Plotting points in Mathematica - wolfram-mathematica

Question

I am trying to plot a few points on the following picture in Mathematica:
ParametricPlot3D[
{{u, v, (Cos[u] + Cos[v])/3}, {u, -1, (Cos[u] + Cos[0])/3},
{5, v, (Cos[4] + Cos[v])/3}}, {u, -4, 4}, {v, 0, 8}, Axes -> False,
Boxed -> False, BoxRatios -> {8, 8, 1.5}]
(they should just look like dots on the surface)
What I was trying to do is enter the coordinates of the points manually on another graph using ListPointPlot3D, and then combine them using Show. But for some reason that isn't working. Suggestions?
Also, I would like to add small vectors tangent to the surface in the x directions for the points I have plotted, but I have no idea on how to do that, so suggestions would be very much appreciated!

Answer 1

Perhaps this will help you get started on a solution. It plots 3 random points on the surface. You can change the number of points by setting nPoints. I don't know how to plot tangents along x. But when you figure that out you can use Arrows, as suggested by #Verbeia.
nPoints = 3;
Show[ParametricPlot3D[{
{u, v, (Cos[u] + Cos[v])/3},
{u, -1, (Cos[u] + Cos[0])/3}, {5, v, (Cos[4] + Cos[v])/3}},
{u, -4, 4}, {v, 0, 8}, Axes -> False,
Boxed -> False, BoxRatios -> {8, 8, 1.5},
PlotStyle -> Directive[Opacity[0.5]]],
Graphics3D[{Red, PointSize[.025],
Point[Table[{u1 = RandomReal[{-3, 3}], v1 = RandomReal[{1, 7}],
(Cos[u1] + Cos[v1])/3}, {nPoints}]]}]]
Edit
The following dynamic variation makes use of #belisarius 's contribution:
Manipulate[
Show[ParametricPlot3D[{{u, v, (Cos[u] + Cos[v])/3} },
{u, -4, 4}, {v, 0, 8}, Axes -> False, Boxed -> False,
BoxRatios -> {8, 8, 1.5},
Mesh -> None,
ImageSize -> {400, 300},
PlotRange -> {{-4, 4}, {0, 8}},
PlotRangePadding -> {{0, 1.4}, {0, 0}},
PlotStyle -> Directive[Opacity[0.5]]],
Graphics3D[({Red, PointSize[.025],
Point#f[pt[[1, 1]], pt[[1, 2]]], Black,
Arrow[{f[pt[[1, 1]], pt[[1, 2]]],
f[pt[[1, 1]], pt[[1, 2]]] + D[f[t, pt[[1, 2]]], t] /.
t -> pt[[1, 1]]}]}]],
Grid[{{
LocatorPane[Dynamic[pt],
Dynamic[Graphics[{},
PlotRange -> {{-4, 4}, {0, 8}},
Frame -> True,
ImageSize -> 160,
FrameTicks -> {Range[-4, 4], Range[0, 8], None, None},
FrameLabel -> {"u", "v"},
GridLines -> {Range[-4, 4], Range[0, 8]},
GridLinesStyle -> Directive[LightGray]]],
{{-4, 0}, {4, 8}}]}}],
{{pt, {{1, 2}}}, ControlType -> None},
Initialization :> {f[u_, v_] := {u, v, (Cos[u] + Cos[v])/3};}]

Answer 2

For the Arrows
f[u_, v_] := {u, v, (Cos[u] + Cos[v])/3};
Show[ParametricPlot3D[{f[u, v]}, {u, -4, 4}, {v, 0, 8},
Axes -> False, Mesh -> None, Boxed -> False, BoxRatios -> {8, 8, 1.5},
PlotStyle -> Directive[Opacity[0.5]]],
Graphics3D#
Table[{Red, PointSize[.025], Point#f[u, v],
Black, Arrow[{f[u, v], f[u, v] + D[f[t, v], t] /. t -> u}]},
{u, -4, 4, 2}, {v, 0, 8, 2}]]
For getting the arrows in any direction a = { a1, a2 } instead of x, you may do:
Dot[{a1,a2}.#] & /# D[f[u, v], {{u, v}}]
(*
-> {a1, a2, -(1/3) a1 Sin[u] - 1/3 a2 Sin[v]}
*)
Edit
Both derivatives and normal:
f[u_, v_] := {u, v, (Cos[u] + Cos[v])/3};
Show[
Graphics3D#
Table[{Red, PointSize[.025], Point#f[u, v], Black, Arrowheads[.02],
Arrow[{f[u, v], f[u, v] + D[f[t, v], t] /. t -> u}],
Arrow[{f[u, v], f[u, v] + D[f[u, t], t] /. t -> v}],
Arrow[{f[u, v], f[u, v] +
Cross[D[f[t, v], t] /. t -> u,
D[f[u, t], t] /. t -> v]}]},
{u, -4, 4, 2}, {v, 0, 8, 2}],
ParametricPlot3D[{f[u, v]}, {u, -4, 4}, {v, 0, 8},
Axes -> False, Mesh -> 3, MeshStyle -> {{Opacity[0.1], LightBlue}},
Boxed -> False, BoxRatios -> {8, 8, 1.5},
PlotStyle -> Directive[Opacity[0.5]]]]

Answer 3

You can combine the plot with points using Graphics3D[listofpoints], where listofpoints is a T*3 matrix list, and the arrows using constructs like Graphics3D[Arrow[{{1, 1, -1}, {2, 2, 0}, {3, 3, -1}, {4, 4, 0}}]]. If they are all Graphics3D objects, you should be able to combine them with Show.
Sorry, I am not near a Mathematica installation to provide you with an example just now.