Skip to main content

plotting - High-quality RegionPlot3D for logical combinations of predicates



Unlike RegionPlot, RegionPlot3D copes poorly with logical combinations of predicates (&&, ||), which should result in sharp edges in the region to be plotted. Instead, these edges are usually drawn rounded and sometimes with severe aliasing artifacts. This has been observed in many posts on this site:



One solution, as noted by Silvia, by halirutan, and most recently by Jens, is to use a ContourPlot3D instead with an appropriate RegionFunction, as this produces much higher-quality results. I think it would be useful to have a general-purpose solution along these lines. That is, we want a single function that can be used as a drop-in replacement for RegionPlot3D and will automatically produce high-quality results by setting up the appropriate instances of ContourPlot3D.


Here is a test example, inspired by this post:


RegionPlot3D[1/4 <= x^2 + y^2 + z^2 <= 1 && (x <= 0 || y >= 0),
 {x, -1, 1}, {y, -1, 1}, {z, -1, 1}]

enter image description here


It should look more like this (created by increasing PlotPoints, and even then the edges are not perfectly sharp):


enter image description here




Answer



This is based on Rahul's ideas, but a different implementation:


contourRegionPlot3D[region_, {x_, x0_, x1_}, {y_, y0_, y1_}, {z_, z0_, z1_}, 
  opts : OptionsPattern[]] := Module[{reg, preds},
  reg = LogicalExpand[region && x0 <= x <= x1 && y0 <= y <= y1 && z0 <= z <= z1];
  preds = Union@Cases[reg, _Greater | _GreaterEqual | _Less | _LessEqual, -1];
  Show @ Table[ContourPlot3D[
     Evaluate[Equal @@ p], {x, x0, x1}, {y, y0, y1}, {z, z0, z1}, 
     RegionFunction -> Function @@ {{x, y, z}, Refine[reg, p] && Refine[! reg, ! p]},
     opts], {p, preds}]]


Examples:


contourRegionPlot3D[
 (x < 0 || y > 0) && 0.5 <= x^2 + y^2 + z^2 <= 0.99,
 {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, Mesh -> None]

enter image description here


contourRegionPlot3D[
 x^2 + y^2 + z^2 <= 2 && x^2 + y^2 <= (z - 1)^2 && Abs@x >= 1/4,
 {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, Mesh -> None]


enter image description here


contourRegionPlot3D[
 x^2 + y^2 + z^2 <= 0.4 || 0.01 <= x^2 + y^2 <= 0.05,
 {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, Mesh -> None, PlotPoints -> 50]

enter image description here



Firstly LogicalExpand is used to split up multiple inequalities into a combination of single inequalities, for example to convert 0 < x < 1 into 0 < x && x < 1.


Like Rahul's code, an inequality like x < 1 is converted to the equality x == 1 to define a part of the surface enclosing the region. We do not generally want the entire x == 1 plane though, only that part for which the true/false value of the region function is determined solely by the true/false value of x < 1.



This is done by plotting the surface with a RegionFunction like this:


Refine[reg, p] && Refine[! reg, ! p]

which is equivalant to the predicate "reg is true when p is true, and reg is false when p is false"


Comments

Post a Comment

Popular posts from this blog

plotting - Filling between two spheres in SphericalPlot3D

Manipulate[ SphericalPlot3D[{1, 2 - n}, {θ, 0, Pi}, {ϕ, 0, 1.5 Pi}, Mesh -> None, PlotPoints -> 15, PlotRange -> {-2.2, 2.2}], {n, 0, 1}] I cant' seem to be able to make a filling between two spheres. I've already tried the obvious Filling -> {1 -> {2}} but Mathematica doesn't seem to like that option. Is there any easy way around this or ... Answer There is no built-in filling in SphericalPlot3D . One option is to use ParametricPlot3D to draw the surfaces between the two shells: Manipulate[ Show[SphericalPlot3D[{1, 2 - n}, {θ, 0, Pi}, {ϕ, 0, 1.5 Pi}, PlotPoints -> 15, PlotRange -> {-2.2, 2.2}], ParametricPlot3D[{ r {Sin[t] Cos[1.5 Pi], Sin[t] Sin[1.5 Pi], Cos[t]}, r {Sin[t] Cos[0 Pi], Sin[t] Sin[0 Pi], Cos[t]}}, {r, 1, 2 - n}, {t, 0, Pi}, PlotStyle -> Yellow, Mesh -> {2, 15}]], {n, 0, 1}]
Post a Comment
Read more

plotting - Plot 4D data with color as 4th dimension

I have a list of 4D data (x position, y position, amplitude, wavelength). I want to plot x, y, and amplitude on a 3D plot and have the color of the points correspond to the wavelength. I have seen many examples using functions to define color but my wavelength cannot be expressed by an analytic function. Is there a simple way to do this? Answer Here a another possible way to visualize 4D data: data = Flatten[Table[{x, y, x^2 + y^2, Sin[x - y]}, {x, -Pi, Pi,Pi/10}, {y,-Pi,Pi, Pi/10}], 1]; You can use the function Point along with VertexColors . Now the points are places using the first three elements and the color is determined by the fourth. In this case I used Hue, but you can use whatever you prefer. Graphics3D[ Point[data[[All, 1 ;; 3]], VertexColors -> Hue /@ data[[All, 4]]], Axes -> True, BoxRatios -> {1, 1, 1/GoldenRatio}]
Post a Comment
Read more

plotting - Mathematica: 3D plot based on combined 2D graphs

I have several sigmoidal fits to 3 different datasets, with mean fit predictions plus the 95% confidence limits (not symmetrical around the mean) and the actual data. I would now like to show these different 2D plots projected in 3D as in but then using proper perspective. In the link here they give some solutions to combine the plots using isometric perspective, but I would like to use proper 3 point perspective. Any thoughts? Also any way to show the mean points per time point for each series plus or minus the standard error on the mean would be cool too, either using points+vertical bars, or using spheres plus tubes. Below are some test data and the fit function I am using. Note that I am working on a logit(proportion) scale and that the final vertical scale is Log10(percentage). (* some test data *) data = Table[Null, {i, 4}]; data[[1]] = {{1, -5.8}, {2, -5.4}, {3, -0.8}, {4, -0.2}, {5, 4.6}, {1, -6.4}, {2, -5.6}, {3, -0.7}, {4, 0.04}, {5, 1.0}, {1, -6.8}, {2, -4.7}, {3, -1.
Post a Comment
Read more