Assigning a Color to each Point on the Plane

Question

I have a dataset consisting of points on the plane and a corresponding color. These colors divide the plane into a few distinct regions. I would like to make a plot or diagram that nicely shows this breakdown into colored regions.

To generate some example data,

incircle[x_, y_] := Piecewise[{{Red, x^2 + y^2 <= 25 }, {Blue, x^2 + y^2 > 25 }}];
data = Table[{i, j, incircle[i, j]}, {i, -10, 10, 1}, {j, -10, 10, 1}];
data = ArrayReshape[data, {441, 3}];

(*{{-10,-10,Blue},{-10,-9,Blue},{-10,-8,Blue},{-10,-7,Blue},...*)

This gives a list where each entry is a pair of x,y coordinates, and then red or blue depending on whether or not the coordinates are within a circle of radius five.

I can then make a list of coordinates and list of colors, and plot them accordingly using listplot:

pdat = {{#[[1]], #[[2]]}} & /@ data;
pcol = #[[3]] & /@ data;
ListPlot[pdat, PlotStyle -> pcol, PlotMarkers -> "\[FilledSquare]", ImageSize -> {250, 250}]

Which gives me a red circular region. Is there anyway to color each region according to the list more nicely and continuously, preserving a definite boundary between each region? My actual data is much more sporadic so the boundary between regions can be much more complex than a circle.

Answer 1

Convex hull mesh

coords = CoordinateBoundsArray[{{-10, 10}, {-10, 10}}];
in = Select[Flatten[coords, 1], Norm@# <= 5 &];
out = Select[Flatten[coords, 1], Norm@# > 5 &];

p1 = Cases[Normal@ConvexHullMesh[out]["Graphics"], _Polygon, Infinity];
p2 = Cases[Normal@ConvexHullMesh[in]["Graphics"], _Polygon, Infinity];

Graphics[{
  ColorData[97, 1], First@p1,
  ColorData[97, 2], First@p2
  }]

If anybody knows of a better way to turn a convex hull mesh into a polygon, please let me know.

Interpolation

Interpolation with interpolation order 0.

data = Join[{#, 1} & /@ in, {#, 2} & /@ out];
interp = Interpolation[data, InterpolationOrder -> 0];

DensityPlot[
 interp[x, y],
 {x, -10, 10},
 {y, -10, 10},
 PlotPoints -> 100
 ]

Using a higher interpolation order and rounding to get smoothed boundaries:

interp = Interpolation[data, InterpolationOrder -> 1];
DensityPlot[
 Round@interp[x, y],
 {x, -10, 10},
 {y, -10, 10},
 PlotPoints -> 100
 ]

Answer 2

VoronoiMesh method

points = Drop[data, None, -1];
mesh = VoronoiMesh[points];
polygons = MeshPrimitives[mesh, 2];
coloredpolygons = Map[{incircle @@ RegionCentroid[#], #} &, polygons];
Graphics[coloredpolygons]

VoronoiMesh automatically chooses the coordinates of the boundaries of the mesh, but you can also specify the boundaries via a second argument (VoronoiMesh[points, {{xmin, xmax}, {ymin, ymax}}]) if you want.

This method generalizes nicely to irregularly located points as well.

points = RandomReal[{-10, 10}, {200, 2}];
(* Remaining code as above *)

Note, however, that there is a bug in this code: the centroid of a Voronoi cell is not necessarily its "base point". This may make a difference in some cases, particularly if your base points are not uniformly distributed.

I will have to think if there's an easy way to color the Voronoi cells based on their base point, rather than their centroid. Complicating matters is that MeshPrimitives[VoronoiMesh[points]] appears to shuffle the order of the resulting polygons (i.e., the $i$th polygon in the result does not necessarily contain the $i$th element of points.)

Answer 3

This looks a bit cleaner, and it does what you want:

pdat1=List/@data[[All,;;2]];
pcol1=List/@data[[All,3]];
ListPlot[pdat1, PlotStyle -> pcol1, PlotMarkers -> "\[FilledSquare]", ImageSize -> {250, 250}]

Same as original output from OP.

This uses VertexColors, but is in Graphics using Point:

Graphics[{PointSize[Medium],Point[data[[All,;;2]],VertexColors->data[[All,3]]]}, ImageSize -> {250, 250},Axes->True]

I’ll add other methods once I am back to a PC.