Skip to main content

matrix - How to transform general Ellipsoid into rotated axis-oriented one?


Recently, I discovered a long-standing bug in the rendering of Disk and Circle primitives after applying GeometricTransformation with a matrix as the second argument. This bug affects rendering of general ellipsoids of the form Ellipsoid[p, Σ], but does not affect axes-oriented ellipsoids of the form Ellipsoid[p, {r1, …}], because the latter are directly translated into the corresponding Disk objects with the same syntax and arguments. Applying Rotate (or GeometricTransformation with RotationTransform as the second argument) to the axis-oriented ellipsoids is a workaround for the bug. But for this we should be able to obtain from the matrix Σ the corresponding semiaxes lengths {r1, …} and the rotation angle Θ (for the 3D case, we also need the rotation axis w).


There is an elegant and efficient built-in way to perform the opposite task. For the 2D case, we have:


TransformedRegion[Ellipsoid[{x, y}, {r1, r2}], RotationTransform[Θ, {x, y}]]


Ellipsoid[{x, y}, 
   {{r1^2 Cos[Θ]^2 + r2^2 Sin[Θ]^2, r1^2 Cos[Θ] Sin[Θ] - r2^2 Cos[Θ] Sin[Θ]}, 
    {r1^2 Cos[Θ] Sin[Θ]] - r2^2 Cos[Θ] Sin[Θ], r2^2 Cos[Θ]^2 + r1^2 Sin[Θ]^2}}]


But I failed to find a way to transform Ellipsoid[p, Σ] into Rotate[Ellipsoid[p, {r1, …}], Θ, p] (or another suitable syntax form of Rotate or RotationTransform). Is it possible to do this in an elegant and efficient way (2D and 3D cases)?



Answer



We can employ Eigensystem in order to find the principal axes. While the rotation angle can be easily found in 2D case, the 3D case involves also the detection of the rotation axis; this is done by utilizing NullSpace.


2D case


SeedRandom[666];
Σ = RandomReal[{-1, 1}, {2, 2}];
Σ = Transpose[Σ].Σ;
p = RandomReal[{-1, 1}, {2}]; 

{λ, U} = Eigensystem[Σ];

r = Sqrt[λ];
rot = Det[U] Transpose[U];
θ = ArcTan @@ (rot[[All, 1]]);
Max[Abs[RotationMatrix[θ] - rot]]
Graphics[{
  FaceForm[ColorData[97][2]], Ellipsoid[p, Σ],
  EdgeForm[{Thickness[0.015], ColorData[97][1]}], 
  Rotate[#, θ, p] &@Ellipsoid[p, r]
  }]



0.



enter image description here


3D case


SeedRandom[666];
Σ = RandomReal[{-1, 1}, {3, 3}];
Σ = Transpose[Σ].Σ;
p = RandomReal[{-1, 1}, {3}];    


{λ, U} = Eigensystem[Σ];
r = Sqrt[λ];
rot = Det[U] Transpose[U];
w = NullSpace[rot - IdentityMatrix[3]][[1]];
V = RotationMatrix[{{1, 0, 0}, w}];
θ = -ArcTan @@ (Transpose[V].rot.V)[[2, 2 ;; 3]];
Max[Abs[RotationMatrix[θ, w] - rot]]
Show[
 Graphics3D[{
   FaceForm[ColorData[97][2]], 

   Ellipsoid[p, Σ],
   ColorData[97][1],
   Rotate[#, θ, w, p] &@Ellipsoid[p, r]
   }, Lighting -> "Neutral"]
 ]


7.77156*10^-16



enter image description here



Since RotationMatrix is the bottleneck here, I also link to this post of mine.


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 - 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]],
Post a Comment
Read more