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Dr. belisarius
Dr. belisarius
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u = {x, x^3, 0};
v = {0, 0, z2};
l = (u - v) t + v;
w = l /. Solve[xs xs + ys ys + (zs - 1)^2 == 1 /. Thread[{xs, ys, zs} -> l], t][[2]];
Manipulate[Show[
  ParametricPlot3D[{Cos[u] Sin[v], Sin[u] Sin[v], 1 - Cos[v]}, {v, ArcCos[1 - z1], 0}, 
                  {u, 0, 2 Pi}, PlotStyle -> {Opacity[.3], FaceForm[Red, Yellow]}, Mesh -> False, 
                  PlotRange -> {{-1, 1}, {-1, 1}, {0, 2}}], 
  Graphics3D[{Opacity[.3], Cuboid[{-1, -1, -.01}, {1, 1, .01}]}], 
  ParametricPlot3D[{u, u^3, 0}, {u, -1, 1}], 
  ParametricPlot3D[w /. z2 -> z1, {x, -10^3, 10^3}, PlotPoints -> 300]], 
{z1, 0.1, 2}]

Mathematica graphics

The same with a spiral instead of a cubic:

Mathematica graphics

u = {x, x^3, 0};
v = {0, 0, z2};
l = (u - v) t + v;
w = l /. Solve[xs xs + ys ys + (zs - 1)^2 == 1 /. Thread[{xs, ys, zs} -> l], t][[2]];
Manipulate[Show[
  ParametricPlot3D[{Cos[u] Sin[v], Sin[u] Sin[v], 1 - Cos[v]}, {v, ArcCos[1 - z1], 0}, 
                  {u, 0, 2 Pi}, PlotStyle -> {Opacity[.3], FaceForm[Red, Yellow]}, Mesh -> False, 
                  PlotRange -> {{-1, 1}, {-1, 1}, {0, 2}}], 
  Graphics3D[{Opacity[.3], Cuboid[{-1, -1, -.01}, {1, 1, .01}]}], 
  ParametricPlot3D[{u, u^3, 0}, {u, -1, 1}], 
  ParametricPlot3D[w /. z2 -> z1, {x, -10^3, 10^3}, PlotPoints -> 300]], 
{z1, 0.1, 2}]

Mathematica graphics

u = {x, x^3, 0};
v = {0, 0, z2};
l = (u - v) t + v;
w = l /. Solve[xs xs + ys ys + (zs - 1)^2 == 1 /. Thread[{xs, ys, zs} -> l], t][[2]];
Manipulate[Show[
  ParametricPlot3D[{Cos[u] Sin[v], Sin[u] Sin[v], 1 - Cos[v]}, {v, ArcCos[1 - z1], 0}, 
                  {u, 0, 2 Pi}, PlotStyle -> {Opacity[.3], FaceForm[Red, Yellow]}, Mesh -> False, 
                  PlotRange -> {{-1, 1}, {-1, 1}, {0, 2}}], 
  Graphics3D[{Opacity[.3], Cuboid[{-1, -1, -.01}, {1, 1, .01}]}], 
  ParametricPlot3D[{u, u^3, 0}, {u, -1, 1}], 
  ParametricPlot3D[w /. z2 -> z1, {x, -10^3, 10^3}, PlotPoints -> 300]], 
{z1, 0.1, 2}]

Mathematica graphics

The same with a spiral instead of a cubic:

Mathematica graphics

added 77 characters in body
Source Link
Dr. belisarius
Dr. belisarius
  • 116.2k
  • 13
  • 205
  • 456
u = {x, x^3, 0};
v = {0, 0, z2};
l = (u - v) t + v;
w = l /. Solve[xs xs + ys ys + (zs - 1)^2 == 1 /. Thread[{xs, ys, zs} -> l], t][[2]]; 
Manipulate[Manipulate[Show[
 Show[ParametricPlot3D[ ParametricPlot3D[{Cos[u] Sin[v], Sin[u] Sin[v], 1 - Cos[v]}, {v, z1 Pi/2,ArcCos[1 0},- {uz1], 0, 2 Pi}, 
                   {u, 0, 2 Pi}, PlotStyle -> {Opacity[.3], FaceForm[Red, Yellow]}, Mesh -> False]False, 
                  PlotRange -> {{-1, 1}, {-1, 1}, {0, 2}}], 
  Graphics3D[{Opacity[.3], Cuboid[{-1, -1, -.01}, {1, 1, .01}]}], 
      ParametricPlot3D[{u, u^3, 0}, {u, -1, 1}], 
      ParametricPlot3D[w /. z2 -> z1, {x, -10^210^3, 10^210^3}]], PlotPoints -> 300]], 
{z1, 0.1, 2, 1}]

Mathematica graphicsMathematica graphics

u = {x, x^3, 0};
v = {0, 0, z2};
l = (u - v) t + v;
w = l /. Solve[xs xs + ys ys + (zs - 1)^2 == 1 /. Thread[{xs, ys, zs} -> l], t][[2]]; 
Manipulate[
 Show[ParametricPlot3D[{Cos[u] Sin[v], Sin[u] Sin[v], 1 - Cos[v]}, {v, z1 Pi/2, 0}, {u, 0, 2 Pi}, 
                       PlotStyle -> {Opacity[.3], FaceForm[Red, Yellow]}, Mesh -> False], 
      Graphics3D[{Opacity[.3], Cuboid[{-1, -1, -.01}, {1, 1, .01}]}], 
      ParametricPlot3D[{u, u^3, 0}, {u, -1, 1}], 
      ParametricPlot3D[w /. z2 -> z1, {x, -10^2, 10^2}]], {z1, 0.1, 2, 1}]

Mathematica graphics

u = {x, x^3, 0};
v = {0, 0, z2};
l = (u - v) t + v;
w = l /. Solve[xs xs + ys ys + (zs - 1)^2 == 1 /. Thread[{xs, ys, zs} -> l], t][[2]];
Manipulate[Show[
  ParametricPlot3D[{Cos[u] Sin[v], Sin[u] Sin[v], 1 - Cos[v]}, {v, ArcCos[1 - z1], 0}, 
                  {u, 0, 2 Pi}, PlotStyle -> {Opacity[.3], FaceForm[Red, Yellow]}, Mesh -> False, 
                  PlotRange -> {{-1, 1}, {-1, 1}, {0, 2}}], 
  Graphics3D[{Opacity[.3], Cuboid[{-1, -1, -.01}, {1, 1, .01}]}], 
  ParametricPlot3D[{u, u^3, 0}, {u, -1, 1}], 
  ParametricPlot3D[w /. z2 -> z1, {x, -10^3, 10^3}, PlotPoints -> 300]], 
{z1, 0.1, 2}]

Mathematica graphics

Source Link
Dr. belisarius
Dr. belisarius
  • 116.2k
  • 13
  • 205
  • 456

u = {x, x^3, 0};
v = {0, 0, z2};
l = (u - v) t + v;
w = l /. Solve[xs xs + ys ys + (zs - 1)^2 == 1 /. Thread[{xs, ys, zs} -> l], t][[2]]; 
Manipulate[
 Show[ParametricPlot3D[{Cos[u] Sin[v], Sin[u] Sin[v], 1 - Cos[v]}, {v, z1 Pi/2, 0}, {u, 0, 2 Pi}, 
                       PlotStyle -> {Opacity[.3], FaceForm[Red, Yellow]}, Mesh -> False], 
      Graphics3D[{Opacity[.3], Cuboid[{-1, -1, -.01}, {1, 1, .01}]}], 
      ParametricPlot3D[{u, u^3, 0}, {u, -1, 1}], 
      ParametricPlot3D[w /. z2 -> z1, {x, -10^2, 10^2}]], {z1, 0.1, 2, 1}]

Mathematica graphics