plotting - Modular surface of tri-focal Cassini curve ContourPlot3D missing feet

I am wondering why the following fails to cover the surface at points near $k = 0$.

c[z_] := (z + 1) (z - 1) (z + 1 + I);


ContourPlot3D[Abs[c[x + I y]] == k^3, {x, -2.5, 2}, {y, -2, 2}, 
{k, 0, 1.75}, Background -> White, AxesLabel -> {"x", "y", "k"}]

Mathematica graphics

Furthermore, when the left hand side is complex expanded and the equation is rearranged with $2xy+k^3$ on the right, the curve looks significantly different, with one large and one small foot, both touching at zero. Why is that?

Answer

ContourPlot3D is not very good at resolving thin features, because it only knows that the feature exists when one of the sampling points happens to land inside it. In general, one thing you can do is to increase PlotPoints, which improves the plot but takes a very long time.

ContourPlot3D[
 Abs[c[x + I y]] == k^3, {x, -2.5, 2}, {y, -2, 2}, {k, 0, 1.75}, 
 Background -> White, AxesLabel -> {"x", "y", "k"}, PlotPoints -> 20]

enter image description here

In this particular case, though, your plot is equivalent to $k = |c(x+iy)|^{1/3}$, so you could just use Plot3D instead. This is much faster because it only has to sample the two-dimensional $xy$ plane rather than the three-dimensional $xyk$ space. Then you can afford to make MaxRecursion quite large and it's still really quick to plot.

Plot3D[Abs[c[x + I y]]^(1/3), {x, -2.5, 2}, {y, -2, 2}, PlotRange -> {0, 1.75},
 Background -> White, AxesLabel -> {"x", "y", "k"}, ClippingStyle -> None,
 BoxRatios -> 1, MeshFunctions -> {#1 &, #2 &, #3 &}, MaxRecursion -> 5]

enter image description here

(I've added a few options to make it look like your original plot.)

Comments

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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}]

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}]

plotting - Adding a thick curve to a regionplot

Suppose we have the following simple RegionPlot: f[x_] := 1 - x^2 g[x_] := 1 - 0.5 x^2 RegionPlot[{y < f[x], f[x] < y < g[x], y > g[x]}, {x, 0, 2}, {y, 0, 2}] Now I'm trying to change the curve defined by $y=g[x]$ into a thick black curve, while leaving all other boundaries in the plot unchanged. I've tried adding the region $y=g[x]$ and playing with the plotstyle, which didn't work, and I've tried BoundaryStyle, which changed all the boundaries in the plot. Now I'm kinda out of ideas... Any help would be appreciated! Answer With f[x_] := 1 - x^2 g[x_] := 1 - 0.5 x^2 You can use Epilog to add the thick line: RegionPlot[{y < f[x], f[x] < y < g[x], y > g[x]}, {x, 0, 2}, {y, 0, 2}, PlotPoints -> 50, Epilog -> (Plot[g[x], {x, 0, 2}, PlotStyle -> {Black, Thick}][[1]]), PlotStyle -> {Directive[Yellow, Opacity[0.4]], Directive[Pink, Opacity[0.4]],

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