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

def

anim 1,  sl 1,2

kugla[r_, a_, b_, c_] := 
      ParametricPlot3D[{r*Cos[u] Sin[v] + a, r*Sin[u] Sin[v] + b, 
          r*Cos[v] + c, FaceForm[Black]}, {u, 0, Pi, Pi/16}, {v, 0, 2  Pi, 
          Pi/16}, 
DisplayFunction -> 
          Identity]

Paraboloid[a_]:=Plot3D[{-a (x^2 + y^2), Hue[0.66, 0.3, 0.9]},{x, -1, 1},{y, -1, 1},PlotPoints -> {20, 20},
DisplayFunction -> Identity]

krivulja[a_][u_] := 
      ParametricPlot3D[{x, 
          x*Tan[u], -a (x^2 + (x*Tan[u])^2), {Hue[1], 
            AbsoluteThickness[2]}}, {x, -1, 1, 0.1}, 
        DisplayFunction -> Identity]

krivulja1[a_][u_] := 
      ParametricPlot3D[{t*Cos[u], 
          t*Sin[u], -a*t^2 + 0.01, {Hue[1], 
            AbsoluteThickness[2]}}, {t, -1.5, 1.5, 0.1}, 
        DisplayFunction -> Identity]

tangenta[u_] := 
      Graphics3D[{AbsoluteThickness[3], 
          Line[{{0.5  Cos[u], 0.5  Sin[u], 
                0.01}, {-0.5 Cos[u], -0.5 Sin[u], 
                0.01}}]}]

RotLemniscate[a_][t_, u_]:=ParametricPlot3D[Append[a/(1 + Sin[t]^2){Sin[t]*Cos[t]*Cos[u], Sin[t]*Cos[t]*Sin[u], 
                      Cos[t]},Hue[0.66, 0.3, 0.9]]//Evaluate,{u, 0, Pi},{t, 0, 2  Pi, Pi/36},
DisplayFunction -> Identity]

TangStozac[u_, z_] := 
      ParametricPlot3D[{z*Cos[u], z*Sin[u], z, Hue[0.14, 0.5, 1]}, {u, 0, 
          2  Pi, Pi/12}, {z, -0.25, 0.25, 
          0.25}, 
DisplayFunction -> 
          Identity]

sl 3

fleksija[krivulja_][
        t_] := 
Simplify[
              Factor[
Dot[
Cross[
                    D[krivulja[tt], tt], D[krivulja[tt], {tt, 2}]], 
                  Cross[D[krivulja[tt], tt], 
                    D[krivulja[tt], {tt, 2}]]]]]^(1/
                2)/
Simplify[
              Factor[
Dot[D[krivulja[tt], tt], 
                  D[krivulja[tt], tt]]]]^(3/2) /. tt -> t

elipsa[a_, b_][t_] := {a*Cos[t], b*Sin[t]}

elipsa1[a_, b_][t_] := {a*Cos[t], b*Sin[t], 0}

kruznica[p_, q_, r_][t_] := {r*Cos[t] + p, r*Sin[t] + q}

kruznica1[p_, q_, r_][t_] := {r*Cos[t] + p, r*Sin[t] + q, 
        0}

fleksija[elipsa1[3, 2]][0]

fleksija[elipsa1[3, 2]][Pi/2]

fleksija[elipsa1[3, 2]][Pi/6]

sl 4

ElPar[a_, b_][x_, y_]:=ParametricPlot3D[{x, y, -x^2/a^2 - y^2/b^2, 
              FaceForm[Hue[0.5], Hue[0.7]]},{x, -1, 1},{y, -1, 1},
DisplayFunction -> Identity]

paraboleX[a_, b_][i_][y_]:=ParametricPlot3D[{i, y, -i^2/a^2 - y^2/b^2},{y, -1, 1, 0.05},
DisplayFunction -> Identity]

paraboleY[a_, b_][i_][x_]:=ParametricPlot3D[{x, i, -x^2/a^2 - i^2/b^2},{x, -1, 1, 0.05},
DisplayFunction -> Identity]

paraboleX1[a_, b_][i_][y_]:=ParametricPlot3D[{i, y, -i^2/a^2 - y^2/b^2},{y, 0, 1, 0.05},
DisplayFunction -> Identity]

paraboleY1[a_, b_][i_][x_]:=ParametricPlot3D[{x, i, -x^2/a^2 - i^2/b^2},{x, 0, 1, 0.05},
DisplayFunction -> Identity]

paraboleX2[a_, b_][i_][y_]:=ParametricPlot3D[{i, y, -i^2/a^2 - y^2/b^2},{y, -1, 0, 0.05},
DisplayFunction -> Identity]

paraboleY2[a_, b_][i_][x_]:=ParametricPlot3D[{x, i, -x^2/a^2 - i^2/b^2},{x, -1, 0, 0.05},
DisplayFunction -> Identity]

Solve[x - ((4 (x^2))/9) == 0, x]

NSolve[y == 2  y^2, y]

Solve[y*Sqrt[2] == 2  y^2 (4/9 + 1), y]

(9/(13 (Sqrt[2])))*Sqrt[2]

N[%]

Simplify[-x^2/a^2 - y^2/b^2 /. {a -> 3/2, 
          b -> 1}]

Simplify[-((4 (x^2))/9) - (y^2) /. {x -> t*Cos[u], 
          y -> t*Sin[u]}]

anim 2

normalni[u_] := 
      ParametricPlot3D[{t*Cos[u], t*Sin[u], 
          (1/18) (t^2) (-13 + 5 Cos[2 u]), {Hue[1], 
            AbsoluteThickness[2]}}, {t, -1.5, 1.5, 0.1}, 
        DisplayFunction -> Identity]

definicije gaussove isrednje 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

F[a_, b_][x_, y_]:={x, y, -x^2/a^2 - y^2/b^2}

gcurvature[F[3/2, 1]][0, 0]

meancurvature[F[3/2, 1]][0, 0]

Solve[{x1*x2 == (16/9), 1/2 (x1 + x2) == -(13/9)}, {x1, x2}] // 
      Simplify

zakrivljenost[x_] := 
      Cos[x]^2*(-2) + Sin[x]^2*(-(8/9))

sl 6

hipparX[i_][y_]:=ParametricPlot3D[{i, y, i^2 - y^2},{y, -1, 1, 0.05},
DisplayFunction -> Identity]

hipparY[i_][x_]:=ParametricPlot3D[{x, i, x^2 - i^2},{x, -1, 1, 0.05},
DisplayFunction -> Identity]

sl 7,8,9,10

normalniElPar[u_] := 
      ParametricPlot3D[{t*Cos[u], t*Sin[u], 
          (1/18) (t^2) (-13 + 5 Cos[2 u])}, {t, -1.5, 1.5, 
          0.1}, DisplayFunction -> Identity]

normalniHipPar[u_] := 
      ParametricPlot3D[{t*Cos[u], t*Sin[u], t^2*Cos[2  u]}, {t, -2, 2, 
          0.3}, DisplayFunction -> Identity]

normalniValjak[v_]:=ParametricPlot3D[{Cos[u], Tan[v]*Cos[u], Sin[u]},{u, 0, Pi},DisplayFunction -> Identity]

sl 11

kugla1[r_, a_, b_, c_] := 
      ParametricPlot3D[{r*Cos[u] Sin[v] + a, r*Sin[u] Sin[v] + b, 
          r*Cos[v] + c, FaceForm[Black]}, {u, 0, Pi, Pi/6}, {v, 0, 2  Pi, 
          Pi/6}, 
DisplayFunction -> 
          Identity]

torus[a_, b_][u_, 
        v_] := {(a + b*Cos[v])*Cos[u], (a + b*Cos[v])*Sin[u], 
        b*Sin[v]}

tocka[a_, b_][u0_, v0_] := torus[a, b][u0, v0]

TangRav[a_, b_][u0_, v0_] := tocka[a, b][u0, v0] + 
        t1*(D[torus[a, b][u, v], u] /. {u -> u0, v -> v0}) + 
        t2*(D[torus[a, b][u, v], v] /. {u -> u0, 
                v -> v0})

normala[a_, b_][u0_, 
        v0_] := tocka[a, b][u0, v0] + 
          t*Cross[D[torus[a, b][u, v], u], 
              D[torus[a, b][u, v], v]] /. {u -> u0, 
          v -> v0}

dirna ravnina u regularnoj tocki

\(Table[
      Show[
kugla[0.04, 0, 0, 0], 
        Paraboloid[2/3] // 
          Evaluate, 
ParametricPlot3D[{x, y, 0, 
            Hue[0.14, 0.5, 1]}, {x, \(-0.5\), 0.5, 1}, {y, \(-0.5\), 0.5, 1}, 
          DisplayFunction -> 
            Identity], 

\(krivulja1[
              2/3]\)[u] // Evaluate, 
tangenta[u] // 
          Evaluate, 


Graphics3D[
          Text[FontForm["ϕ", {"Symbol-Bold", 
                16}], {1, \(-1\), \(-1\)}]], 
Graphics3D[
          Text[FontForm["T", {"Ariel-Bold", 14}], {0, 0, 
              0.125}]], 
Graphics3D[
          Text[FontForm["τ", {"Symbol-Bold", 
                16}], {0.5, \(-0.5\), 
              0.1}]], 

PlotRange -> {{\(-1. \), 1. }, {\(-1. \), 1. }, {\(-1.5\), 0.25}}, 
        DisplayFunction -> $DisplayFunction, 
        Lighting -> False, 
Axes -> None, 
        Boxed -> False, Background -> GrayLevel[0.8], 
        ImageSize -> {300, 300}], {u, \(-Pi\)/2, Pi/2 - Pi/36, 
        Pi/36}]\)

singularne tocke

Show[
      kugla[0.03, 0, 0, 0], RotLemniscate[1. ][t, u] // 
        Evaluate, 
TangStozac[u, z] // 
        Evaluate, 

PlotRange -> {{-0.45, 0.45}, {-0.45, 0.45}, {-1, 1}}, 
      DisplayFunction -> $DisplayFunction, 
      ViewPoint -> {1.5, 2, 1.5}, 
      Lighting -> False, 
Axes -> None, 
      Boxed -> False, Background -> GrayLevel[0.8], 
      ImageSize -> {150, 180}]

Show[
ParametricPlot3D[{Cos[u]*Cos[2  u]*Sin[v], 
          Sin[u]*Cos[2  u]*Sin[v], Cos[2  u]*Cos[v] - Cos[2  u], 
          Hue[0.66, 0.3, 0.9]}, 
{u, -Pi, Pi, 
          Pi/36}, {v, -Pi, Pi, 
          Pi/24}, 
DisplayFunction -> 
          Identity], 

ParametricPlot3D[{{x, x, z, 
            Hue[0.14, 0.5, 1]}, {x, -x, z, Hue[0.14, 0.5, 1]}, {x, z, 0, 
            Hue[0.14, 0.5, 1]}}, {x, -0.25, 0.25, 0.5}, {z, -0.25, 
          0.25, 0.5}, 
        DisplayFunction -> 
          Identity], 

ParametricPlot3D[{{x, 1.4  x, z, 
            Hue[0.14, 0.5, 1]}, {x, -1.4 x, z, 
            Hue[0.14, 0.5, 1]}}, {x, -0.25, 0.25, 0.5}, {z, 0.75, 1.25, 
          0.5}, DisplayFunction -> 
          Identity], 

Graphics3D[{Line[{{0, 0, -1.5}, {0, 
                0, 1.5}}], 
          Line[{{-0.25, -0.25, 0}, {0.25, 0.25, 0}}], 
          Line[{{-0.25, 0.25, 0}, {0.25, -0.25, 
                0}}]}], 


kugla[0.05, 0, 0, 0], 
kugla[0.05, 0, 0, 
        1], 

PlotRange -> {{-1.4, 1.4}, {-1, 1}, {-2, 
            2}}, 

DisplayFunction -> $DisplayFunction, 
      ViewPoint -> {-2.8, 1.5, 
          1.3}, 
Lighting -> 
        False, 
Axes -> None, Boxed -> False, 
      Background -> GrayLevel[0.8], 
      ImageSize -> {150, 180}]

radijus zakrivljenosti

ParametricPlot[elipsa[3, 2][t] // Evaluate, {t, 0, 
        2  Pi}]

1/fleksija[elipsa1[3, 2]][Pi/4]

Show[
      ParametricPlot[elipsa[3, 2][t] // Evaluate, {t, -Pi/3, 
          2  Pi/3}, 
        DisplayFunction -> 
          Identity], 
ParametricPlot[kruznica[
              (5/(6 (Sqrt[2]))), -(5/(4 (Sqrt[2]))), ((13 (Sqrt[(13/2)]))/12)][t] // Evaluate, {t, 0, 2  Pi}, DisplayFunction -> Identity, 
        PlotStyle -> {Hue[
              1]}], 

ImplicitPlot[
        2  x + 3  y - 6*Sqrt[2] == 0, {x, 0, 3}, 
        DisplayFunction -> Identity], 
ImplicitPlot[
        3/2  x - y - 5/4  Sqrt[2] == 0, {x, 0, 3}, 
        DisplayFunction -> 
          Identity], 


Graphics[{AbsolutePointSize[7], 
          Point[{3*Sqrt[2]/2, 
              Sqrt[2]}]}], 
DisplayFunction -> $DisplayFunction, AspectRatio -> Automatic]

Solve[{3/2  x - y - 5/4  Sqrt[2] == 
            0, (3*Sqrt[2]/2 - x)^2 + (Sqrt[2] - 
                    y)^2 == (((13 (Sqrt[(13/2)]))/12))^2}, {x, 
          y}] // Simplify

Solve[3/2  x - y - 5/4  Sqrt[2] == 0 /. x -> 3 , y]

\(Show[
      ParametricPlot[\(elipsa[3, 2]\)[t] // Evaluate, {t, \(-Pi\)/3, 
          2  Pi/3}, DisplayFunction -> Identity, 
        PlotStyle -> {AbsoluteThickness[
              3]}], 
ParametricPlot[\(kruznica[
              5\/\(6\ \@2\), \(-\(5\/\(4\ \@2\)\)\), \(13\ \@\(13\/2\)\)\/12]\)[t] // Evaluate, {t, 0, 2  Pi}, DisplayFunction -> Identity, 
        PlotStyle -> {AbsoluteThickness[1], 
            Hue[1]}], 

ImplicitPlot[
        2  x + 3  y - 6*Sqrt[2] == 0, {x, 1.2, 3}, 
        DisplayFunction -> Identity], 
ImplicitPlot[
        3/2  x - y - 5/4  Sqrt[2] == 0, {x, 5\/\(6\ \@2\), 3\ \@2/2}, 
        PlotStyle -> {AbsoluteThickness[3], Hue[1]}, 
        DisplayFunction -> 
          Identity], 




Graphics[{AbsoluteThickness[2], Hue[0.7], 
          Arrow[{3\ \@2/2, \@2}, {3, 
              9\/2 - 5\/\(2\ \@2\)}]}], 

Graphics[{AbsolutePointSize[12], 
          Point[{3*Sqrt[2]/2, 
              Sqrt[2]}]}], 
Graphics[{AbsolutePointSize[7], 
          Hue[1], Point[{3*Sqrt[2]/2, 
              Sqrt[2]}]}], 
Graphics[{AbsolutePointSize[7], 
          Point[{5\/\(6\ \@2\), \(-\(5\/\(4\ \@2\)\)\)}]}], 
Graphics[{Text[
            FontForm["T", {"Arial-Bold", 16}], {2.45, 1.45}, {\(-1\), 
              0}]}], 
Graphics[{Text[
            FontForm["R", {"Arial-Bold", 16}], {1.3, 0}, {\(-1\), 
              0}]}], 
Graphics[{Text[
            FontForm[\*""k=-(1/R)"", {"Arial-Bold", 
                16}], {\(-3.5\), 1.5}, {\(-1\), 
              0}]}], 
Graphics[{Text[
            FontForm[\*""(Overscript[n,  ->]) "", {"Arial-Bold", 
                16}], {2.15, 2.5}, {\(-1\), 
              0}]}], 

PlotRange -> {{\(-4\), 4}, {\(-3.7\), 
            2.5}}, 

DisplayFunction -> $DisplayFunction, AspectRatio -> Automatic, Axes -> None, 
      Background -> GrayLevel[0.8], 
      ImageSize -> {300, 300}]\)

normalne zakrivljenosti slika

Show[kugla[0.02, 0, 0, 0],
Table[paraboleX1[1.5, 1][i][
              y], {i, -1, 1, 0.05}] // Evaluate,
Table[paraboleX2[1.5, 1][i][
              y], {i, -1, 0, 0.05}] // Evaluate,
Table[paraboleY2[1.5, 1][i][
              x], {i, -1, 1, 0.05}] // Evaluate,
Table[paraboleY1[1.5, 1][i][x], {i, 
              0, 1, 0.05}] // Evaluate,

ParametricPlot3D[{x, y, 0, 
            Hue[0.14, 0.5, 1]}, {x, -0.25, 0.25, 0.5}, {y, -0.25, 
            0.25, 0.5}, DisplayFunction -> Identity],

ParametricPlot3D[{x, y, -x^2/1.5^2 - y^2, 
                FaceForm[Hue[0.5], Hue[0.7]]},{x, -0.5, 0.5, 0.025},{y, -0.5, -0.475, 0.025},
DisplayFunction -> Identity],
ParametricPlot3D[{x, y, -x^2/1.5^2 - y^2, 
                FaceForm[Hue[0.5], Hue[0.7]]},{x, -0.5, 0.5, 0.025},{y, 0.475, 0.5, 0.025},
DisplayFunction -> Identity],
ParametricPlot3D[{x, y, -x^2/1.5^2 - y^2, 
                FaceForm[Hue[0.5], Hue[0.7]]},{x, -0.5, -0.475, 0.025},{y, -0.5, 0.5, 0.025},
DisplayFunction -> Identity],
ParametricPlot3D[{x, y, -x^2/1.5^2 - y^2, 
                FaceForm[Hue[0.5], Hue[0.7]]},{x, 0.475, 0.5, 0.025},{y, -0.5, 0.5, 0.025},
DisplayFunction -> Identity],



ParametricPlot3D[{0, 
            y, -y^2, {AbsoluteThickness[3]}}, {y, -1, 1, 0.05}, 
          DisplayFunction -> Identity],

ParametricPlot3D[{x, -(Sqrt[x - ((4 (x^2))/9)]), -x, {AbsoluteThickness[2]}}, {x, 0, 1, 0.025}, 
          DisplayFunction -> Identity],

ParametricPlot3D[{x, (Sqrt[x - ((4 (x^2))/9)]), -x, {AbsoluteThickness[2]}}, {x, 0, 1, 0.025}, 
          DisplayFunction -> Identity],

Graphics3D[{Hue[1], AbsoluteThickness[2], 
            Line[{{0, 0, 0}, {0, 0, -0.5}}], 
            Line[{{0, 0, 0}, {1/4, 0, -1/4}}]}],
Graphics3D[{Hue[1], AbsoluteThickness[1], 
            AbsoluteDashing[{2, 5}], 
            Line[{{1/4, 0, -1/4}, {0, 0, -0.5}}]}],

ParametricPlot3D[{0, 0.5  Cos[u], 
            0.5  Sin[u] - 1/2, {Hue[0.7]}}, {u, 0, 2  Pi}, 
          DisplayFunction -> Identity],
ParametricPlot3D[{0.25  Cos[u] + 0.25, 
            0.25  Sin[u], -0.25 Cos[u] - 0.25, {Hue[0.7]}}, {u, 0, 
            2  Pi}, DisplayFunction -> Identity],
Graphics3D[{Hue[0.7], AbsoluteThickness[2], 
            Line[{{0, 0, 0}, {0, 0, 0.32}}]}],
Graphics3D[{AbsoluteThickness[2], 
            Line[{{0, -0.25, 0}, {0, 0.25, 0}}]}],
ParametricPlot3D[{0, y, z, 
            White}, {y, -0.2, 0.2, 0.4}, {z, 0, 0.25, 0.25}, 
          DisplayFunction -> Identity],

kugla[0.01, 0, 0, -0.5],
kugla[0.01, 0.25, 0, -0.25],

ParametricPlot3D[{0.3  Cos[u], 0, 
            0.3  Sin[u]}, {u, -Pi/2, -Pi/4}, 
          DisplayFunction -> Identity],
Graphics3D[Point[{0.22, 0, -0.25}]],
Graphics3D[{Black, 
            Polygon[{{0, 0, 0.35}, {-0.01, -0.01, 0.32}, {0.01, 0.01, 
                  0.32}}]}],

\(Graphics3D[
          Text[FontForm[\*""\!\(R\_t\)"", {"Ariel-Bold", 
                14}], {\(-0.06\), 0, \(-0.4\)}]]\),
\(Graphics3D[
          Text[FontForm[\*""(Overscript[n, ⇀])"", {"Ariel-Bold", 14}], {0.05, 0, 0.3}]]\),
Graphics3D[
          Text[FontForm["t", {"Ariel-Bold", 
                15}], {-0.1, -0.15, 0.01}]],
\(Graphics3D[
          Text[FontForm[\*""(Subscript[r, t])(ϑ)"", {"Ariel-Bold", 15}], {0.3, 0, \(-0.13\)}]]\),
Graphics3D[
          Text[FontForm["ϑ", {"Symbol-Bold", 15}], {0.07, 
              0, -0.2}]],

PlotRange -> {{-0.5, 
              0.5}, {-0.5, 0.5}, {-1, 0.4}},DisplayFunction -> $DisplayFunction,
Boxed -> False,Axes -> None,Lighting -> False,Background -> GrayLevel[0.8],ViewPoint -> {1. , -1.4, 0.8},ImageSize -> {300, 300}]

normalne zakrivljenosti animacije

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

Graphics[{Hue[1], 
          AbsoluteThickness[2], AbsoluteDashing[{2, 5}], 
          Line[{{0, 0}, {0, \(-2\)}}], Line[{{Pi, 0}, {Pi, \(-2\)}}], 
          Line[{{Pi/2, 
                0}, {Pi/2, \(-8\)/
                  9}}]}], 

Graphics[{Hue[
            1], AbsolutePointSize[6], Point[{0, 0}], Point[{Pi, 0}], 
          Point[{Pi/2, 
              0}]}], 


Graphics[{AbsolutePointSize[6], Point[{0, \(-2\)}], Point[{Pi, \(-2\)}], 
          Point[{Pi/2, \(-8\)/
                9}]}], 

Graphics[{{Arrow[{0, 1.3}, {0, 1.4}, HeadWidth -> 0.25, 
              HeadLength -> 0.04]}, 
{Arrow[{3.7, 
                0}, {3.8, 0}, HeadWidth -> 0.25, 
              HeadLength -> 
                0.04]}}], 

Graphics[{Text[
            FontForm[\*""normalna zakrivljenost    (K=(1/R))"", {"Arial-Bold", 14}], {0.1, 1.2}, {\(-1\), 
              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, \(-1\)}, {\(-1\), 
              0}]}], 

Graphics[{Hue[1], 
          AbsolutePointSize[8], 
          Point[{2.2, 
              0.8}]}], 

Graphics[{Text[
            FontForm["-  glavni smjerovi", {"Arial", 12}], {2.3, 
              0.8}, {\(-1\), 
              0}]}], 

Background -> 
        GrayLevel[
          0.8], 
Ticks -> {{0, Pi/4, 1, Pi/2, 3  Pi/4, 
            Pi}, {\(-2\), \(-8\)/9, 0, 1/2}}, AspectRatio -> 1, 
      PlotRange -> {{\(-0.5\), 3.8}, {\(-2.2\), 
            
            1.4}}, 
DisplayFunction -> $DisplayFunction, ImageSize -> {400, 400}]\)

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

Graphics[{AbsoluteThickness[2], 
            AbsoluteDashing[{2, 5}], Line[{{0, 0}, {0, \(-2\)}}], 
            Line[{{Pi, 0}, {Pi, \(-2\)}}], 
            Line[{{Pi/2, 
                  0}, {Pi/2, \(-8\)/
                    9}}]}], 

Graphics[{Hue[1], AbsolutePointSize[6], 
            Point[{0, 0}], Point[{Pi, 0}], 
            Point[{Pi/2, 
                0}]}], 


Graphics[{AbsolutePointSize[6], Point[{0, \(-2\)}], 
            Point[{Pi, \(-2\)}], 
            Point[{Pi/2, \(-8\)/
                  9}]}], 

Graphics[{{Arrow[{0, 1.3}, {0, 1.4}, HeadWidth -> 0.25, 
                HeadLength -> 0.04]}, 
{Arrow[{3.7, 
                  0}, {3.8, 0}, HeadWidth -> 0.25, 
                HeadLength -> 
                  0.04]}}], 

Graphics[{Text[
              FontForm[\*""normalna zakrivljenost    (K=-(1/R))"", {"Arial-Bold", 14}], {0.1, 1.2}, {\(-1\), 
                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, \(-1\)}, {\(-1\), 
                0}]}], 

Graphics[{Hue[
              1], AbsolutePointSize[8], 
            Point[{2.2, 
                0.8}]}], 

Graphics[{Hue[
              0.7], AbsolutePointSize[8], 
            Point[{u, 
                zakrivljenost[u] // 
                  Evaluate}]}], 

Graphics[{Hue[1], 
            Line[{{u, 0}, {u, 
                  zakrivljenost[u] // 
                    Evaluate}}]}], 

Graphics[{Text[
              FontForm["-  glavni smjerovi", {"Arial", 12}], {2.3, 
                0.8}, {\(-1\), 
                0}]}], 

Background -> GrayLevel[
            0.8], 
Ticks -> {{0, Pi/4, 1, Pi/2, 
              3  Pi/4, Pi}, {\(-2\), \(-8\)/9, 0, 1/2}}, 
        AspectRatio -> 1, 
        PlotRange -> {{\(-0.5\), 3.8}, {\(-2.2\), 
              1.4}}, 
DisplayFunction -> $DisplayFunction, ImageSize -> {400, 400}], {u, 0, Pi, 
        Pi/36}]\)

Table[

Show[kugla[0.04, 0, 0, 0],
Table[paraboleX1[1.5, 1][i][
                  y], {i, -1, 1, 0.1}] // Evaluate,
Table[paraboleX2[1.5, 1][i][
                  y], {i, -1, 0, 0.1}] // Evaluate,
Table[paraboleY2[1.5, 1][i][
                  x], {i, -1, 1, 0.1}] // Evaluate,
Table[paraboleY1[1.5, 1][i][
                  x], {i, 0, 1, 0.1}] // Evaluate,

ParametricPlot3D[{x, y, 0, 
                Hue[0.14, 0.5, 1]}, {x, -0.35, 0.4, 0.75}, {y, -0.35, 
                0.35, 0.7}, DisplayFunction -> Identity],

ParametricPlot3D[{x, y, -x^2/1.5^2 - y^2, 
                    FaceForm[Hue[0.5], Hue[0.7]]},{x, -1, 1, 0.05},{y, -1, -0.95, 0.05},
DisplayFunction -> Identity],
ParametricPlot3D[{x, y, -x^2/1.5^2 - y^2, 
                    FaceForm[Hue[0.5], Hue[0.7]]},{x, -1, 1},{y, 0.95, 1, 0.05},
DisplayFunction -> Identity],
ParametricPlot3D[{x, y, -x^2/1.5^2 - y^2, 
                    FaceForm[Hue[0.5], Hue[0.7]]},{x, -1, -0.95, 0.05},{y, -1, 1, 0.05},
DisplayFunction -> Identity],
ParametricPlot3D[{x, y, -x^2/1.5^2 - y^2, 
                    FaceForm[Hue[0.5], Hue[0.7]]},{x, 0.95, 1, 0.05},{y, -1, 1, 0.05},
DisplayFunction -> Identity],

normalni[u] // Evaluate,

ParametricPlot3D[{t*Cos[u], t*Sin[u], z, 
                White}, {t, -0.35, 0.35, 0.7}, {z, -1.5, 0.5, 2}, 
              DisplayFunction -> Identity],
ParametricPlot3D[{t*Cos[u], t*Sin[u], 
                0}, {t, -0.35, 0.35, 0.35}, 
              DisplayFunction -> Identity],

Graphics3D[{Hue[1], 
                AbsoluteThickness[3], 
                Line[{{0, 0, 0}, {0, 0, 
                      1/zakrivljenost[u + Pi/2] // Evaluate}}]}],

Graphics3D[{Black, 
                Polygon[{{0, 0, 0.75}, {-0.02, -0.02, 0.67}, {0.02, 
                      0.02, 0.67}}]}],
Graphics3D[{Hue[0.7], 
                AbsoluteThickness[2], Line[{{0, 0, 0}, {0, 0, 0.67}}]}],

kugla[0.04, 0, 0, 
              1/zakrivljenost[u + Pi/2] // Evaluate],

Graphics3D[
              Text[FontForm["R", {"Ariel-Bold", 14}], {0. , 
                  0, -0.3}]],
\(Graphics3D[
              Text[FontForm[\*""(Overscript[n, ⇀])"", {"Ariel-Bold", 14}], {0.2, 0, 0.7}]]\),


PlotRange -> {{-1. , 
                  1. }, {-1. , 1. }, {-1.5, 0.75}},DisplayFunction -> $DisplayFunction,
AxesLabel -> {x, y, z},Lighting -> False,Boxed -> False,Axes -> None,ImageSize -> {400, 400},
AxesLabel -> {x, y, z},Background -> GrayLevel[0.8],ViewPoint -> {2, -1.5, 0.75}],{u, Pi/4, 5  Pi/4, Pi/36}]

Table[Show[GraphicsArray[{Show[kugla[0.04, 0, 0, 0],
Table[paraboleX1[1.5, 
                                  1][i][y], {i, -1, 1, 0.1}] // 
                          Evaluate,
Table[paraboleX2[1.5, 
                                  1][i][y], {i, -1, 0, 0.1}] // 
                          Evaluate,
Table[paraboleY2[1.5, 
                                  1][i][x], {i, -1, 1, 0.1}] // 
                          Evaluate,
Table[paraboleY1[1.5, 
                                  1][i][x], {i, 0, 1, 0.1}] // Evaluate,

ParametricPlot3D[{x, y, 0, 
                            Hue[0.14, 0.5, 1]}, {x, -0.35, 0.4, 
                            0.75}, {y, -0.35, 0.35, 0.7}, 
                          DisplayFunction -> Identity],

ParametricPlot3D[{x, y, -x^2/1.5^2 - y^2, 
                                FaceForm[Hue[0.5], Hue[0.7]]},{x, -1, 1, 0.05},{y, -1, -0.95, 0.05},
DisplayFunction -> 
                              Identity],
ParametricPlot3D[{x, y, -x^2/1.5^2 - y^2, 
                                FaceForm[Hue[0.5], Hue[0.7]]},{x, -1, 1},{y, 0.95, 1, 0.05},
DisplayFunction -> 
                              Identity],
ParametricPlot3D[{x, y, -x^2/1.5^2 - y^2, 
                                FaceForm[Hue[0.5], Hue[0.7]]},{x, -1, -0.95, 0.05},{y, -1, 1, 0.05},
DisplayFunction -> 
                              Identity],
ParametricPlot3D[{x, y, -x^2/1.5^2 - y^2, 
                                FaceForm[Hue[0.5], Hue[0.7]]},{x, 0.95, 1, 0.05},{y, -1, 1, 0.05},
DisplayFunction -> 
                              Identity],

normalni[u] // Evaluate,

ParametricPlot3D[{t*Cos[u], 
                            t*Sin[u], z, White}, {t, -0.35, 0.35, 
                            0.7}, {z, -1.5, 0.5, 2}, 
                          DisplayFunction -> Identity],
ParametricPlot3D[{t*Cos[u], 
                            t*Sin[u], 0}, {t, -0.35, 0.35, 0.35}, 
                          DisplayFunction -> Identity],
ParametricPlot3D[{0, t, 
                            0}, {t, -0.35, 0, 0.35}, 
                          DisplayFunction -> Identity],
\(If[u == Pi/2, 
                          Graphics3D[
                            Text[FontForm[\*""(Overscript[n, ⇀])"", {"Ariel-Bold", 14}], {0.2, 0, 0.7}]], 
                          ParametricPlot3D[{0.3*Cos[v], 0.3*Sin[v], 
                              0}, {v, \(-Pi\)/2, u - Pi, Pi/36}, 
                            DisplayFunction -> Identity]]\),

Graphics3D[{Hue[1], 
                            AbsoluteThickness[3], 
                            Line[{{0, 0, 0}, {0, 0, 
                                  1/zakrivljenost[u + Pi/2] // 
                                    Evaluate}}]}],

Graphics3D[{Black, 
                            Polygon[{{0, 0, 0.75}, {-0.02, -0.02, 
                                  0.67}, {0.02, 0.02, 0.67}}]}],
Graphics3D[{Hue[0.7], 
                            AbsoluteThickness[2], 
                            Line[{{0, 0, 0}, {0, 0, 0.67}}]}],

kugla[0.03, 0, 0, 
                          1/zakrivljenost[u + Pi/2] // Evaluate],

\(Graphics3D[
                          Text[FontForm["R(φ)", {"Ariel-Bold", 14}], {\(-0.3\), 0, \(-0.4\)}]]\),
\(Graphics3D[
                          Text[FontForm[\*""(Overscript[n, ⇀])"", {"Ariel-Bold", 14}], {0.2, 0, 0.7}]]\),


PlotRange -> {{-1. , 1. }, {-1. , 1. }, {-1.5, 0.75}},
AxesLabel -> {x, y, 
                            z},Lighting -> False,Boxed -> False,Axes -> None,
AxesLabel -> {x, 
                            y, z},Background -> GrayLevel[0.8],ViewPoint -> {2, -1.5, 0.75}],



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

Graphics[{AbsoluteThickness[2], 
                          AbsoluteDashing[{2, 5}], 
                          Line[{{Pi, 0}, {Pi, \(-2\)}}], 
                          Line[{{Pi/2, 
                                0}, {Pi/2, \(-8\)/
                                  9}}]}], 

Graphics[{Hue[1], AbsolutePointSize[6], Point[{0, 0}], 
                          Point[{Pi, 0}], 
                          Point[{Pi/2, 
                              0}]}], 


Graphics[{AbsolutePointSize[6], Point[{0, \(-2\)}], 
                          Point[{Pi, \(-2\)}], 
                          Point[{Pi/2, \(-8\)/
                                9}]}], 

Graphics[{{Arrow[{0, 1.3}, {0, 1.4}, 
                              HeadWidth -> 0.25, 
                              HeadLength -> 
                                0.04]}, 
{Arrow[{3.7, 
                                0}, {3.8, 0}, HeadWidth -> 0.25, 
                              HeadLength -> 
                                0.04]}}], 

Graphics[{Text[
                            FontForm[\*""normalna zakrivljenost    K(φ)=-(1/(R (φ)))"", {"Arial-Bold", 
                                14}], {0.1, 1.2}, {\(-1\), 
                              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, \(-1\)}, {\(-1\), 
                              0}]}], 
Graphics[{Text[
                            FontForm["K(φ)", {"Arial-Bold", 
                                12}], {\(-0.5\), 1.2}, {\(-1\), 
                              0}]}], 



Graphics[{Hue[0.7], 
                          AbsolutePointSize[8], 
                          Point[{u - Pi/2, 
                              zakrivljenost[u - Pi/2] // 
                                Evaluate}]}], 

Graphics[{AbsolutePointSize[6], 
                          Point[{0, 
                              zakrivljenost[u - Pi/2] // 
                                Evaluate}]}], 

Graphics[{Hue[1], AbsoluteDashing[{2, 5}], 
                          Line[{{u - Pi/2, 0}, {u - Pi/2, 
                                zakrivljenost[u - Pi/2] // 
                                  Evaluate}}]}], 
Graphics[{AbsoluteThickness[2], 
                          Line[{{0, \(-0.05\)}, {0, 
                                zakrivljenost[u - Pi/2] // 
                                  Evaluate}}]}], 
Graphics[{AbsoluteDashing[{2, 5}], 
                          Line[{{0, 
                                zakrivljenost[u - Pi/2] // Evaluate}, {u - 
                                  Pi/2, 
                                zakrivljenost[u - Pi/2] // 
                                  Evaluate}}]}], 

Graphics[{Hue[1], AbsolutePointSize[8], 
                          Point[{1, \(-2\)}]}], 
Graphics[{Text[
                            FontForm["-  glavni smjerovi", {"Arial", 
                                12}], {1.1, \(-2\)}, {\(-1\), 
                              0}]}], 

Background -> 
                        GrayLevel[
                          0.8], 
Ticks -> {{0, Pi/4, 
                            1, Pi/2, 3  Pi/4, Pi}, {0}}, 
                      AspectRatio -> 1, 
                      PlotRange -> {{\(-0.5\), 3.8}, {\(-2.2\), 
                            1.4}}]\)},GraphicsSpacing -> 0],Background -> GrayLevel[0.8],DisplayFunction -> $DisplayFunction,ImageSize -> {600, 300}],
{u, Pi/2, 3  Pi/2, Pi/36}]

glavne zakrivljenosti

Show[kugla[0.025, 0, 0, 0],
Table[paraboleX[1.5, 1][i][
              y], {i, -1, 1, 0.05}] // Evaluate,
Table[paraboleY[1.5, 1][i][
              x], {i, -1, 1, 0.05}] // Evaluate,

ParametricPlot3D[{x, y, 0, 
            Hue[0.14, 0.5, 1]}, {x, -0.35, 0.35, 0.7}, {y, -0.35, 
            0.35, 0.7}, DisplayFunction -> Identity],

ParametricPlot3D[{x, y, -x^2/1.5^2 - y^2, 
                FaceForm[Hue[0.5], Hue[0.7]]},{x, -0.5, 0.5, 0.025},{y, -0.5, -0.475, 0.025},
DisplayFunction -> Identity],
ParametricPlot3D[{x, y, -x^2/1.5^2 - y^2, 
                FaceForm[Hue[0.5], Hue[0.7]]},{x, -0.5, 0.5, 0.025},{y, 0.475, 0.5, 0.025},
DisplayFunction -> Identity],
ParametricPlot3D[{x, y, -x^2/1.5^2 - y^2, 
                FaceForm[Hue[0.5], Hue[0.7]]},{x, -0.5, -0.475, 0.025},{y, -0.5, 0.5, 0.025},
DisplayFunction -> Identity],
ParametricPlot3D[{x, y, -x^2/1.5^2 - y^2, 
                FaceForm[Hue[0.5], Hue[0.7]]},{x, 0.475, 0.5, 0.025},{y, -0.5, 0.5, 0.025},
DisplayFunction -> Identity],

Graphics3D[{Black, 
            Polygon[{{0, 0, 1}, {-0.01, -0.02, 0.94}, {0.02, 0.02, 
                  0.94}}]}],


Graphics3D[{Blue, AbsoluteThickness[2], 
            Line[{{0, 0, -0.5}, {0, 0, 0.32}}]}],

normalni[Pi/2] // Evaluate,
normalni[0] // Evaluate,
ParametricPlot3D[{-(t/(Sqrt[2])), 
            (t/(Sqrt[2])), -((13 (t^2))/18), {AbsoluteThickness[
                2]}}, {t, -1.5, 1.5, 0.1}, 
          DisplayFunction -> Identity],

ParametricPlot3D[{t, 0, z, 
            White}, {t, -0.35, 0.35, 0.35}, {z, -0.2, 0.2, 0.2}, 
          DisplayFunction -> Identity],

ParametricPlot3D[{0, t, z, 
            White}, {t, -0.35, 0.35, 0.35}, {z, -0.2, 0.2, 0.2}, 
          DisplayFunction -> Identity],
ParametricPlot3D[{t*Cos[3  Pi/4], 
            t*Sin[3  Pi/4], z, White}, {t, -0.35, 0.35, 
            0.35}, {z, -0.2, 0.2, 0.2}, 
          DisplayFunction -> Identity],

Graphics3D[{Hue[1], AbsoluteThickness[3], 
            Line[{{-0.4, 0, 0}, {0.4, 0, 0}}]}],
Graphics3D[{Hue[1], AbsoluteThickness[3], 
            Line[{{0, -0.4, 0}, {0, 0.4, 0}}]}],
Graphics3D[{AbsoluteThickness[3], 
            Line[{{-0.4, 0.4, 0}, {0.4, -0.4, 0}}]}],

ParametricPlot3D[{0.3  Cos[u], 0.3  Sin[u], 
            0}, {u, -Pi/2, -Pi/4}, 
          DisplayFunction -> Identity],

Graphics3D[{Black, 
            Polygon[{{0, 0, 0.35}, {-0.01, -0.01, 0.32}, {0.01, 0.01, 
                  0.32}}]}],

\(Graphics3D[
          Text[FontForm[\*""(Overscript[n, ⇀])"", {"Ariel-Bold", 14}], {0.05, 0, 0.3}]]\),
\(Graphics3D[
          Text[FontForm[\*""\!\(p\_1\)"", {"Ariel-Bold", 
                15}], {\(-0.1\), \(-0.15\), 0.01}]]\),
\(Graphics3D[
          Text[FontForm[\*""\!\(p\_2\)"", {"Ariel-Bold", 
                15}], {0.25, 0.15, 0.01}]]\),
Graphics3D[
          Text[FontForm["t", {"Ariel-Bold", 15}], {0.35, -0.35, 
              0.04}]],
Graphics3D[
          Text[FontForm["φ", {"Symbol-Bold", 
                15}], {0.1, -0.17, 0.01}]],


PlotRange -> {{-0.5, 
              0.5}, {-0.5, 0.5}, {-0.4, 0.5}},DisplayFunction -> $DisplayFunction,
Boxed -> False,Axes -> None,Lighting -> False,
Background -> GrayLevel[0.8],ViewPoint -> {1, -1.5, 0.35},ImageSize -> {300, 300}]

asimptotski

Show[kugla[0.04, 0, 0, 0],
Table[hipparX[i][y], {i, -1, 1, 
              0.05}] // Evaluate,
Table[hipparY[i][x], {i, -1, 1, 
              0.05}] // Evaluate,

ParametricPlot3D[{x, y, 0, 
            Hue[0.14, 0.5, 1]}, {x, -1, 1, 2}, {y, -1, 1, 2}, 
          DisplayFunction -> Identity],


ParametricPlot3D[{x, y, x^2 - y^2, FaceForm[Hue[0.5], Hue[0.7]]},{x, -1, 1, 0.05},{y, -1, -0.95, 0.05},
DisplayFunction -> Identity],
ParametricPlot3D[{x, y, x^2 - y^2, FaceForm[Hue[0.5], Hue[0.7]]},{x, -1, 1, 0.05},{y, 0.95, 1, 0.05},
DisplayFunction -> Identity],
ParametricPlot3D[{x, y, x^2 - y^2, FaceForm[Hue[0.5], Hue[0.7]]},{x, -1, -0.95, 0.05},{y, -1, 1, 0.05},
DisplayFunction -> Identity],
ParametricPlot3D[{x, y, x^2 - y^2, FaceForm[Hue[0.5], Hue[0.7]]},{x, 0.95, 1, 0.05},{y, -1, 1, 0.05},
DisplayFunction -> Identity],

Graphics3D[{Black, 
            Polygon[{{0, 0, 1}, {-0.01, -0.02, 0.94}, {0.02, 0.02, 
                  0.94}}]}],


Graphics3D[{Blue, AbsoluteThickness[2], 
            Line[{{0, 0, -0.5}, {0, 0, 0.94}}]}],


ParametricPlot3D[{t*Cos[3  Pi/4], 
            t*Sin[3  Pi/4], z, White}, {t, -1.35, 1.35, 
            1.35}, {z, -0.25, 0.25, 0.5}, 
          DisplayFunction -> Identity],
ParametricPlot3D[{t*Cos[Pi/4], t*Sin[Pi/4], 
            z, White}, {t, -1.35, 1.35, 1.35}, {z, -0.25, 0.25, 0.5}, 
          DisplayFunction -> Identity],

Graphics3D[{Hue[1], AbsoluteThickness[3], 
            Line[{{-1, 1, 0}, {1, -1, 0}}], 
            Line[{{1, 1, 0}, {-1, -1, 0}}]}],


\(Graphics3D[
          Text[FontForm[\*""(Overscript[n, ⇀])"", {"Ariel-Bold", 14}], {0.1, 0, 0.9}]]\),
\(Graphics3D[
          Text[FontForm[\*""\!\(a\_1\)"", {"Ariel-Bold", 
                15}], {\(-0.8\), \(-0.8\), 0.1}]]\),
\(Graphics3D[
          Text[FontForm[\*""\!\(a\_2\)"", {"Ariel-Bold", 
                15}], {0.8, \(-0.8\), 0.1}]]\),


PlotRange -> {{-1, 1}, {-1, 
              1}, {-1, 1}},DisplayFunction -> $DisplayFunction,
Boxed -> False,Axes -> None,Lighting -> False,
Background -> GrayLevel[0.8],ViewPoint -> {0.7, -1.5, 0.35},ImageSize -> {300, 300}]

elipticke, hiperbolicke, parabolicke i planarne tocke

Show[kugla[0.035, 0, 0, 0],
Table[
              normalniElPar[u], {u, 0, Pi, Pi/25}] // Evaluate,


ParametricPlot3D[{x, y, 0, 
              Hue[0.14, 0.5, 1]}, {x, -0.35, 0.35, 0.7}, {y, -0.35, 
              0.35, 0.7}, DisplayFunction -> Identity],

ParametricPlot3D[{x, y, -x^2/1.5^2 - y^2, 
                  FaceForm[Hue[0.5], Hue[0.7]]},{x, -1, 1, 0.05},{y, -1, -0.95, 0.05},
DisplayFunction -> Identity],
ParametricPlot3D[{x, y, -x^2/1.5^2 - y^2, 
                  FaceForm[Hue[0.5], Hue[0.7]]},{x, -1, 1},{y, 0.95, 1, 0.05},
DisplayFunction -> Identity],
ParametricPlot3D[{x, y, -x^2/1.5^2 - y^2, 
                  FaceForm[Hue[0.5], Hue[0.7]]},{x, -1, -0.95, 0.05},{y, -1, 1, 0.05},
DisplayFunction -> Identity],
ParametricPlot3D[{x, y, -x^2/1.5^2 - y^2, 
                  FaceForm[Hue[0.5], Hue[0.7]]},{x, 0.95, 1, 0.05},{y, -1, 1, 0.05},
DisplayFunction -> Identity],

Graphics3D[{Black, 
              Polygon[{{0, 0, 0.7}, {-0.02, -0.02, 0.64}, {0.02, 
                    0.02, 0.64}}]}],


Graphics3D[{Blue, AbsoluteThickness[2], 
              Line[{{0, 0, 0}, {0, 0, 0.63}}]}],

normalni[Pi/2] // Evaluate,
ParametricPlot3D[{x, 
                  0, -x^2/1.5^2, {Hue[1], AbsoluteThickness[2]}},{x, -1, 1, 0.05},
DisplayFunction -> Identity],
PlotRange -> {{-1. , 
                1. }, {-1. , 1. }, {-1.5, 0.72}},DisplayFunction -> $DisplayFunction,
Boxed -> False,Axes -> None,Lighting -> False,Background -> GrayLevel[0.8],ViewPoint -> {0.7, -1.5, 0.75},ImageSize -> {300, 300}]

Show[kugla[0.04, 0, 0, 0],
Table[
              normalniHipPar[u], {u, 0, Pi, Pi/39}] // Evaluate,

ParametricPlot3D[{t, 0, 
              t^2 + 0.01, {Hue[1], AbsoluteThickness[2]}}, {t, -2, 2, 
              0.1}, DisplayFunction -> Identity],
ParametricPlot3D[{0, 
              t, -t^2, {Hue[1], AbsoluteThickness[2]}}, {t, -2, 2, 
              0.1}, DisplayFunction -> Identity],


ParametricPlot3D[{x, y, 0, 
              Hue[0.14, 0.5, 1]}, {x, -0.35, 0.35, 0.7}, {y, -0.35, 
              0.35, 0.7}, DisplayFunction -> Identity],

ParametricPlot3D[{x, y, x^2 - y^2, FaceForm[Hue[0.5], Hue[0.7]]},{x, -1, 1, 0.05},{y, -1, -0.97, 0.03},
DisplayFunction -> Identity],
ParametricPlot3D[{x, y, x^2 - y^2, FaceForm[Hue[0.5], Hue[0.7]]},{x, -1, 1, 0.05},{y, 0.97, 1, 0.03},
DisplayFunction -> Identity],
ParametricPlot3D[{x, y, x^2 - y^2, FaceForm[Hue[0.5], Hue[0.7]]},{x, -1, -0.97, 0.03},{y, -1, 1, 0.05},
DisplayFunction -> Identity],
ParametricPlot3D[{x, y, x^2 - y^2, FaceForm[Hue[0.5], Hue[0.7]]},{x, 0.97, 1, 0.03},{y, -1, 1, 0.05},
DisplayFunction -> Identity],

Graphics3D[{Black, 
              Polygon[{{0, 0, 1}, {-0.02, -0.02, 0.94}, {0.02, 0.02, 
                    0.94}}]}],


Graphics3D[{Blue, AbsoluteThickness[2], 
              Line[{{0, 0, 0}, {0, 0, 0.93}}]}],

PlotRange -> {{-1. , 
                1. }, {-1. , 1. }, {-1. , 1. }},DisplayFunction -> $DisplayFunction,
Lighting -> False,
Boxed -> False,Axes -> None,Background -> GrayLevel[0.8],ViewPoint -> {0.7, -1.5, 1. },ImageSize -> {300, 300}]

Show[kugla[0.035, 0, 0, 1],
Table[
              normalniValjak[v], {v, 0, Pi, Pi/35}] // Evaluate,



ParametricPlot3D[{x, y, 1, 
              Hue[0.14, 0.5, 1]}, {x, -0.35, 0.35, 0.7}, {y, -0.35, 
              0.35, 0.7}, DisplayFunction -> Identity],

ParametricPlot3D[{x, y, Sqrt[1 - x^2], 
                  FaceForm[Hue[0.5], Hue[0.7]]},{x, -1, 1, 0.05},{y, -1, -0.95, 0.05},
DisplayFunction -> Identity],
ParametricPlot3D[{x, y, Sqrt[1 - x^2], 
                  FaceForm[Hue[0.5], Hue[0.7]]},{x, -1, 1, 0.05},{y, 0.95, 1, 0.05},
DisplayFunction -> Identity],
ParametricPlot3D[{x, y, Sqrt[1 - x^2], 
                  FaceForm[Hue[0.5], Hue[0.7]]},{x, -1, -0.99, 0.01},{y, -1, 1, 0.05},
DisplayFunction -> Identity],
ParametricPlot3D[{x, y, Sqrt[1 - x^2], 
                  FaceForm[Hue[0.5], Hue[0.7]]},{x, 0.99, 1, 0.01},{y, -1, 1, 0.05},
DisplayFunction -> Identity],

ParametricPlot3D[{Cos[u], 0, 
              Sin[u], {Hue[1], AbsoluteThickness[2]}}, {u, 0, Pi}, 
            DisplayFunction -> Identity],

Graphics3D[{Black, 
              Polygon[{{0, 0, 2}, {0, 0.03, 1.9}, {0, -0.03, 1.9}}]}],
Graphics3D[{Blue, 
              AbsoluteThickness[2], Line[{{0, 0, 1}, {0, 0, 1.9}}]}],

Graphics3D[{Red, AbsoluteThickness[2], 
              Line[{{0, -1, 1}, {0, 1, 1}}]}],

PlotRange -> {{-1. , 
                1. }, {-1. , 1. }, {-0.2, 2}},DisplayFunction -> $DisplayFunction,
Lighting -> False,
Boxed -> False,Axes -> None,Background -> GrayLevel[0.8],ViewPoint -> {2, 1, 1.25},ImageSize -> {300, 300}]

Show[kugla[0.02, 0, 0, 0],

ParametricPlot3D[{x, y, 0, 
            Hue[0.14, 0.5, 1]}, {x, -0.15, 0.15, 0.3}, {y, -0.15, 
            0.15, 0.3}, DisplayFunction -> Identity],
ParametricPlot3D[
          Table[{r*Cos[i], r*Sin[i], 
                (r^3) Cos[i] (-1 + 2 Cos[2 i])}, {i, 0, Pi, 
                Pi/36}] // Evaluate, {r, -1, 1}, 
          DisplayFunction -> Identity],

ParametricPlot3D[{x, y, x^3 - 3  x*y^2, FaceForm[Hue[0.5], Hue[0.7]]},{x, -0.5, 0.5, 0.025},{y, -0.5, -0.475, 0.025},
DisplayFunction -> Identity],
ParametricPlot3D[{x, y, x^3 - 3  x*y^2, FaceForm[Hue[0.5], Hue[0.7]]},{x, -0.5, 0.5, 0.025},{y, 0.475, 0.5, 0.025},
DisplayFunction -> Identity],
ParametricPlot3D[{x, y, x^3 - 3  x*y^2, FaceForm[Hue[0.5], Hue[0.7]]},{x, -0.5, -0.475, 0.025},{y, -0.5, 0.5, 0.025},
DisplayFunction -> Identity],
ParametricPlot3D[{x, y, x^3 - 3  x*y^2, FaceForm[Hue[0.5], Hue[0.7]]},{x, 0.475, 0.5, 0.025},{y, -0.5, 0.5, 0.025},
DisplayFunction -> Identity],

Graphics3D[{Black, 
            Polygon[{{0, 0, 0.6}, {-0.01, -0.01, 0.56}, {0.01, 0.01, 
                  0.56}}]}],


Graphics3D[{Blue, AbsoluteThickness[2], 
            Line[{{0, 0, 0}, {0, 0, 0.56}}]}],




PlotRange -> {{-0.5, 
              0.5}, {-0.5, 0.5}, {- .3, 0.7}},DisplayFunction -> $DisplayFunction,
Axes -> None,Boxed -> False,Lighting -> False,
Background -> GrayLevel[0.8],ViewPoint -> {0.9, -1.7, 0.7},ImageSize -> {300, 300}]

torus

Show[
ParametricPlot3D[
        Append[torus[2, 1][u, v], FaceForm[Hue[0.3, 0.7, 0.7]]] // 
          Evaluate, {u, 0, 2  Pi}, {v, Pi/2, 3  Pi/2}, 
        PlotPoints -> {48, 24}, 
        DisplayFunction -> 
          Identity], 
ParametricPlot3D[
        Append[torus[2, 1][u, v], FaceForm[Hue[0.8, 0.7, 0.7]]] // 
          Evaluate, {u, 0, 2  Pi}, {v, -Pi/2, Pi/2}, 
        PlotPoints -> {48, 24}, 
        DisplayFunction -> 
          Identity], 



kugla1[0.08, (5/2), 0, ((Sqrt[3])/2)], 
kugla1[0.08, -(3/2), 
        0, ((Sqrt[3])/2)], 
kugla1[0.08, 0, 2, 
        1], 

ParametricPlot3D[
        Append[TangRav[2, 1][0, Pi/3], FaceForm[Hue[0.13]]] // 
          Evaluate, {t1, -0.5, 0.5, 1}, {t2, -0.5, 0.5, 1}, 
        DisplayFunction -> 
          Identity], 
ParametricPlot3D[
        Append[TangRav[2, 1][Pi, 2  Pi/3], FaceForm[Hue[0.13]]] // 
          Evaluate, {t1, -1, 1, 2}, {t2, -0.75, 0.75, 1.5}, 
        DisplayFunction -> 
          Identity], 
ParametricPlot3D[
        Append[TangRav[2, 1][Pi/2, Pi/2], FaceForm[Hue[0.13]]] // 
          Evaluate, {t1, -0.5, 0.5, 1}, {t2, -0.5, 0.5, 1}, 
        DisplayFunction -> 
          Identity], 

ParametricPlot3D[{-(3/2) + (((Sqrt[3]) ((Sqrt[3]) - 2 Sin[v]))/2), -((3 (-(2/3) (2 + Cos[v]) Sqrt[
                        1 - (-(((Sqrt[3]) Sin[v])/(2 + 
                                      Cos[v])))^2]))/2), ((Sqrt[3])/2) - (((Sqrt[3]) - 2 Sin[v])/2), {AbsoluteThickness[2]}}, {v, Pi/4, Pi - Pi/6, 
          Pi/32}, DisplayFunction -> 
          Identity], 

ParametricPlot3D[{-(3/2) + (((Sqrt[3]) ((Sqrt[3]) - 2 Sin[v]))/2), -((3 ((2/3) (2 + 
                          Cos[v]) 
                      Sqrt[1 - (-(((Sqrt[3]) Sin[v])/(2 + 
                                      Cos[v])))^2]))/2), ((Sqrt[3])/2) - (((Sqrt[3]) - 2 Sin[v])/2), {AbsoluteThickness[2]}}, {v, Pi/2.5, Pi/1.15, 
          Pi/32}, 
        DisplayFunction -> 
          Identity], 

ParametricPlot3D[{2  Cos[t], 2  Sin[t], 
          1, {Hue[1], AbsoluteThickness[3]}}, {t, Pi/2 + Pi/6, 2  Pi + Pi/3, 
          Pi/32}, DisplayFunction -> 
          Identity], 

ParametricPlot3D[{2  Cos[t], 2  Sin[t], 
          1, {AbsoluteThickness[2]}}, {t, 2  Pi + Pi/3, 2  Pi + 2  Pi/3, 
          Pi/24}, DisplayFunction -> 
          Identity], 

Axes -> None, 
      Boxed -> False, DisplayFunction -> $DisplayFunction, 
      Background -> GrayLevel[0.8], ImageSize -> {300, 250}, 
      Lighting -> False]