Vizualizacije Gaussove i srednje zakrivljenosti ploha

Definicije

paketi

<< Graphics`ParametricPlot3D`

definicije funkcija Gaussove i srednje zakrivljenosti

ee[x_][u_, v_] := 
      FullSimplify[D[x[uu, vv], uu] . D[x[uu, vv], uu]] /. {uu -> u, 
          vv -> v}

ff[x_][u_, v_] := 
      FullSimplify[D[x[uu, vv], uu] . D[x[uu, vv], vv]] /. {uu -> u, 
          vv -> v}

gg[x_][u_, v_] := 
      FullSimplify[D[x[uu, vv], vv] . D[x[uu, vv], vv]] /. {uu -> u, 
          vv -> v}

N0[x_][u_, v_] := 
      FullSimplify[
          Cross[D[x[uu, vv], uu], D[x[uu, vv], vv]]/
            Sqrt[Cross[D[x[uu, vv], uu], D[x[uu, vv], vv]] . 
                Cross[D[x[uu, vv], uu], D[x[uu, vv], vv]]]] /. {uu -> u, 
          vv -> v}

ll[x_][u_, v_] := 
      FullSimplify[
          D[x[uu, vv], uu, uu] . N0[x][uu, vv]] /. {uu -> u, 
          vv -> v}

mm[x_][u_, v_] := 
      FullSimplify[
          D[x[uu, vv], uu, vv] . N0[x][uu, vv]] /. {uu -> u, 
          vv -> v}

nn[x_][u_, v_] := 
      FullSimplify[
          D[x[uu, vv], vv, vv] . N0[x][uu, vv]] /. {uu -> u, 
          vv -> v}

GZ[x_][u_, v_] := 
      FullSimplify[(ll[x][uu, vv]*nn[x][uu, vv] - mm[x][uu, 
                    vv]*mm[x][uu, vv])/(ee[x][uu, vv]*gg[x][
                    uu, vv] - ff[x][uu, vv]*ff[x][uu, 
                    vv])] /. {uu -> u, vv -> v}

SZ[x_][u_, v_] := 
      FullSimplify[
          1/2 (ee[x][uu, vv]*nn[x][uu, vv] - 
                  2  mm[x][uu, vv]*ff[x][uu, vv] + gg[x][uu, 
                      vv]*ll[x][uu, vv])/(ee[x][uu, vv]*gg[x][
                      uu, vv] - ff[x][uu, vv]*ff[x][uu, 
                      vv])] /. {uu -> u, vv -> v}

definicije grafičkih prikaza

dip[ins_][g_] := $DisplayFunction[
        Insert[g, ins, {1, 1}]]

ploha[f_][u_, v_][u0_, u1_, du_][v0_, v1_, dv_][
        opts___]:=ParametricPlot3D[f[u, v],{u, u0, u1, du},{v, v0, v1, dv},DisplayFunction -> 
            dip[EdgeForm[AbsoluteThickness[0.1]]],opts]

grafgauss[f_][u_, v_][u0_, u1_][v0_, v1_][
        opts___]:=
Plot3D[GZ[f][u, v], {u, u0, u1}, {v, v0, v1}, 
        Mesh -> False, opts]

grafsrednja[f_][u_, v_][u0_, u1_][v0_, v1_][
        opts___]:=
Plot3D[SZ[f][u, v], {u, u0, u1}, {v, v0, v1}, 
        Mesh -> False, opts]

bojagauss[f_][u_, v_][u0_, u1_, du_][v0_, v1_, 
          dv_][opts___]:=ParametricPlot3D[Append[f[u, v], Hue[GZ[f][u, v]]],{u, u0, u1, du},{v, v0, v1, dv},DisplayFunction -> 
            dip[EdgeForm[AbsoluteThickness[0.1]]],
Lighting -> False,opts]

bojasrednja[f_][u_, v_][u0_, u1_, du_][v0_, v1_, 
          dv_][opts___]:=ParametricPlot3D[Append[f[u, v], Hue[3  SZ[f][u, v]]],{u, u0, u1, du},{v, v0, v1, dv},Lighting -> False,DisplayFunction -> 
            dip[EdgeForm[AbsoluteThickness[0.1]]],opts]

definicije grafičkih prikaza s potpisom

\(\(\(\(\(prvo[f_]\)[u_, v_]\)[u0_, u1_, du_]\)[v0_, v1_, dv_]\)[
          opts1___] := \(\(\(\(ploha[f]\)[u, v]\)[u0, u1, du]\)[v0, v1, dv]\)[
            opts1] && 
          Print[\*""Prikaz plohe f(u,v) u podrucju [(Subscript[u, 0]),(Subscript[u, 1])]×[(Subscript[v, 0]),(Subscript[v, 1])]""]\)

\(\(\(\(\(drugo[f_]\)[u_, v_]\)[u0_, u1_, du_]\)[v0_, v1_, dv_]\)[
        opts2___] := \(\(\(\(grafgauss[f]\)[u, v]\)[u0, u1]\)[v0, v1]\)[
          opts2] && 
        Print[\*""Graf funkcije Gussove zakrivljenosti plohe f(u,v) nad podrucjem [(Subscript[u, 0]),(Subscript[u, 1])]×[(Subscript[v, 0]),(Subscript[v, 1])]""]\)

\(\(\(\(\(trece[f_]\)[u_, v_]\)[u0_, u1_, du_]\)[v0_, v1_, dv_]\)[
        opts3___] := \(\(\(\(bojagauss[f]\)[u, v]\)[u0, u1, du]\)[v0, v1, 
            dv]\)[opts3] && 
        Print[\*""Prikaz plohe f(u,v) u podrucju [(Subscript[u, 0]),(Subscript[u, 1])]×[(Subscript[v, 0]),(Subscript[v, 1])] obojan bojom Hue(GZ)""]\)

\(\(\(\(\(cetvrto[f_]\)[u_, v_]\)[u0_, u1_, du_]\)[v0_, v1_, dv_]\)[
        opts4___] := \(\(\(\(grafsrednja[f]\)[u, v]\)[u0, u1]\)[v0, v1]\)[
          opts4] && 
        Print[\*""Graf funkcije srednje zakrivljenosti plohe f(u,v) nad podrucjem [(Subscript[u, 0]),(Subscript[u, 1])]×[(Subscript[v, 0]),(Subscript[v, 1])]""]\)

\(\(\(\(\(peto[f_]\)[u_, v_]\)[u0_, u1_, du_]\)[v0_, v1_, dv_]\)[
        opts5___] := \(\(\(\(bojasrednja[f]\)[u, v]\)[u0, u1, du]\)[v0, v1, 
            dv]\)[opts5] && 
        Print[\*""Prikaz plohe f(u,v) u podrucju [(Subscript[u, 0]),(Subscript[u, 1])]×[(Subscript[v, 0]),(Subscript[v, 1])] obojan bojom Hue(SZ)""]\)

Primjeri

ALGEBARSKE PLOHE 2. REDA

1. hipar (z=x*y)

parametrizacija

xuv[u_, v_] := u;
yuv[u_, v_] := v;
zuv[u_, v_] := u*v;
u0 = -2;
u1 = 2;
du = 0.1;
v0 = -2;
v1 = 2;
dv = 0.1;

crtanje

f[u_, v_] := {xuv[u, v], yuv[u, v], zuv[u, v]}

GZ[f][u, v]

SZ[f][u, v]

prvo[f][u, v][u0, u1, du][v0, v1, dv][
      BoxRatios -> {1, 1, 1}]

drugo[f][u, v][u0, u1, du][v0, v1, dv][PlotPoints -> 100,AxesLabel -> {u, v, GZ},AxesEdge -> {{-1, -1}, {1, -1}, {-1, -1}},
BoxRatios -> {1, 1, 1},PlotRange -> {{-2, 2}, {-2, 2}, {-1, 
              0}},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}},FaceGrids -> All]

trece[f][u, v][u0, u1, du][v0, v1, dv][
BoxRatios -> {1, 1, 1},AxesLabel -> {x, y, z},
DisplayFunction -> $DisplayFunction,ViewPoint -> {1.1,  -3,  1},Lighting -> False]

cetvrto[f][u, v][u0, u1, du][v0, v1, dv][PlotPoints -> 100,AxesLabel -> {u, v, SZ},AxesEdge -> {{-1, -1}, {1, -1}, {-1, -1}},
BoxRatios -> {1, 1, 1},PlotRange -> {{-2, 2}, {-2, 2}, {-0.25, 
              0.25}},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}},FaceGrids -> All]

peto[f][u, v][u0, u1, du][v0, v1, dv][BoxRatios -> {1, 1, 1},AxesLabel -> {x, y, z},
DisplayFunction -> $DisplayFunction,ViewPoint -> {1.1,  -3,  1},Lighting -> False]

2. hipar (z=x^2-y^2)

parametrizacija

xuv[u_, v_] := u;
yuv[u_, v_] := v;
zuv[u_, v_] := u^2 - v^2;
u0 = -1;
u1 = 1;
du = 0.05;
v0 = -1;
v1 = 1;
dv = 0.05;

crtanje

f[u_, v_] := {xuv[u, v], yuv[u, v], zuv[u, v]}

GZ[f][u, v]

SZ[f][u, v]

prvo[f][u, v][u0, u1, du][v0, v1, dv][]

drugo[f][u, v][u0, u1, du][v0, v1, dv][]

trece[f][u, v][u0, u1, du][v0, v1, dv][]

cetvrto[f][u, v][u0, u1, du][v0, v1, dv][]

peto[f][u, v][u0, u1, du][v0, v1, dv][]

3. kugla

parametrizacija

xuv[u_, v_] := Sin[v]*Cos[u];
yuv[u_, v_] := Sin[v]*Sin[u];
zuv[u_, v_] := Cos[v];
u0 = 0;
u1 = 2  Pi;
du = Pi/24;
v0 = 0.01;
v1 = Pi - 0.01;
dv = Pi/24;

crtanje

f[u_, v_] := {xuv[u, v], yuv[u, v], zuv[u, v]}

GZ[f][u, v]

SZ[f][u, v]

prvo[f][u, v][u0, u1, du][v0, v1, dv][]

drugo[f][u, v][u0, u1, du][v0, v1, dv][PlotRange -> {{0, 2  Pi}, {0, Pi}, {0, 1.3}},
AxesLabel -> {u, v, GZ},BoxRatios -> {1, 1, 0.8},
AxesEdge -> {{-1, -1}, {1, -1}, {-1, -1}}]

trece[f][u, v][u0, u1, du][v0, v1, dv][AxesLabel -> {x, y, z},
AxesEdge -> {{-1, -1}, {1, -1}, {-1, -1}}]

cetvrto[f][u, v][u0, u1, du][v0, v1, dv][]

peto[f][u, v][u0, u1, du][v0, v1, dv][]

4. rotacijski elipsoid

parametrizacija

xuv[u_, v_] := 2*Sin[v]*Cos[u];
yuv[u_, v_] := 2*Sin[v]*Sin[u];
zuv[u_, v_] := Cos[v];
u0 = 0;
u1 = 2  Pi;
du = Pi/36;
v0 = 0.01;
v1 = Pi - 0.01;
dv = Pi/42;

crtanje

f[u_, v_] := {xuv[u, v], yuv[u, v], zuv[u, v]}

GZ[f][u, v]

SZ[f][u, v]

prvo[f][u, v][u0, u1, du][v0, v1, dv][]

drugo[f][u, v][u0, u1, du][v0, v1, dv][PlotPoints -> 100,PlotRange -> {{0, 2  Pi}, {0, Pi}, {0, 1. }},
AxesLabel -> {u, v, GZ},BoxRatios -> {1, 1, 0.8},
AxesEdge -> {{-1, -1}, {1, -1}, {-1, -1}},Ticks -> {{0, (π/2), π, ((3 π)/2), 
              2 π}, {0, (π/2), π, ((3 π)/2), 2 π}, 
            Automatic}]

trece[f][u, v][u0, u1, du][v0, v1, dv][AxesLabel -> {x, y, z},
AxesEdge -> {{-1, -1}, {1, -1}, {-1, -1}}]

cetvrto[f][u, v][u0, u1, du][v0, v1, dv][]

peto[f][u, v][u0, u1, du][v0, v1, dv][]

5. troosni elipsoid

parametrizacija

xuv[u_, v_] := 5/2*Sin[v]*Cos[u];
yuv[u_, v_] := 4*Sin[v]*Sin[u];
zuv[u_, v_] := 2  Cos[v];
u0 = 0;
u1 = 2  Pi;
du = Pi/48;
v0 = 0.01;
v1 = Pi - 0.01;
dv = Pi/48;

crtanje

f[u_, v_] := {xuv[u, v], yuv[u, v], zuv[u, v]}

GZ[f][u, v]

SZ[f][u, v]

prvo[f][u, v][u0, u1, du][v0, v1, dv][]

drugo[f][u, v][u0, u1, du][v0, v1, dv][PlotRange -> {{0, 2  Pi}, {0, Pi}, {0, 0.6}},
AxesLabel -> {u, v, GZ},BoxRatios -> {1, 1, 0.8},
AxesEdge -> {{-1, -1}, {1, -1}, {-1, -1}},Ticks -> {{0, (π/2), π, ((3 π)/2), 
              2 π}, {0, (π/2), π, ((3 π)/2), 2 π}, 
            Automatic},PlotPoints -> 100]

trece[f][u, v][u0, u1, du][v0, v1, dv][AxesLabel -> {x, y, z},
AxesEdge -> {{-1, -1}, {1, -1}, {-1, -1}}]

cetvrto[f][u, v][u0, u1, du][v0, v1, dv][]

peto[f][u, v][u0, u1, du][v0, v1, dv][]

6. jednokrilni hiperboloid

parametrizacija

xuv[u_, v_] := 3  Cosh[v]*Cos[u];
yuv[u_, v_] := 2  Cosh[v]*Sin[u];
zuv[u_, v_] := 3  Sinh[v];
u0 = 0;
u1 = 2  Pi;
du = Pi/24;
v0 = -2;
v1 = 2;
dv = 0.2;

crtanje

f[u_, v_] := {xuv[u, v], yuv[u, v], zuv[u, v]}

GZ[f][u, v]

SZ[f][u, v]

prvo[f][u, v][u0, u1, du][v0, v1, dv][]

drugo[f][u, v][u0, u1, du][v0, v1, dv][]

trece[f][u, v][u0, u1, du][v0, v1, dv][]

cetvrto[f][u, v][u0, u1, du][v0, v1, dv][]

peto[f][u, v][u0, u1, du][v0, v1, dv][]

7. eliptički paraboloid (z=x^2+y^2)

parametrizacija

xuv[u_, v_] := u;
yuv[u_, v_] := v;
zuv[u_, v_] := -u^2 - v^2;
u0 = -1;
u1 = 1;
du = 0.05;
v0 = -1;
v1 = 1;
dv = 0.05;

crtanje

f[u_, v_] := {xuv[u, v], yuv[u, v], zuv[u, v]}

GZ[f][u, v]

SZ[f][u, v]

prvo[f][u, v][u0, u1, du][v0, v1, dv][
      Ticks -> {{-1, 0, 1}, {-1, 0, 1}, {-1, 0, 
            1}}]

drugo[f][u, v][u0, u1, du][v0, v1, dv][
      PlotPoints -> 50, 
      Ticks -> {{-1, 0, 1}, {-1, 0, 1}, {-1, 0, 
            1}}]

trece[f][u, v][u0, u1, du][v0, v1, dv][]

cetvrto[f][u, v][u0, u1, du][v0, v1, dv][
      PlotPoints -> 50, 
      Ticks -> {{-1, 0, 1}, {-1, 0, 1}, {-1, 0, 
            1}}]

peto[f][u, v][u0, u1, du][v0, v1, dv][]

ALGEBARSKE PLOHE 3. REDA

8.parabolički konoid 3. reda

parametrizacija

xuv[u_, v_] := u;
yuv[u_, v_] := v;
zuv[u_, v_] := 
        0.5 (v^2 - 1) (3 - u);
u0 = 1;
u1 = 5;
du = 0.1;
v0 = -2;
v1 = 2;
dv = 0.1;

crtanje

f[u_, v_] := {xuv[u, v], yuv[u, v], zuv[u, v]}

GZ[f][u, v]

SZ[f][u, v]

prvo[f][u, v][u0, u1, du][v0, v1, dv][
      Axes -> None, Boxed -> False, 
      BoxRatios -> {1, 1, 0.7}]

drugo[f][u, v][u0, u1, du][v0, v1, dv][]

trece[f][u, v][u0, u1, du][v0, v1, dv][
      Axes -> None, Boxed -> False, 
      BoxRatios -> {1, 1, 0.7}]

cetvrto[f][u, v][u0, u1, du][v0, v1, dv][]

peto[f][u, v][u0, u1, du][v0, v1, dv][
      Axes -> None, Boxed -> False, 
      BoxRatios -> {1, 1, 0.7}]

9. majmunsko sedlo

parametrizacija

xuv[u_, v_] := v*Cos[u];
yuv[u_, v_] := v*Sin[u];
zuv[u_, v_] := 
        (v^3) Cos[
            u] (-1 + 2 Cos[2 u]);
u0 = 0;
u1 = Pi;
du = Pi/24;
v0 = -1;
v1 = 1;
dv = 0.1;

crtanje

f[u_, v_] := {xuv[u, v], yuv[u, v], zuv[u, v]}

GZ[f][u, v]

SZ[f][u, v]

prvo[f][u, v][u0, u1, du][v0, v1, dv][
      ViewPoint -> {1, 1, 1}, 
      PlotRange -> {{-1, 1}, {-1, 1}, {-1, 1}}]

drugo[f][u, v][u0, u1, du][v0, v1, dv][]

trece[f][u, v][u0, u1, du][v0, v1, dv][]

cetvrto[f][u, v][u0, u1, du][v0, v1, dv][]

peto[f][u, v][u0, u1, du][v0, v1, dv][]

ALGEBARSKE PLOHE 4. REDA

10. kružni konoid 4. reda

parametrizacija

xuv[u_, v_] := Cos[u];
yuv[u_, v_] := Sin[u] (1 - v);
zuv[u_, v_] := v;
u0 = 0;
u1 = 2  Pi;
du = Pi/42;
v0 = -1;
v1 = 2.1;
dv = 0.1;

crtanje

f[u_, v_] := {xuv[u, v], yuv[u, v], zuv[u, v]}

GZ[f][u, v]

SZ[f][u, v]

\(\(\(\(\(prvo[f]\)[u, v]\)[u0, u1, du]\)[v0, v1, dv]\)[
      ViewPoint -> {1, 0.5, 
          0.5}, 			PlotRange -> {{\(-1.1\), 1.1}, {\(-2\), 2}, {\(-1\), 
            2.1}}]\)

drugo[f][u, v][u0, u1, du][v0, v1, dv][PlotPoints -> 100,AxesLabel -> {u, v, GZ},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},FaceGrids -> All]

\(\(\(\(\(trece[f]\)[u, v]\)[u0, u1, du]\)[v0, v1, dv]\)[
      ViewPoint -> {1, 0.3, 
          0.7}, 			PlotRange -> {{\(-1.1\), 1.1}, {\(-2\), 2}, {\(-1\), 
            2.1}}, Axes -> None, Boxed -> False]\)

cetvrto[f][u, v][u0, u1, du][v0, v1, dv][PlotPoints -> 100,AxesLabel -> {u, v, SZ},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},FaceGrids -> All]

\(\(\(\(\(peto[f]\)[u, v]\)[u0, u1, du]\)[v0, v1, dv]\)[
      ViewPoint -> {1, 0.3, 
          0.7}, 			PlotRange -> {{\(-1.1\), 1.1}, {\(-2\), 2}, {\(-1\), 
            2.1}}, Axes -> None, Boxed -> False]\)

11. torus

parametrizacija

xuv[u_, v_] := (8 + 3  Cos[v]) 
          Cos[u];
yuv[u_, v_] := (8 + 3  Cos[v]) 
          Sin[u];
zuv[u_, v_] := 3  Sin[v];
u0 = 0;
u1 = 2  Pi;
du = Pi/24;
v0 = 0;
v1 = 2  Pi;
dv = Pi/24;

crtanje

f[u_, v_] := {xuv[u, v], yuv[u, v], zuv[u, v]}

GZ[f][u, v]

SZ[f][u, v]

prvo[f][u, v][u0, u1, du][v0, v1, dv][]

drugo[f][u, v][u0, u1, du][v0, v1, dv][]

trece[f][u, v][u0, u1, du][v0, v1, dv][]

cetvrto[f][u, v][u0, u1, du][v0, v1, dv][]

peto[f][u, v][u0, u1, du][v0, v1, dv][]

12. Plückerov konoid 4. reda

parametrizacija

xuv[u_, v_] := v*Cos[u];
yuv[u_, v_] := v*Sin[u];
zuv[u_, v_] := 2  Sin[u];
u0 = 0;
u1 = 2  Pi;
du = Pi/48;
v0 = -2;
v1 = 2;
dv = 0.1;

crtanje

f[u_, v_] := {xuv[u, v], yuv[u, v], zuv[u, v]}

GZ[f][u, v]

SZ[f][u, v]

prvo[f][u, v][u0, u1, du][v0, v1, dv][
      Boxed -> False, Axes -> False, 
      ViewPoint -> {1.3, -1, 2}]

drugo[f][u, v][u0, u1, du][v0, v1, dv][
      PlotPoints -> 50]

trece[f][u, v][u0, u1, du][v0, v1, dv][
      Boxed -> False, Axes -> False, 
      ViewPoint -> {1.3, -1, 2}]

cetvrto[f][u, v][u0, u1, du][v0, v1, dv][
      PlotPoints -> 50]

peto[f][u, v][u0, u1, du][v0, v1, dv][
      Boxed -> False, Axes -> False, 
      ViewPoint -> {1.3, -1, 2}]

PARAMETRIZACIJE POMOĆU TRIGONOMETRIJSKIH I HIPERBOLNIH FUNKCIJA

13. Sin[u],Sin[v],Cos[u]*Cos[v]

parametrizacija

xuv[u_, v_] := Sin[u];
yuv[u_, v_] := Sin[v];
zuv[u_, v_] := Cos[u] Cos[v];
u0 = -Pi;
u1 = Pi;
du = Pi/48;
v0 = -Pi;
v1 = Pi;
dv = Pi/48;

crtanje

f[u_, v_] := {xuv[u, v], yuv[u, v], zuv[u, v]}

GZ[f][u, v]

SZ[f][u, v]

prvo[f][u, v][u0, u1, du][v0, v1, dv][
      Axes -> None, Boxed -> False]

drugo[f][u, v][u0, u1, du][v0, v1, dv][
      PlotPoints -> 50]

trece[f][u, v][u0, u1, du][v0, v1, dv][
      Axes -> None, Boxed -> False]

cetvrto[f][u, v][u0, u1, du][v0, v1, dv][
      PlotPoints -> 50]

peto[f][u, v][u0, u1, du][v0, v1, dv][
      Axes -> None, Boxed -> False]

14. u,v,Cos[u]*Cos[v]

parametrizacija

xuv[u_, v_] := u;
yuv[u_, v_] := v;
zuv[u_, v_] := Cos[u] Cos[v];
u0 = -Pi;
u1 = Pi;
du = Pi/24;
v0 = 0;
v1 = 2  Pi;
dv = Pi/24;

crtanje

f[u_, v_] := {xuv[u, v], yuv[u, v], zuv[u, v]}

GZ[f][u, v]

SZ[f][u, v]

prvo[f][u, v][u0, u1, du][v0, v1, dv][]

drugo[f][u, v][u0, u1, du][v0, v1, dv][
      PlotPoints -> 100, AxesLabel -> {u, v, GZ}]

trece[f][u, v][u0, u1, du][v0, v1, dv][
      AxesLabel -> {x, y, z}]

cetvrto[f][u, v][u0, u1, du][v0, v1, dv][
      PlotPoints -> 100, AxesLabel -> {u, v, SZ}]

peto[f][u, v][u0, u1, du][v0, v1, dv][
      AxesLabel -> {x, y, z}]

15. Cos[u],Sin[u+v],-Cosh[v]

parametrizacija

xuv[u_, v_] := Cos[u]
yuv[u_, v_] := Sin[u + v];
zuv[u_, v_] := -Cosh[v];
u0 = -Pi;
u1 = Pi;
du = Pi/48;
v0 = -Pi/2;
v1 = Pi/2;
dv = Pi/48;

crtanje

f[u_, v_] := {xuv[u, v], yuv[u, v], zuv[u, v]}

GZ[f][u, v]

SZ[f][u, v]

prvo[f][u, v][u0, u1, du][v0, v1, dv][
      Axes -> None, Boxed -> False]

drugo[f][u, v][u0, u1, du][v0, v1, dv][
      PlotPoints -> 50]

trece[f][u, v][u0, u1, du][v0, v1, dv][
      Axes -> None, Boxed -> False]

cetvrto[f][u, v][u0, u1, du][v0, v1, dv][]

peto[f][u, v][u0, u1, du][v0, v1, dv][
      Axes -> None, Boxed -> False]

16. Cos[v]+Sin[u],Cos[u]+Sin[v],Cos[v+u]+Sin[u+v]

parametrizacija

xuv[u_, v_] := Cos[v] + Sin[u];
yuv[u_, v_] := Cos[u] + Sin[v];
zuv[u_, v_] := Cos[v + u] + Sin[u + v];
u0 = -Pi;
u1 = Pi;
du = Pi/24;
v0 = -Pi;
v1 = Pi;
dv = Pi/24;

crtanje

f[u_, v_] := {xuv[u, v], yuv[u, v], zuv[u, v]}

gausszak[f][u, v]

srednjazak[f][u, v]

prvo[f][u, v][u0, u1, du][v0, v1, dv][
      PlotRange -> {{-2, 2}, {-1, 2}, {-1.5, 1.5}}, 
      Boxed -> False, Axes -> False]

drugo[f][u, v][u0, u1, du][v0, v1, dv][
      PlotPoints -> 50]

trece[f][u, v][u0, u1, du][v0, v1, dv][
      PlotRange -> {{-2, 2}, {-1, 2}, {-1.5, 1.5}}, 
      Boxed -> False, Axes -> False]

cetvrto[f][u, v][u0, u1, du][v0, v1, dv][]

peto[f][u, v][u0, u1, du][v0, v1, dv][
      PlotRange -> {{-2, 2}, {-1, 2}, {-1.5, 1.5}}, 
      Boxed -> False, Axes -> False]

17. Sin[u],Sin[v],Sin[u+v]

parametrizacija

xuv[u_, v_] := Sin[u];
yuv[u_, v_] := Sin[v];
zuv[u_, v_] := Sin[u + v];
u0 = -Pi;
u1 = Pi;
du = Pi/36;
v0 = -Pi;
v1 = Pi;
dv = Pi/36;

crtanje

f[u_, v_] := {xuv[u, v], yuv[u, v], zuv[u, v]}

GZ[f][u, v]

SZ[f][u, v]

prvo[f][u, v][u0, u1, du][v0, v1, dv][
      Boxed -> False, Axes -> False]

drugo[f][u, v][u0, u1, du][v0, v1, dv][
      PlotPoints -> 50]

trece[f][u, v][u0, u1, du][v0, v1, dv][
      Boxed -> False, Axes -> False]

cetvrto[f][u, v][u0, u1, du][v0, v1, dv][
      PlotPoints -> 50]

peto[f][u, v][u0, u1, du][v0, v1, dv][
      Boxed -> False, Axes -> False]

18. u,v,Sin[u/Cos[v]]

parametrizacija

xuv[u_, v_] := u;
yuv[u_, v_] := v;
zuv[u_, v_] := Sin[(u/Cos)[v]];
u0 = -1;
u1 = 2.4;
du = 0.068;
v0 = -1;
v1 = 1;
dv = 0.05;

crtanje

f[u_, v_] := {xuv[u, v], yuv[u, v], zuv[u, v]}

GZ[f][u, v]

SZ[f][u, v]

ploha[f][u, v][u0, u1, du][v0, v1, dv][
      ViewPoint -> {1.3, -3, 2.6}, BoxRatios -> {2, 2, 1}, 
      Axes -> None, Boxed -> False]

drugo[f][u, v][u0, u1, du][v0, v1, dv][
      PlotPoints -> 50]

trece[f][u, v][u0, u1, du][v0, v1, dv][
      ViewPoint -> {1.3, -3, 2.6}, BoxRatios -> {2, 2, 1}, 
      Axes -> None, Boxed -> False]

cetvrto[f][u, v][u0, u1, du][v0, v1, dv][
      PlotPoints -> 50]

peto[f][u, v][u0, u1, du][v0, v1, dv][
      ViewPoint -> {1.3, -3, 2.6}, BoxRatios -> {2, 2, 1}, 
      Axes -> None, Boxed -> False]

19. u,v,-Cosh[u*v]

parametrizacija

xuv[u_, v_] := u;
yuv[u_, v_] := v;
zuv[u_, v_] := -Cosh[u*v];
u0 = -1;
u1 = 1;
du = 0.05;
v0 = -1;
v1 = 1;
dv = 0.05;

crtanje

f[u_, v_] := {xuv[u, v], yuv[u, v], zuv[u, v]}

GZ[f][u, v]

SZ[f][u, v]

ploha[f][u, v][u0, u1, du][v0, v1, dv][
      BoxRatios -> {1, 1, 1.5}, Axes -> None, Boxed -> False, 
      PlotRange -> {{-1, 1}, {-1, 
            1}, {-2, -0.9}}]

drugo[f][u, v][u0, u1, du][v0, v1, dv][
      PlotPoints -> 50]

trece[f][u, v][u0, u1, du][v0, v1, dv][
      BoxRatios -> {1, 1, 1.5}, Axes -> None, Boxed -> False, 
      PlotRange -> {{-1, 1}, {-1, 
            1}, {-2, -0.9}}]

cetvrto[f][u, v][u0, u1, du][v0, v1, dv][
      PlotPoints -> 50]

peto[f][u, v][u0, u1, du][v0, v1, dv][
      BoxRatios -> {1, 1, 1.5}, Axes -> None, Boxed -> False, 
      PlotRange -> {{-1, 1}, {-1, 
            1}, {-2, -0.9}}]