Plotting the surface gained by joining every point of a line to every point of another line in 3D space

Question

How can I write code for displaying two lines in 3D space at random and connecting all points of line A with all points of line B and displaying the resulting surface in Mathematica (what would it's equation be).

Can someone please randomize the placement of the two lines A and B in 3D space several times and show me the different 3D lines and surface outputs?

I need this for a topological investigation I need to carry out exploring this concept first.

I am new to Mathematica and have never used it before.

I also do not have access to it because it is expensive (and I'm at home, not on a campus).

Can someone please help me with the visualization.

I would be extremely thankful to you if you could help me.

Thanks.

• Michael E2: Maybe line1 = RandomReal[{}, {2, 3}]; line2 = RandomReal[{}, {2, 3}]; ConvexHullMesh[Join[line1, line2]]?
• b3m2a1: It's unclear exactly you mean by "connecting all points", do you want this to be by nearest distance? Do you want to just connect the end points? Do you want to somehow discretize and just linearly connect by the parametrization?
• user118161: This exactly describes a hyperboloid of one sheet!

Answer 1

I think you're just trying to make a closed surface from four points? Here's a fast way to do that which you could easily run in any system (e.g. python with X3D)

First we make the triangulation

lines = RandomReal[{}, {2, 2, 3}];
tris = Flatten[
 Table[
 Append[lines[[i]], lines[[Mod[i + 1, 2, 1], j]]],
 {i, 2},
 {j, 2}
 ],
 1
 ];

and then we'll visualize the faces

Graphics3D[
 {
 {Thick, Red, Line@lines[[1]]},
 {Thick, Blue, Line@lines[[2]]},
 MapThread[
 {#, Triangle[#2]} &,
 {
 {Green, Orange, Purple, Pink},
 tris
 }
 ]
 },
 Boxed -> False,
 Lighting -> "Neutral"
 ]

enter image description here

Maybe you want to discretize between endpoints and connect, though, here's what that would look like

subdivs = 100;
lines = RandomReal[{}, {2, 2, 3}];
linePoints = Subdivide[#, #2, subdivs] & @@@ lines;
Graphics3D[{
 {
 Red,
 Point@linePoints[[1]]
 },
 {
 Blue,
 Point@linePoints[[2]]
 },
 MapThread[Line@*List, linePoints]
 },
 Boxed -> False
 ]

enter image description here

Here's another interpretation, taking the nearest points from one line to another. This might have an analytic form, but it's fast to discretize

subdivs = 100;
lines = RandomReal[{}, {2, 2, 3}];
linePoints = Subdivide[#, #2, subdivs] & @@@ lines;
lineFriends = Nearest[linePoints[[2]], linePoints[[1]]][[;; , 1]];
Graphics3D[{
 {
 Red,
 Point@linePoints[[1]]
 },
 {
 Blue,
 Point@linePoints[[2]]
 },
 MapThread[Line@*List, {linePoints[[1]], lineFriends}]
 },
 Boxed -> False
 ]

enter image description here

Or who knows, maybe you want to get the planes defined by the nearest points between the lines, here's how you'd get that

u1 = Subtract @@ lines[[1]] // Normalize;
u2 = Subtract @@ lines[[2]] // Normalize;
n = Cross[u1, u2] // Normalize;
d = Inverse[{u1, -u2, n}] . (lines[[2, 1]] - lines[[1, 1]]);
mid1 = lines[[1, 1]] + d[[1]]*u1;
mid2 = lines[[2, 1]] + d[[2]]*u2;
planeNormal1 = Cross[u1, mid1 - mid2] // Normalize;
planeNormal2 = Cross[u2, mid2 - mid1] // Normalize;

which we visualize with Hyperplane

enter image description here

Answer 2

• We use parameter u to represent the position of the lines and use s to join the lines, after that we use ParametricPlot3D.

Clear["Global`*"];
{a1, b1} = RandomReal[{-10, 10}, {2, 3}];
{a2, b2} = RandomReal[{-10, 10}, {2, 3}];
f[u_, s_] := {1 - s, s} . {a1 + u*(b1 - a1), a2 + u*(b2 - a2)};
plot = ParametricPlot3D[f[u, s], {u, 0, 1}, {s, 0, 1}]; lines = 
 Graphics3D[{AbsoluteThickness[5], Red, Line[{a1, b1}], Blue, 
 Line[{a2, b2}]}];
Show[plot, lines, Boxed -> False,PlotRange -> All]

enter image description here

• Get four random pictures.

Table[({a1, b1} = RandomReal[{-10, 10}, {2, 3}];
 {a2, b2} = RandomReal[{-10, 10}, {2, 3}];
 f[u_, s_] := {1 - s, s} . {a1 + u*(b1 - a1), a2 + u*(b2 - a2)};
 plot = ParametricPlot3D[f[u, s], {u, 0, 1}, {s, 0, 1}]; 
 lines = Graphics3D[{AbsoluteThickness[3], Red, Line[{a1, b1}], Blue,
 Line[{a2, b2}]}];
 Show[plot, lines, Boxed -> False,PlotRange -> All]), 4]

enter image description here

• To get non-uniform grid, we could add two functions as factors.( two functions join {0,0} and {1,1}, for example λ = #^2 &;μ = Sqrt[#] &;

Clear["Global`*"];
{a1, b1} = RandomReal[{-10, 10}, {2, 3}];
{a2, b2} = RandomReal[{-10, 10}, {2, 3}];
λ = #^2 &;
μ = Sqrt[#] &;
f[u_, s_] := {1 - s, s} . {a1 + λ@u*(b1 - a1), 
 a2 + μ@u*(b2 - a2)};
plot = ParametricPlot3D[f[u, s], {u, 0, 1}, {s, 0, 1}]; lines = 
 Graphics3D[{AbsoluteThickness[5], Red, Line[{a1, b1}], Blue, 
 Line[{a2, b2}]}];
Show[plot, lines, Boxed -> False,PlotRange -> All]

Answer 3

I may only guess what OP wants because of confusing description but when mentioning surface and its equation (not a solid object) then I think OP wanted the following.

lines = RandomReal[{}, {2, 2, 3}];
n = 25;
Graphics3D[{Thin, Opacity[0.5], 
 Table[Line[#1 + p (#2 - #1) & @@@ lines], {p, 0, 1, 1/n}], Thick, 
 Riffle[{Red, Blue}, Line /@ lines]}]

---

enter image description here