Extracting points from 3D plot that lie along an arbitrarily oriented line

Question

Starting from i.e. the following 3d plot:

d = RandomReal[1, {100, 3}];
ListPlot3D[d]

Is it possible to extract points that lay along an arbitrarily oriented line, i.e. like this:

Show[{ListPlot3D[d], Graphics3D[Line[{{0, -.5, 1}, {0.5, 1, 1}}]]}]

?

EDIT: Z values of the plot along the line projected onto the x-y plane

Answer 1

SeedRandom[5]
d = RandomReal[1, {100, 3}];

You can use -.5 + 3 # - #2 & (or Function[{x, y}, -.5 + 3 x - y]) as the setting for MeshFunctions in ListPlot3D:

Show[lp3d = ListPlot3D[d, MeshFunctions -> {-.5 + 3 # - #2 &}, 
    Mesh -> {{0}}, MeshStyle -> Directive[Red, Thick], BoundaryStyle -> None], 
  Graphics3D[{Thick , Blue, Line[{{0, -.5, 1}, {0.5, 1, 1}}], 
    Opacity[.5, Yellow], EdgeForm @ None, 
    InfinitePlane[{{0, -.5, 0}, {0, -.5, 1}, {0.5, 1, 1}}]}]]

To extract the points on the red line:

Cases[Normal @ lp3d, Line[x_] :> x, All][[1]]

{{0.492655, 0.977959, 0.559503}, {0.491386, 0.973947,    0.484116}, {0.477966, 0.933211, 0.303948}, {0.476062, 0.92816,    0.451244}, {0.459746, 0.878862, 0.640324}, {0.457492, 0.872273,    0.586974}, {0.454029, 0.861943, 0.568448}, {0.441994, 0.825895,    0.39336}, {0.396855, 0.690417, 0.20754}, {0.395551, 0.686462,    0.255852}, {0.392548, 0.677572, 0.17963}, {0.350269, 0.550753,    0.612022}, {0.341512, 0.524435, 0.760695}, {0.313441, 0.440292,    0.52727}, {0.304016, 0.411343, 0.164743}, {0.300216, 0.400556,    0.123016}, {0.293312, 0.379549, 0.431375}, {0.279583, 0.33874,    0.571648}, {0.267421, 0.302245, 0.850239}, {0.266698, 0.299818,    0.85685}, {0.226072, 0.177329, 0.737677}, {0.217928, 0.152775,    0.650557}, {0.192837, 0.0784445, 0.831587}, {0.187349, 0.0619587,    0.854096}, {0.174868, 0.024533, 0.602241}}

Answer 2

You can use the interpolation that ListPlot uses, which you can then evaluate at any point on the line (within the domain of the interpolation):

zFN = Interpolation[d, InterpolationOrder -> 1, 
   "ExtrapolationHandler" -> {Indeterminate &, "WarningMessage" -> False}];

ClearAll[xyline, zSect];
xyline[x_] = {(1 - 2 x), 2 x}.{{0, -.5}, {0.5, 1}};
zSect[x_] := zFN @@ xyline[x];

zSect[0.25]  (* test a value *)
(*  0.654833  *)

Show[
 ListPlot3D[d],
 ParametricPlot3D[Append[xyline[x], zSect[x]], {x, 0, 1}]
 ]