Change Style of Points

Question

I was wondering if it's possible to change the style of aPoint with another symbol e.g.: cross, x etc...

It may sound weird-it is a Point after all, but I would like to have crosses instead of points in the following code

  Manipulate[Module[{data, inside, insidepts}, SeedRandom[n];
  data = RandomReal[{-1, 1}, {m, 2}];
  insidepts = Cases[data, {x_, y_} /; x^2 + y^2 < 1];
  inside = Length[insidepts];
  Text@
   Style[
    Column[
     {Graphics[{PointSize[0.0001], Red, Disk[{0, 0}, 1], Blue,
        Point[data]}, ImageSize -> If[format, 500, 350]],
      Row[{"inside: ", inside, "\toutside: ", m - inside, "\ttotal: ",
         m}],
      Row[{"π=4 x ", inside, "/", m, " = ", 4. inside/m}]}], "Label"]],
 {{n, 1, "random seed"}, 1, 1000, 1, Appearance -> "Labeled"},
 {{m, 1, "sample size"}, 1, 100000, 1, Appearance -> "Labeled"},
 {{format, False, "large format"}, {True, False}},
 AutorunSequencing -> {1, 2}]

Answer 1

It's a bit faster to create one marker, and use GeometricTransform to position it where needed than to use individual placements with Text or Inset:

Manipulate[SeedRandom[n];
 data = RandomReal[{-1, 1}, {m, 2}];
 insidepts = Cases[data, {x_, y_} /; x^2 + y^2 < 1];
 inside = Length[insidepts];
 Text@Style[
   Column[{Graphics[{PointSize[0.0001], Red, Disk[{0, 0}, 1], Blue, 
       GeometricTransformation[
        Line[{{Offset[{-2, 2}, {0, 0}], 
           Offset[{2, -2}, {0, 0}]}, {Offset[{-2, -2}, {0, 0}], 
           Offset[{2, 2}, {0, 0}]}}], Dynamic[data]]}, 
      ImageSize -> If[format, 500, 350]], 
     Row[{"inside: ", inside, "\toutside: ", m - inside, "\ttotal: ", 
       m}], Row[{"\[Pi]=4 x ", inside, "/", m, " = ", 4. inside/m}]}],
    "Label"], {{n, 1, "random seed"}, 1, 1000, 1, 
  Appearance -> "Labeled"}, {{m, 1, "sample size"}, 1, 100000, 1, 
  Appearance -> "Labeled"}, {{format, False, "large format"}, {True, 
   False}}, AutorunSequencing -> {1, 2}]

I used Line with Offset coordinates to construct a cross, but you could use Inset["\[Cross]", {0,0}] instead.

Answer 2

Given your example "point" symbols (cross, plus) and the need for speed if you want live updating in Manipulate, I recommend that you format your symbols as a series of lines, and display all of these lines in a single Line expression.

The size of the X's is controlled by the parameter of makeXfun.

You can use Opacity to reduce the inevitable visual density of the "points" in your graphic.

makeXfun[r_] := 
 Compile[{{xy, _Real, 1}},
   {{xy + {-r, -r}, xy + {r, r}}, {xy + {-r, r}, xy + {r, -r}}}]

Manipulate[
 Graphics[{
   {Red, Disk[{0, 0}, 1]},
   {Blue, Opacity[0.5],
     Line[Join @@ makeXfun[0.015] /@ RandomReal[{-1, 1}, {Round[samp^2], 2}]]},
   {Black, Text["Sample size: " <> ToString@Round[samp^2], {0, -1.2}]}
   }, ImageSize -> 400],
 {samp, 1, 300, 1}
]

For plus symbols instead of X's you would use something like:

makePlusfun[r_] := 
  Compile[{{xy, _Real, 1}},
    {{xy + {-r, 0}, xy + {r, 0}}, {xy + {0, r}, xy + {0, -r}}}]

Answer 3

You can use Text, but it gets much slower than using Point.

Manipulate[Module[{data, inside, insidepts}, SeedRandom[n];
  data = RandomReal[{-1, 1}, {m, 2}];
  insidepts = Cases[data, {x_, y_} /; x^2 + y^2 < 1];
  inside = Length[insidepts];
  Text@Style[
    Column[{Graphics[{Red, Disk[{0, 0}, 1], Blue, 
        Text[Style["\[Cross]", 12], #] & /@ data}, 
       ImageSize -> If[format, 500, 350]], 
      Row[{"inside: ", inside, "\toutside: ", m - inside, "\ttotal: ", m}],
      Row[{"\[Pi]=4 x ", inside, "/", m, " = ", 4. inside/m}]}], "Label"]],
  {{n, 1, "random seed"}, 1, 1000, 1, Appearance -> "Labeled"},
  {{m, 1, "sample size"}, 1, 100000, 1, Appearance -> "Labeled"},
  {{format, False, "large format"}, {True, False}},
  AutorunSequencing -> {1, 2}]

Answer 4

There is no styling for Point as for plot markers. Hence you can use Show with ListPlot added to your Graphics. I use the standard "x" for the x. I think from Paletts there might be a better looking cross symbol to use if you search there.

Manipulate[
 Module[{data, inside, insidepts},
  SeedRandom[n];
  data = RandomReal[{-1, 1}, {m, 2}];     
  insidepts = Cases[data, {x_, y_} /; x^2 + y^2 < 1];
  inside = Length[insidepts];

  Text@Style[Column[{
      Show[
       Graphics[
        {
         PointSize[0.0001],
         Red, Disk[{0, 0}, 1]
         },
        ImageSize -> If[format, 500, 350]
        ], 

        ListPlot[data, PlotMarkers -> Style["x", 9]]
       ],

      Row[{"inside: ", inside, "\toutside: ", m - inside, "\ttotal: ",m}], 
      Row[{"\[Pi]=4 x ", inside, "/", m, " = ", 4. inside/m}]}], "Label"]
  ],

 {{n, 1, "random seed"}, 1, 1000, 1,Appearance -> "Labeled"}, 
 {{m, 1, "sample size"}, 1, 100000, 1, Appearance -> "Labeled"}, 
 {{format, False, "large format"}, {True,False}},

 SynchronousUpdating -> False,
 SynchronousInitialization -> False,
 ContinuousAction -> False,

 AutorunSequencing -> {1, 2}
 ]

But with so many points you have, I do not see what is the point of this really, as it will not make any difference after few points, it is hard to see what is there any way. (I also noticed it is a little slower that Point