]/2 - (Sqrt[(3/7)]) Sin[i]))^2)))}

Eliminate[((3 u)/5) + ((2 v Sin[i])/((Sqrt[7]) (Sqrt[((3 Cos[i]^2)/4) + ((4 Sin[i]^2)/7) + (((Cos[i]/2 - (Sqrt[(3/7)]) Sin[i]))^2)]))) == z, u]

Simplify[{(1/2), (1/2) (Sqrt[3]) , 0} + u*{(2/5), ((2 (Sqrt[3]))/5), (3/5)} + 
        v*{-(((Sqrt[3]) Cos[
                      i])/(2 (Sqrt[((3 Cos[i]^2)/4) + ((4 Sin[i]^2)/7) + (((Cos[i]/2 - (Sqrt[(3/7)]) Sin[i]))^2)]))), ((Cos[i]/2 - (Sqrt[(3/7)]) Sin[i])/(Sqrt[((3 Cos[i]^2)/4) + ((4 Sin[i]^2)/7) + (((Cos[i]/2 - (Sqrt[(3/7)]) Sin[i]))^2)])), ((2 Sin[i])/((Sqrt[7]) (Sqrt[((3 Cos[i]^2)/4) + ((4 Sin[i]^2)/7) + (((Cos[i]/2 - (Sqrt[(3/7)]) Sin[i]))^2)])))}, 
      7 Cos[i]^2 - (Sqrt[21]) Cos[i] Sin[i] + 7 Sin[i]^2 ≠ 
        0]

ravNormal1[
        i_] := {(1/2) + ((2 u)/5) - ((v Cos[i])/(Sqrt[(4/3) - ((2 Sin[2 i])/(Sqrt[21]))])), ((Sqrt[3])/2) + ((2 (Sqrt[3]) u)/5) + ((v (7 Cos[i] - 2 (Sqrt[21]) Sin[i]))/(Sqrt[196 - 14 (Sqrt[21]) Sin[2 i]])), ((3 u)/5) + ((2 v Sin[i])/(Sqrt[7 - (Sqrt[21]) Cos[i] Sin[i]]))}

normalni presjeci

Simplify[
      Solve[{(1/2) + ((2 u)/5) - ((v Cos[i])/(Sqrt[(4/3) - ((2 Sin[2 i])/(Sqrt[21]))])) == 
            Cos[t], ((Sqrt[3])/2) + ((2 (Sqrt[3]) u)/5) + ((v (7 Cos[i] - 2 (Sqrt[21]) Sin[i]))/(Sqrt[196 - 14 (Sqrt[21]) Sin[2 i]])) == 
            Sin[t] (1 - 
                  z), ((3 u)/5) + ((2 v Sin[i])/(Sqrt[7 - (Sqrt[21]) Cos[i] Sin[i]])) == z}, {u, v, z}]]

Simplify[(3 (Sqrt[7]) Cos[i]^3 (-3 (Sqrt[3]) + 14 (Sqrt[3]) Cos[t] - 
                    8 Sin[t]) + 
              189 Cos[i]^2 Sin[i] (-2 + Cos[t] + (Sqrt[3]) Sin[t]) + 
              3 (Sqrt[7]) Cos[
                  i] (59 (Sqrt[3]) + 
                    21 (Sqrt[3]) (-3 + Cos[2 i]) Cos[t] - 76 Sin[t] + 
                    3 Sin[i]^2 (3 (Sqrt[3]) + 8 Sin[t])) - 
              7 Sin[i] (54 + 3 Cos[t] (-59 + 3 Sin[i]^2) + 
                    23 (Sqrt[3]) Sin[t] + 
                    9 Sin[i]^2 (-2 + (Sqrt[3]) Sin[
                              t])))/(12 (Sqrt[7]) Cos[i]^3 ((Sqrt[3]) - 
                    2 Sin[t]) - 
              4 (Sqrt[7]) Cos[
                  i] (59 (Sqrt[3]) + 9 Sin[i]^2 ((Sqrt[3]) - 2 Sin[t]) + 
                    57 Sin[t]) + 
              63 Cos[i]^2 Sin[i] (8 + 3 (Sqrt[3]) Sin[t]) - 
              7 Sin[i] (-72 + 23 (Sqrt[3]) Sin[t] + 
                    3 Sin[i]^2 (8 + 3 (Sqrt[3]) Sin[t]))), {Sin[i]^2 + 
              Cos[i]^2 == 1, 
          Sin[t]^2 + Cos[t]^2 == 1}];

Z[t_][
        i_] := (3 (Sqrt[7]) Cos[i]^3 (-3 (Sqrt[3]) + 14 (Sqrt[3]) Cos[t] - 
                  8 Sin[t]) + 
            189 Cos[i]^2 Sin[i] (-2 + Cos[t] + (Sqrt[3]) Sin[t]) + 
            3 (Sqrt[7]) Cos[
                i] (59 (Sqrt[3]) + 21 (Sqrt[3]) (-3 + Cos[2 i]) Cos[t] - 
                  76 Sin[t] + 3 Sin[i]^2 (3 (Sqrt[3]) + 8 Sin[t])) - 
            7 Sin[i] (54 + 3 Cos[t] (-59 + 3 Sin[i]^2) + 
                  23 (Sqrt[3]) Sin[t] + 
                  9 Sin[i]^2 (-2 + (Sqrt[3]) Sin[
                            t])))/(12 (Sqrt[7]) Cos[i]^3 ((Sqrt[3]) - 
                  2 Sin[t]) - 
            4 (Sqrt[7]) Cos[
                i] (59 (Sqrt[3]) + 9 Sin[i]^2 ((Sqrt[3]) - 2 Sin[t]) + 
                  57 Sin[t]) + 
            63 Cos[i]^2 Sin[i] (8 + 3 (Sqrt[3]) Sin[t]) - 
            7 Sin[i] (-72 + 23 (Sqrt[3]) Sin[t] + 
                  3 Sin[i]^2 (8 + 3 (Sqrt[3]) Sin[t])))

krivulj[t_][i_] := {Cos[t], 
        Sin[t] (1 - Z[t][i]), Z[t][i]}

Z1[t_][
        i_] := (-3 (Sqrt[7]) Cos[
                i] (7 (Sqrt[3]) (-2 + Cos[2 i]) Cos[t] + 
                  7 (2 (Sqrt[3]) - 3 Sin[t]) + 
                  Sin[i]^2 (3 (Sqrt[3]) + 8 Sin[t])) + 
            7 Sin[i] (27 + 
                  Cos[t] (-51 + 9 Sin[i]^2) - (Sqrt[3]) Sin[t] + 
                  9 Sin[i]^2 (-2 + (Sqrt[3]) Sin[t])))/((Sqrt[7]) Cos[
                i] (12 Sin[i]^2 ((Sqrt[3]) - 2 Sin[t]) + 
                  7 (8 (Sqrt[3]) + 9 Sin[t])) + 
            7 Sin[i] (-36 - (Sqrt[3]) Sin[t] + 
                  3 Sin[i]^2 (8 + 3 (Sqrt[3]) Sin[t])))

krivulj1[t_][i_] := {Cos[t], 
        Sin[t] (1 - Z1[t][i]), Z1[t][i]}

tockice[i_] := 
        Table[{Cos[t], 
            Sin[t] (1 - Z1[t][i] // Evaluate), Z1[t][i] // 
              Evaluate}, {t, 0, 2  Pi, Pi/48}];

krivulje-posebne

asimptRav1 = {(1/2) + ((2 u)/5), ((Sqrt[3])/2) + ((2 (Sqrt[3]) u)/5) - (Sqrt[(3/7)]) v, ((3 u)/5) + ((2 v)/(Sqrt[7]))}

asimptRav2 = {(1/2) + ((2 u)/5) + ((3 v)/(2 (Sqrt[13]))), ((Sqrt[3])/2) + ((2 (Sqrt[3]) u)/5) - (3/2) (Sqrt[(3/13)]) v, ((3 u)/5) + ((2 v)/(Sqrt[13]))}

glavRav1 = {(1/2) + ((2 u)/5) + ((3 v)/(2 (Sqrt[13]))), 
        (1/910) (Sqrt[3]) (364 u - 
              5 (-91 + 26 (Sqrt[7]) v + 
                    21 (Sqrt[13]) v)), ((3 u)/5) + ((2 v)/(Sqrt[7])) + ((2 v)/(Sqrt[13]))}

glavRav2 = {(1/2) + ((2 u)/5) - ((3 v)/(2 (Sqrt[13]))), 
        (1/910) (Sqrt[3]) (364 u + 
              5 (91 - 26 (Sqrt[7]) v + 
                    21 (Sqrt[13]) v)), ((3 u)/5) + ((2 v)/(Sqrt[7])) - ((2 v)/(Sqrt[13]))}

kr1

Solve[{(1/2) + ((2 u)/5) == 
          Cos[t], ((Sqrt[3])/2) + ((2 (Sqrt[3]) u)/5) - (Sqrt[(3/7)]) v == 
          Sin[t] (1 - z), ((3 u)/5) + ((2 v)/(Sqrt[7])) == z}, {u, v, 
        z}]

kr1[t_] := {Cos[t], 
        Sin[t] (1 + ((3 (Sqrt[3]) - 14 (Sqrt[3]) Cos[t] + 8 Sin[t])/(4 ((Sqrt[3]) - 2 Sin[t])))), -((3 (Sqrt[3]) - 14 (Sqrt[3]) Cos[t] + 
                8 Sin[t])/(4 ((Sqrt[3]) - 2 Sin[t])))}

kr2

Solve[{(1/2) + ((2 u)/5) + ((3 v)/(2 (Sqrt[13]))) == 
          Cos[t], ((Sqrt[3])/2) + ((2 (Sqrt[3]) u)/5) - (3/2) (Sqrt[(3/13)]) v == 
          Sin[t] (1 - z), ((3 u)/5) + ((2 v)/(Sqrt[13])) == z}, {u, 
        v, z}]

kr2[t_] := {Cos[t], 
        Sin[t] (1 + ((9 (Sqrt[3]) - 17 (Sqrt[3]) Cos[t] - Sin[t])/(12 (Sqrt[3]) + Sin[t]))), -((9 (Sqrt[3]) - 17 (Sqrt[3]) Cos[t] - Sin[t])/(12 (Sqrt[3]) + 
                Sin[t]))}

kr3

Solve[{(1/2) + ((2 u)/5) + ((3 v)/(2 (Sqrt[13]))) == Cos[t], 
        (1/910) (Sqrt[3]) (364 u - 
                5 (-91 + 26 (Sqrt[7]) v + 21 (Sqrt[13]) v)) == 
          Sin[t] (1 - 
                z), ((3 u)/5) + ((2 v)/(Sqrt[7])) + ((2 v)/(Sqrt[13])) == 
          z}, {u, v, z}]

kr3[t_] := {Cos[t], 
        Sin[t] (1 + ((39 (Sqrt[21]) + 63 (Sqrt[39]) - 182 (Sqrt[21]) Cos[t] - 119 (Sqrt[39]) Cos[t] + 104 (Sqrt[7]) Sin[t] - 7 (Sqrt[13]) Sin[t])/(52 (Sqrt[21]) + 84 (Sqrt[39]) - 104 (Sqrt[7]) Sin[t] + 7 (Sqrt[13]) Sin[t]))), -((39 (Sqrt[21]) + 63 (Sqrt[39]) - 
                182 (Sqrt[21]) Cos[t] - 119 (Sqrt[39]) Cos[t] + 104 (Sqrt[7]) Sin[t] - 
                7 (Sqrt[13]) Sin[t])/(52 (Sqrt[21]) + 84 (Sqrt[39]) - 104 (Sqrt[7]) Sin[t] + 
                7 (Sqrt[13]) Sin[t]))}

kr4

Solve[{(1/2) + ((2 u)/5) - ((3 v)/(2 (Sqrt[13]))) == Cos[t], 
        (1/910) (Sqrt[3]) (364 u + 
                5 (91 - 26 (Sqrt[7]) v + 21 (Sqrt[13]) v)) == 
          Sin[t] (1 - 
                z), ((3 u)/5) + ((2 v)/(Sqrt[7])) - ((2 v)/(Sqrt[13])) == 
          z}, {u, v, z}]

kr4[t_] := {Cos[t], 
        Sin[t] (1 + ((39 (Sqrt[21]) - 63 (Sqrt[39]) - 182 (Sqrt[21]) Cos[t] + 119 (Sqrt[39]) Cos[t] + 104 (Sqrt[7]) Sin[t] + 7 (Sqrt[13]) Sin[t])/(52 (Sqrt[21]) - 84 (Sqrt[39]) - 104 (Sqrt[7]) Sin[t] - 7 (Sqrt[13]) Sin[t]))), -((39 (Sqrt[21]) - 63 (Sqrt[39]) - 
                182 (Sqrt[21]) Cos[t] + 119 (Sqrt[39]) Cos[t] + 104 (Sqrt[7]) Sin[t] + 
                7 (Sqrt[13]) Sin[t])/(52 (Sqrt[21]) - 84 (Sqrt[39]) - 104 (Sqrt[7]) Sin[t] - 
                7 (Sqrt[13]) Sin[t]))}

glavne zakrivljenosti i normalna zakrivljenost

Solve[{K1*K2 == g, 1/2 (K1 + K2) == m}, {K1, K2}]

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

definicije gaussove i srednje zakrivljenosti

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}

racun zakrivljenosti

gcurvature[F][Pi/3, 0]

meancurvature[F][Pi/3, 0]

Solve[{x1*x2 == -(12/121), 
          1/2 (x1 + x2) == -((17 (Sqrt[(2/11)]))/11)}, {x1, x2}] // 
      Simplify

N[-(1/11) (Sqrt[(2/11)]) (17 + (Sqrt[355]))]

1/%

N[(1/11) (Sqrt[(2/11)]) (-17 + (Sqrt[355]))]

1/%

Solve[
        Cos[x]^2*((1/11) (Sqrt[(2/11)]) (-17 + (Sqrt[355]))) + 
            Sin[x]^2*(-(1/11) (Sqrt[(2/11)]) (17 + (Sqrt[355]))) == 0, x, InverseFunctions -> True] // Simplify

N[ArcCos[-(Sqrt[(1/2) + (17/(2 (Sqrt[355])))])]]

N[ArcCos[(Sqrt[(1/2) + (17/(2 (Sqrt[355])))])]]

{{phi1 -> 
          ArcCos[-(Sqrt[(1/2) + (17/(2 (Sqrt[355])))])]}, {phi2 -> 
          ArcCos[(Sqrt[(1/2) + (17/(2 (Sqrt[355])))])]}}

zakrivljenost[x_] := 
      Cos[x]^2*((1/11) (Sqrt[(2/11)]) (-17 + (Sqrt[355]))) + 
        Sin[x]^2*(-(1/11) (Sqrt[(2/11)]) (17 + (Sqrt[355])))

crtrezi1

\(Show[
      ParametricPlot3D[
        F[u, v] // Evaluate, 
	{u, 0, 2  Pi, Pi/24}, {v, \(-1\), 2, 0.1}, 
        DisplayFunction -> 
          Identity], 
Graphics3D[{AbsolutePointSize[10], 
          Point[{1/2, Sqrt[3]/2, 0}]}], 
	
      Axes -> None, Boxed -> False, 
      ViewPoint -> {1, 0.5, 
          0.7}, 
			PlotRange -> {{\(-1.1\), 1.1}, {\(-2\), 2}, {\(-1\), 
            2}}, DisplayFunction -> $DisplayFunction, 
      Background -> GrayLevel[0.8], 
      ImageSize -> {400, 330}]\)

\(Show[
Table[
        ParametricPlot3D[
          F[u, z] /. {u -> i} // Evaluate, 
	{z, \(-0.9\), 
            1.9}, 
				DisplayFunction -> Identity], {i, 0, 2  Pi, 
          Pi/48}], 

ParametricPlot3D[{Cos[u], 
          Sin[u] \((1 - z)\), z, 
          FaceForm[Hue[0.66, 0.5, 0.9], Hue[0.66, 0.2, 0.9]]}, 
	{u, 0, 
          2  Pi, Pi/24}, {z, \(-1\), \(-0.9\), 0.1}, 
			DisplayFunction -> 
          Identity], 
ParametricPlot3D[{Cos[u], 
          Sin[u] \((1 - z)\), z, 
          FaceForm[Hue[0.66, 0.2, 0.9], Hue[0.66, 0.5, 0.9]]}, 
	{u, 0, 
          2  Pi, Pi/24}, {z, 1.9, 2, 0.1}, 
			DisplayFunction -> 
          Identity], 

ParametricPlot3D[
        Append[ravTang, Hue[0.14, 0.5, 1]] // Evaluate, {u, \(-1\), 2, 
          3}, {v, \(-1\), 2, 3}, 
        DisplayFunction -> 
          Identity], 

ParametricPlot3D[\(Kriv1[0]\)[z] // 
          Evaluate, {z, \(-1\), 2}, 
        DisplayFunction -> 
          Identity], 

Kugla[
        0], 
ParametricPlot3D[
        Append[normala, {AbsoluteThickness[2], Hue[0.7]}] // 
          Evaluate, {t, \(-1.5\), 2.5}, 
        DisplayFunction -> 
          Identity], 

Table[
        ParametricPlot3D[
          Append[tang[i], Hue[1]] // Evaluate, {t, \(-0.7\), 0.7}, 
          DisplayFunction -> Identity], {i, 0, 2  Pi, 
          Pi/12}], 

Graphics3D[
        Text[FontForm["n", {"Ariel-Bold", 14}], {0.5, 1.8, 
            0.7}, {\(-1\), 0}]], 
Graphics3D[
        Text[FontForm["T", {"Ariel-Bold", 14}], {0.5, 0.85, 
            0.3}]], 
Graphics3D[
        Text[FontForm["τ", {"Symbol-Bold", 16}], {0.9, \(-1.2\), 
            1.8}]], 
	
	Axes -> None, 
      Boxed -> False, 
      ViewPoint -> {1, 0.5, 
          0.7}, 
			PlotRange -> {{\(-1.1\), 1.2}, {\(-2.5\), 
            2.7}, {\(-1. \), 2}}, DisplayFunction -> $DisplayFunction, 
      Background -> GrayLevel[0.8], Lighting -> False, 
      ImageSize -> {400, 330}]\)

anim1

N[prvaGZ[F][Pi/3, 0]]

N[drugaGZ[F][Pi/3, 0]]

Table[
      N[normalna[F][Pi/3, 0][i]], {i, 0, Pi - Pi/45, 
        Pi/45}]

\(Table[
      Show[
Table[
          ParametricPlot3D[
            F[u, z] /. {u -> i} // Evaluate, 
	{z, \(-0.9\), 
              1.9}, 
				DisplayFunction -> Identity], {i, 0, 2  Pi, 
            Pi/36}], 

ParametricPlot3D[{Cos[u], 
            Sin[u] \((1 - z)\), z, 
            FaceForm[Hue[0.66, 0.5, 0.9], Hue[0.66, 0.2, 0.9]]}, 
	{u, 0, 
            2  Pi, Pi/24}, {z, \(-1\), \(-0.9\), 
            0.1}, 
			DisplayFunction -> 
            Identity], 
ParametricPlot3D[{Cos[u], 
            Sin[u] \((1 - z)\), z, 
            FaceForm[Hue[0.66, 0.2, 0.9], Hue[0.66, 0.5, 0.9]]}, 
	{u, 0, 
            2  Pi, Pi/24}, {z, 1.9, 2, 0.1}, 
			DisplayFunction -> 
            Identity], 

ParametricPlot3D[
          Append[ravNormal1[i + Pi/2 + 0.223], White] // 
            Evaluate, {u, \(-0.75\), 0.75, 0.75}, {v, \(-0.75\), 0.75, 0.75}, 
          DisplayFunction -> 
            Identity], 

Kugla[
          0], 
ParametricPlot3D[
          Append[normala, {AbsoluteThickness[3], Hue[0.7]}] // 
            Evaluate, {t, \(-1.5\), 1.5}, 
          DisplayFunction -> 
            Identity], 

ScatterPlot3D[
          tockice[i + Pi/2 + 0.223] // Evaluate, 
          PlotStyle -> {Hue[1], AbsolutePointSize[4]}, 
          DisplayFunction -> 
            Identity], 

Graphics3D[{Text[
              FontForm[\*""(1/R)="", {"Ariel-Bold", 
                  10}], {1.1, \(-2.5\), 2}, {\(-1\), 
                0}]}], 
Graphics3D[{Text[
              FontForm[
                N[\(\(normalna[F]\)[Pi/3, 0]\)[i] // 
                    Evaluate], {"Ariel-Bold", 10}], {0.85, \(-2.25\), 
                1.95}, {\(-1\), 
                0}]}], 

Graphics3D[
          Text[FontForm["n", {"Ariel-Bold", 14}], {0.5, 1.8, 
              0.7}, {\(-1\), 0}]], 
	
	
        Axes -> None, Boxed -> False, 
        ViewPoint -> {1, 0.5, 
            0.7}, 
			PlotRange -> {{\(-1.1\), 1.5}, {\(-2.5\), 
              2.7}, {\(-1. \), 2}}, DisplayFunction -> $DisplayFunction, 
        Background -> GrayLevel[0.8], Lighting -> False, 
        ImageSize -> {300, 250}], {i, 0, Pi - Pi/45, Pi/45}]\)

anim2

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

Graphics[{Hue[
              1], AbsoluteThickness[2], AbsoluteDashing[{2, 5}], 
            Line[{{0, 0}, {0, 0.0713}}], Line[{{Pi, 0}, {Pi, 0.0713}}], 
            Line[{{Pi/2, 
                  0}, {Pi/
                    2, \(-1.389\)}}]}], 

Graphics[{Hue[1], AbsolutePointSize[6], 
            Point[{0, 0}], Point[{Pi, 0}], 
            Point[{Pi/2, 0}]}], 
Graphics[{Hue[0.5], 
            AbsolutePointSize[6], 
            Point[{ArcCos[\(-\@\(1\/2 + 17\/\(2\ \@355\)\)\)], 0}], 
            Point[{ArcCos[\@\(1\/2 + 17\/\(2\ \@355\)\)], 
                0}]}], 

Graphics[{AbsolutePointSize[6], Point[{0, 0.0713813}], Point[{Pi, 0.0713813}], 
            Point[{Pi/
                  2, \(-1.38935\)}]}], 

Graphics[{{Arrow[{0, 0.7}, {0, 0.8}, 
                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, 
                0.7}, {\(-1\), 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, \(-0.5\)}, {\(-1\), 
                0}]}], 
Graphics[{Hue[1], 
            Text[FontForm[\*""(Subscript[K, 1])"", {"Arial-Bold", 
                  12}], {Pi + 0.05, 0.05}, {\(-1\), 
                0}]}], 

Graphics[{Hue[
              0.5], AbsolutePointSize[8], 
            Point[{1.75, 0.5}]}], 
Graphics[{Hue[1], 
            AbsolutePointSize[8], 
            Point[{1.75, 
                0.35}]}], 

Graphics[{Text[
              FontForm["-  asymptotic  directions", {"Arial", 
                  10}], {1.9, 0.5}, {\(-1\), 
                0}]}], 
Graphics[{Text[
              FontForm["-  principal  directions", {"Arial", 
                  10}], {1.9, 0.35}, {\(-1\), 
                0}]}], 

Background -> GrayLevel[
            0.8], 
Ticks -> {{0, Pi/4, 1, Pi/2, 
              3  Pi/4, Pi}, {\(-1.5\), \(-1.38935\), \(-1\), 0.0713813, 
              0.5}}, AspectRatio -> 1, 
        PlotRange -> {{\(-0.5\), 3.8}, {\(-1.6\), 
              0.8}}, 
DisplayFunction -> $DisplayFunction, ImageSize -> {300, 250}], {i, 0, Pi - Pi/45, 
        Pi/45}]\)

crtez2

\(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, 0.0713}}], Line[{{Pi, 0}, {Pi, 0.0713}}], 
          Line[{{Pi/2, 
                0}, {Pi/
                  2, \(-1.389\)}}]}], 

Graphics[{Hue[1], AbsolutePointSize[6], 
          Point[{0, 0}], Point[{Pi, 0}], 
          Point[{Pi/2, 0}]}], 
Graphics[{Hue[0.5], 
          AbsolutePointSize[6], 
          Point[{ArcCos[\(-\@\(1\/2 + 17\/\(2\ \@355\)\)\)], 0}], 
          Point[{ArcCos[\@\(1\/2 + 17\/\(2\ \@355\)\)], 
              0}]}], 

Graphics[{AbsolutePointSize[6], Point[{0, 0.0713813}], Point[{Pi, 0.0713813}], 
          Point[{Pi/
                2, \(-1.38935\)}]}], 

Graphics[{{Arrow[{0, 0.7}, {0, 0.8}, 
              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, 
              0.7}, {\(-1\), 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, \(-0.5\)}, {\(-1\), 
              0}]}], 
Graphics[{Hue[1], 
          Text[FontForm[\*""(Subscript[K, 1])"", {"Arial-Bold", 
                12}], {Pi + 0.05, 0.05}, {\(-1\), 
              0}]}], 

Graphics[{Hue[
            0.5], AbsolutePointSize[8], 
          Point[{1.75, 0.5}]}], 
Graphics[{Hue[1], 
          AbsolutePointSize[8], 
          Point[{1.75, 
              0.35}]}], 

Graphics[{Text[
            FontForm["-  asymptotic  directions", {"Arial", 
                10}], {1.9, 0.5}, {\(-1\), 
              0}]}], 
Graphics[{Text[
            FontForm["-  principal  directions", {"Arial", 
                10}], {1.9, 0.35}, {\(-1\), 
              0}]}], 

Background -> 
        GrayLevel[
          0.8], 
Ticks -> {{0, Pi/4, 1, Pi/2, 3  Pi/4, 
            Pi}, {\(-1.5\), \(-1.38935\), \(-1\), 0.0713813, 0.5}}, 
      AspectRatio -> 1, 
      PlotRange -> {{\(-0.5\), 3.8}, {\(-1.6\), 
            0.8}}, 
DisplayFunction -> $DisplayFunction, ImageSize -> {300, 250}]\)

\(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, 0.0713}}], Line[{{Pi, 0}, {Pi, 0.0713}}], 
          Line[{{Pi/2, 
                0}, {Pi/
                  2, \(-1.389\)}}]}], 

Graphics[{Hue[1], AbsolutePointSize[6], 
          Point[{0, 0}], Point[{Pi, 0}], 
          Point[{Pi/2, 0}]}], 
Graphics[{Hue[0.5], 
          AbsolutePointSize[6], 
          Point[{ArcCos[\(-\@\(1\/2 + 17\/\(2\ \@355\)\)\)], 0}], 
          Point[{ArcCos[\@\(1\/2 + 17\/\(2\ \@355\)\)], 
              0}]}], 

Graphics[{AbsolutePointSize[6], Point[{0, 0.0713813}], Point[{Pi, 0.0713813}], 
          Point[{Pi/
                2, \(-1.38935\)}]}], 

Graphics[{{Arrow[{0, 0.7}, {0, 0.8}, 
              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, 0.7}, {\(-1\), 
              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, \(-0.5\)}, {\(-1\), 
              0}]}], 
Graphics[{Hue[1], 
          Text[FontForm[\*""(Subscript[K, 1])"", {"Arial-Bold", 
                12}], {Pi + 0.05, 0.05}, {\(-1\), 
              0}]}], 

Graphics[{Hue[
            0.5], AbsolutePointSize[8], 
          Point[{2.05, \(-1.2\)}]}], 
Graphics[{Hue[1], 
          AbsolutePointSize[8], 
          Point[{2.05, \(-1.35\)}]}], 

Graphics[{Text[
            FontForm["-  asimptotski smjerovi", {"Arial", 
                10}], {2.15, \(-1.2\)}, {\(-1\), 
              0}]}], 
Graphics[{Text[
            FontForm["-  glavni smjerovi", {"Arial", 
                10}], {2.15, \(-1.35\)}, {\(-1\), 
              0}]}], 

Background -> 
        GrayLevel[
          0.8], 
Ticks -> {{0, Pi/4, 1, Pi/2, 3  Pi/4, 
            Pi}, {\(-1.5\), \(-1.38935\), \(-1\), 0.0713813, 0.5}}, 
      AspectRatio -> 1, 
      PlotRange -> {{\(-0.5\), 3.8}, {\(-1.6\), 
            0.8}}, 
DisplayFunction -> $DisplayFunction, ImageSize -> {300, 250}]\)

crtezi3

\(Show[
Table[
        ParametricPlot3D[
          F[u, z] /. {u -> i} // Evaluate, 
	{z, \(-0.9\), 
            1.9}, 
				DisplayFunction -> Identity], {i, 0, 2  Pi, 
          Pi/48}], 

ParametricPlot3D[{Cos[u], 
          Sin[u] \((1 - z)\), z, 
          FaceForm[Hue[0.66, 0.5, 0.9], Hue[0.66, 0.2, 0.9]]}, 
	{u, 0, 
          2  Pi, Pi/24}, {z, \(-1\), \(-0.9\), 0.1}, 
			DisplayFunction -> 
          Identity], 
ParametricPlot3D[{Cos[u], 
          Sin[u] \((1 - z)\), z, 
          FaceForm[Hue[0.66, 0.2, 0.9], Hue[0.66, 0.5, 0.9]]}, 
	{u, 0, 
          2  Pi, Pi/24}, {z, 1.9, 2, 0.1}, 
			DisplayFunction -> 
          Identity], 

ParametricPlot3D[
        Append[ravTang, Hue[0.14, 0.5, 1]] // Evaluate, {u, \(-1\), 2, 
          3}, {v, \(-1\), 2, 3}, 
        DisplayFunction -> 
          Identity], 



Kugla[0], 
ParametricPlot3D[
        Append[normala, {AbsoluteThickness[2], Hue[0.7]}] // 
          Evaluate, {t, \(-1.5\), 2.5}, 
        DisplayFunction -> 
          Identity], 

ParametricPlot3D[
        Append[izvodnica, {AbsoluteThickness[2]}] // 
          Evaluate, {t, \(-1\), \(-0.7\)}, 
        DisplayFunction -> Identity], 
ParametricPlot3D[
        Append[izvodnica, {AbsoluteThickness[2]}] // Evaluate, {t, 0.7, 2}, 
        DisplayFunction -> 
          Identity], 

ParametricPlot3D[
        Append[\(Kriv1[0]\)[z], {AbsoluteThickness[2]}] // 
          Evaluate, {z, \(-1\), 2}, 
        DisplayFunction -> 
          Identity], 

ParametricPlot3D[
        Append[gltangenta, {AbsoluteThickness[2], Hue[0.5]}] // 
          Evaluate, {t, \(-0.5\), 0.5}, 
        DisplayFunction -> Identity], 
ParametricPlot3D[
        Append[izvodnica, {AbsoluteThickness[2], Hue[0.5]}] // 
          Evaluate, {t, \(-0.7\), 0.7}, 
        DisplayFunction -> 
          Identity], 

Graphics3D[
        Text[
          FontForm["n", {"Ariel-Bold", 14}], {0.5, 1.8, 
            0.7}, {\(-1\), 0}]], 
Graphics3D[
        Text[FontForm["T", {"Ariel-Bold", 14}], {0.5, 0.85, 
            0.3}]], 
Graphics3D[
        Text[FontForm["τ", {"Symbol-Bold", 16}], {0.9, \(-1.2\), 
            1.8}]], 
Graphics3D[
        Text[FontForm[\*""(Subscript[a, 1])"", {"Ariel-Bold", 
              14}], {0.7,  .5, 0.1}]], 
Graphics3D[
        Text[FontForm[\*""(Subscript[a, 2])"", {"Ariel-Bold", 14}], {0.7, 
            1, \(-0.5\)}, {\(-1\), 
            0}]], 
	
	
	
      Axes -> None, Boxed -> False, 
      ViewPoint -> {1, 0.5, 
          0.7}, 
			PlotRange -> {{\(-1.1\), 1.2}, {\(-2.5\), 
            2.7}, {\(-1. \), 2}}, DisplayFunction -> $DisplayFunction, 
      Background -> GrayLevel[0.8], Lighting -> False, 
      ImageSize -> {400, 330}]\)

\(Show[
Table[
        ParametricPlot3D[
          F[u, z] /. {u -> i} // Evaluate, 
	{z, \(-0.9\), 
            1.9}, 
				DisplayFunction -> Identity], {i, 0, 2  Pi, 
          Pi/36}], 

ParametricPlot3D[{Cos[u], 
          Sin[u] \((1 - z)\), z, 
          FaceForm[Hue[0.66, 0.5, 0.9], Hue[0.66, 0.2, 0.9]]}, 
	{u, 0, 
          2  Pi, Pi/24}, {z, \(-1\), \(-0.9\), 0.1}, 
			DisplayFunction -> 
          Identity], 
ParametricPlot3D[{Cos[u], 
          Sin[u] \((1 - z)\), z, 
          FaceForm[Hue[0.66, 0.2, 0.9], Hue[0.66, 0.5, 0.9]]}, 
	{u, 0, 
          2  Pi, Pi/24}, {z, 1.9, 2, 0.1}, 
			DisplayFunction -> 
          Identity], 


Kugla[
        0], 
ParametricPlot3D[
        Append[normala, {AbsoluteThickness[2], Hue[0.7]}] // 
          Evaluate, {t, \(-1.5\), 2.5}, 
        DisplayFunction -> 
          Identity], 

ParametricPlot3D[
        Append[izvodnica, {AbsoluteThickness[1], Hue[0.5]}] // 
          Evaluate, {t, \(-1\), \(-0.7\)}, 
        DisplayFunction -> Identity], 
ParametricPlot3D[
        Append[izvodnica, {AbsoluteThickness[2], Hue[0.5]}] // Evaluate, {t, 
          0.7, 2}, 
        DisplayFunction -> 
          Identity], 


ParametricPlot3D[
        Append[gltangenta, {AbsoluteThickness[2]}] // Evaluate, {t, \(-0.5\), 
          0.5}, DisplayFunction -> 
          Identity], 
ParametricPlot3D[
        Append[izvodnica, {AbsoluteThickness[2]}] // Evaluate, {t, \(-0.7\), 
          0.7}, DisplayFunction -> 
          Identity], 



ParametricPlot3D[
        Append[asimptRav1, White] // Evaluate, {u, \(-0.7\), 0.7, 
          1.4}, {v, \(-0.7\), 2, 2.7}, 
        DisplayFunction -> 
          Identity], 

ParametricPlot3D[
        Append[kr1[t], {AbsoluteThickness[2], Hue[0.5]}] // 
          Evaluate, {t, \(-Pi\), 0}, 
        DisplayFunction -> 
          Identity], 

ParametricPlot3D[
        Append[asimptRav2, White] // Evaluate, {u, \(-0.7\), 0.7, 
          1.4}, {v, \(-0.7\), 1.5, 2.2}, 
        DisplayFunction -> 
          Identity], 

ParametricPlot3D[
        Append[kr2[t], {AbsoluteThickness[2], Hue[0.5]}] // Evaluate, {t, 0, 
          2  Pi}, DisplayFunction -> 
          Identity], 

Graphics3D[
        Text[FontForm["n", {"Ariel-Bold", 14}], {0.5, 1.8, 
            0.6}, {\(-1\), 0}]], 
	
	Axes -> None, 
      Boxed -> False, 
      ViewPoint -> {1, 0.4, 
          0.8}, 
			PlotRange -> {{\(-1.1\), 1.2}, {\(-2.5\), 
            2.7}, {\(-1. \), 2}}, DisplayFunction -> $DisplayFunction, 
      Background -> GrayLevel[0.8], Lighting -> False, 
      ImageSize -> {400, 330}]\)

\(Show[
Table[
        ParametricPlot3D[
          F[u, z] /. {u -> i} // Evaluate, 
	{z, \(-0.9\), 
            1.9}, 
				DisplayFunction -> Identity], {i, 0, 2  Pi, 
          Pi/48}], 

ParametricPlot3D[{Cos[u], 
          Sin[u] \((1 - z)\), z, 
          FaceForm[Hue[0.66, 0.5, 0.9], Hue[0.66, 0.2, 0.9]]}, 
	{u, 0, 
          2  Pi, Pi/24}, {z, \(-1\), \(-0.9\), 0.1}, 
			DisplayFunction -> 
          Identity], 
ParametricPlot3D[{Cos[u], 
          Sin[u] \((1 - z)\), z, 
          FaceForm[Hue[0.66, 0.2, 0.9], Hue[0.66, 0.5, 0.9]]}, 
	{u, 0, 
          2  Pi, Pi/24}, {z, 1.9, 2, 0.1}, 
			DisplayFunction -> 
          Identity], 

ParametricPlot3D[
        Append[ravTang, Hue[0.14, 0.5, 1]] // Evaluate, {u, \(-1\), 2, 
          3}, {v, \(-1\), 2, 3}, 
        DisplayFunction -> 
          Identity], 



Kugla[0], 
ParametricPlot3D[
        Append[normala, {AbsoluteThickness[2], Hue[0.7]}] // 
          Evaluate, {t, \(-1.5\), 2.5}, 
        DisplayFunction -> 
          Identity], 

ParametricPlot3D[
        Append[izvodnica, {AbsoluteThickness[1]}] // 
          Evaluate, {t, \(-1\), \(-0.7\)}, 
        DisplayFunction -> Identity], 
ParametricPlot3D[
        Append[izvodnica, {AbsoluteThickness[1]}] // Evaluate, {t, 0.7, 2}, 
        DisplayFunction -> 
          Identity], 

ParametricPlot3D[
        Append[\(Kriv1[0]\)[z], {AbsoluteThickness[1]}] // 
          Evaluate, {z, \(-1\), 2}, 
        DisplayFunction -> 
          Identity], 

ParametricPlot3D[
        Append[gltangenta, {AbsoluteThickness[2], Hue[0.5]}] // 
          Evaluate, {t, \(-0.5\), 0.5}, 
        DisplayFunction -> Identity], 
ParametricPlot3D[
        Append[izvodnica, {AbsoluteThickness[2], Hue[0.5]}] // 
          Evaluate, {t, \(-0.7\), 0.7}, 
        DisplayFunction -> 
          Identity], 

ParametricPlot3D[
        Append[tangenta1, {AbsoluteThickness[2], Hue[1]}] // 
          Evaluate, {t, \(-0.5\), 0.5}, 
        DisplayFunction -> Identity], 
ParametricPlot3D[
        Append[tangenta2, {AbsoluteThickness[2], Hue[1]}] // 
          Evaluate, {t, \(-1.5\), 1.5}, 
        DisplayFunction -> 
          Identity], 

Graphics3D[
        Text[FontForm["n", {"Ariel-Bold", 14}], {0.5, 1.8, 
            0.7}, {\(-1\), 0}]], 
Graphics3D[
        Text[FontForm["T", {"Ariel-Bold", 14}], {0.5, 0.85, 
            0.3}]], 
Graphics3D[
        Text[FontForm["τ", {"Symbol-Bold", 16}], {0.9, \(-1.2\), 
            1.8}]], 
Graphics3D[
        Text[FontForm[\*""(Subscript[a, 1])"", {"Ariel-Bold", 
              14}], {0.7,  .5, 0.1}]], 
Graphics3D[
        Text[FontForm[\*""(Subscript[a, 2])"", {"Ariel-Bold", 14}], {0.7, 
            1, \(-0.5\)}, {\(-1\), 0}]], 
Graphics3D[
        Text[FontForm[\*""(Subscript[p, 1])"", {"Ariel-Bold", 14}], {1, 
            1.85, 0. }]], 
Graphics3D[
        Text[FontForm[\*""(Subscript[p, 2])"", {"Ariel-Bold", 14}], {0. , 
            1, 0.5}, {\(-1\), 
            0}]], 

	
	
      Axes -> None, Boxed -> False, 
      ViewPoint -> {1, 0.5, 
          0.7}, 
			PlotRange -> {{\(-1.1\), 1.2}, {\(-2.5\), 
            2.7}, {\(-1. \), 2}}, DisplayFunction -> $DisplayFunction, 
      Background -> GrayLevel[0.8], Lighting -> False, 
      ImageSize -> {400, 330}]\)

\(Show[
Table[
        ParametricPlot3D[
          F[u, z] /. {u -> i} // Evaluate, 
	{z, \(-0.9\), 
            1.9}, 
				DisplayFunction -> Identity], {i, 0, 2  Pi, 
          Pi/36}], 

ParametricPlot3D[{Cos[u], 
          Sin[u] \((1 - z)\), z, 
          FaceForm[Hue[0.66, 0.5, 0.9], Hue[0.66, 0.2, 0.9]]}, 
	{u, 0, 
          2  Pi, Pi/24}, {z, \(-1\), \(-0.9\), 0.1}, 
			DisplayFunction -> 
          Identity], 
ParametricPlot3D[{Cos[u], 
          Sin[u] \((1 - z)\), z, 
          FaceForm[Hue[0.66, 0.2, 0.9], Hue[0.66, 0.5, 0.9]]}, 
	{u, 0, 
          2  Pi, Pi/24}, {z, 1.9, 2, 0.1}, 
			DisplayFunction -> 
          Identity], 

Kugla[
        0], 
ParametricPlot3D[
        Append[normala, {AbsoluteThickness[2], Hue[0.7]}] // 
          Evaluate, {t, \(-1.5\), 2.5}, 
        DisplayFunction -> 
          Identity], 




ParametricPlot3D[
        Append[tangenta2, {AbsoluteThickness[2]}] // Evaluate, {t, \(-2\), 
          1}, DisplayFunction -> 
          Identity], 
ParametricPlot3D[
        Append[tangenta1, {AbsoluteThickness[2]}] // Evaluate, {t, \(-0.4\), 
          0.4}, DisplayFunction -> 
          Identity], 



ParametricPlot3D[
        Append[glavRav1, White] // Evaluate, {u, \(-0.7\), 0.5, 
          1.2}, {v, \(-0.4\), 0.9, 1.3}, 
        DisplayFunction -> 
          Identity], 

ParametricPlot3D[
        Append[kr3[t], {AbsoluteThickness[2], Hue[1]}] // 
          Evaluate, {t, \(-Pi\), Pi}, 
        DisplayFunction -> 
          Identity], 

ParametricPlot3D[
        Append[glavRav2, White] // Evaluate, {u, \(-0.7\), 0.5, 
          1.2}, {v, \(-1\), 2.5, 3.5}, 
        DisplayFunction -> 
          Identity], 

ParametricPlot3D[
        Append[kr4[t], {AbsoluteThickness[2], Hue[1]}] // Evaluate, {t, 0, 
          Pi + Pi/4}, 
        DisplayFunction -> Identity], 
Graphics3D[
        Text[FontForm["n", {"Ariel-Bold", 14}], {0.5, 1.8, 
            0.6}, {\(-1\), 0}]], 

	
	
      Axes -> None, Boxed -> False, 
      ViewPoint -> {1, 0.4, 
          0.8}, 
			PlotRange -> {{\(-1.1\), 1.2}, {\(-2.5\), 
            2.7}, {\(-1. \), 2}}, DisplayFunction -> $DisplayFunction, 
      Background -> GrayLevel[0.8], Lighting -> False, 
      ImageSize -> {400, 330}]\)

\(Show[
ParametricPlot3D[
        F[u, v] // Evaluate, 
	{u, 0, 2  Pi, Pi/24}, {v, \(-1\), 2, 0.1}, 
        DisplayFunction -> 
          Identity], 

ParametricPlot3D[
        Append[normala, {AbsoluteThickness[3], Hue[0.7]}] // 
          Evaluate, {t, \(-1.5\), 2.5}, 
        DisplayFunction -> 
          Identity], 

ParametricPlot3D[
        Append[izvodnica, {AbsoluteThickness[3], Hue[0.5]}] // 
          Evaluate, {t, \(-1\), 2}, 
        DisplayFunction -> 
          Identity], 

ParametricPlot3D[
        Append[kr3[t], {AbsoluteThickness[3], Hue[1]}] // 
          Evaluate, {t, \(-Pi\), Pi}, 
        DisplayFunction -> 
          Identity], 

ParametricPlot3D[
        Append[kr4[t], {AbsoluteThickness[3], Hue[1]}] // Evaluate, {t, 0, 
          Pi + Pi/4}, 
        DisplayFunction -> 
          Identity], 

ParametricPlot3D[
        Append[kr1[t], {AbsoluteThickness[3], Hue[0.5]}] // 
          Evaluate, {t, \(-Pi\), 0}, 
        DisplayFunction -> 
          Identity], 

ParametricPlot3D[
        Append[kr2[t], {AbsoluteThickness[3], Hue[0.5]}] // Evaluate, {t, 0, 
          2  Pi}, DisplayFunction -> 
          Identity], 



Graphics3D[{AbsolutePointSize[10], 
          Point[{1/2, Sqrt[3]/2, 0}]}], 
	
      Axes -> None, Boxed -> False, 
      ViewPoint -> {1, 0.5, 
          0.7}, 
			PlotRange -> {{\(-1.1\), 1.1}, {\(-2\), 2}, {\(-1\), 
            2}}, DisplayFunction -> $DisplayFunction, 
      Background -> GrayLevel[0.8], 
      ImageSize -> {400, 330}]\)

crtezi4

\(ParametricPlot3D[
      Append[F[u, v], Hue[\(gcurvature[F]\)[u, v]]] // Evaluate, 
	{u, 0, 
        2  Pi, Pi/36}, {v, \(-1\), 2.1, 0.1}, 
	
      ViewPoint -> {1, 0.2, 0.7}, Axes -> None, 
      Boxed -> 
        False, 
			PlotRange -> {{\(-1.1\), 1.1}, {\(-2\), 2}, {\(-1\), 
            2.1}}, DisplayFunction -> $DisplayFunction, 
      Lighting -> False, Background -> GrayLevel[0.8], 
      ImageSize -> {400, 330}]\)

Plot3D[gcurvature[F][u, v] /. {u -> x, v -> y} // 
          Evaluate,{x, 0, 2  Pi},{y, -1, 2},
PlotPoints -> 100,AxesLabel -> {u, v, gcurvature},AxesEdge -> {{-1, -1}, {-1, 
                1}, {-1, -1}},
BoxRatios -> {1, 1, 1},Mesh -> False,ViewPoint -> {0.9,  -3, 1},Ticks -> {{0, (π/2), π, ((3 π)/2), 
              2 π}, Automatic, Automatic},Background -> GrayLevel[0.8],ImageSize -> {400, 330},AmbientLight -> DeepPink,FaceGrids -> All]

Plot3D[gcurvature[F][u, v] /. {u -> x, v -> y} // 
          Evaluate,{x, 0, 2  Pi},{y, -1, 2},
PlotPoints -> 100,AxesLabel -> {u, v, gcurvature},AxesEdge -> {{-1, -1}, {1, -1}, {-1, -1}},
BoxRatios -> {1, 1, 1},Mesh -> False,ViewPoint -> {0.9,  -3, 1},Ticks -> {{0, (π/2), π, ((3 π)/2), 
              2 π}, Automatic, Automatic},Background -> GrayLevel[0.8],ImageSize -> {400, 330},AmbientLight -> OrangeRed,FaceGrids -> All]

\(ParametricPlot3D[
      Append[F[u, v], Hue[\(meancurvature[F]\)[u, v]]] // Evaluate, 
	{u, 
        0, 2  Pi, Pi/36}, {v, \(-1\), 2.1, 0.1}, 
	
      ViewPoint -> {1, 0.2, 0.7}, Axes -> None, 
      Boxed -> 
        False, 
			PlotRange -> {{\(-1.1\), 1.1}, {\(-2\), 2}, {\(-1\), 
            2.1}}, DisplayFunction -> $DisplayFunction, 
      Lighting -> False, Background -> GrayLevel[0.8], 
      ImageSize -> {400, 330}]\)

Plot3D[meancurvature[F][u, v] /. {u -> x, v -> y} // 
          Evaluate,{x, 0, 2  Pi},{y, -1, 2},
PlotPoints -> 100,AxesLabel -> {u, v, meancurvature},Mesh -> False,AxesEdge -> {{-1, -1}, {1, -1}, {-1, -1}},
BoxRatios -> {1, 1, 1},ViewPoint -> {0.9,  -3, 1},Ticks -> {{0, (π/2), π, ((3 π)/2), 
              2 π}, Automatic, Automatic},Background -> GrayLevel[0.8],ImageSize -> {400, 330},AmbientLight -> Indigo,FaceGrids -> All]