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

Popular posts from this blog

mathematical optimization - Minimizing using indices, error: Part::pkspec1: The expression cannot be used as a part specification

I want to use Minimize where the variables to minimize are indices pointing into an array. Here a MWE that hopefully shows what my problem is. vars = u@# & /@ Range[3]; cons = Flatten@ { Table[(u[j] != #) & /@ vars[[j + 1 ;; -1]], {j, 1, 3 - 1}], 1 vec1 = {1, 2, 3}; vec2 = {1, 2, 3}; Minimize[{Total@((vec1[[#]] - vec2[[u[#]]])^2 & /@ Range[1, 3]), cons}, vars, Integers] The error I get: Part::pkspec1: The expression u[1] cannot be used as a part specification. >> Answer Ok, it seems that one can get around Mathematica trying to evaluate vec2[[u[1]]] too early by using the function Indexed[vec2,u[1]] . The working MWE would then look like the following: vars = u@# & /@ Range[3]; cons = Flatten@{ Table[(u[j] != #) & /@ vars[[j + 1 ;; -1]], {j, 1, 3 - 1}], 1 vec1 = {1, 2, 3}; vec2 = {1, 2, 3}; NMinimize[ {Total@((vec1[[#]] - Indexed[vec2, u[#]])^2 & /@ R...

functions - Get leading series expansion term?

Given a function f[x] , I would like to have a function leadingSeries that returns just the leading term in the series around x=0 . For example: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x)] x and leadingSeries[(1/x + 2 + (1 - 1/x^3)/4)/(4 + x)] -(1/(16 x^3)) Is there such a function in Mathematica? Or maybe one can implement it efficiently? EDIT I finally went with the following implementation, based on Carl Woll 's answer: lds[ex_,x_]:=( (ex/.x->(x+O[x]^2))/.SeriesData[U_,Z_,L_List,Mi_,Ma_,De_]:>SeriesData[U,Z,{L[[1]]},Mi,Mi+1,De]//Quiet//Normal) The advantage is, that this one also properly works with functions whose leading term is a constant: lds[Exp[x],x] 1 Answer Update 1 Updated to eliminate SeriesData and to not return additional terms Perhaps you could use: leadingSeries[expr_, x_] := Normal[expr /. x->(x+O[x]^2) /. a_List :> Take[a, 1]] Then for your examples: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x), x] leadingSeries[Exp[x], x] leadingSeries[(1/x + 2 + (1 - 1/x...

What is and isn't a valid variable specification for Manipulate?

I have an expression whose terms have arguments (representing subscripts), like this: myExpr = A[0] + V[1,T] I would like to put it inside a Manipulate to see its value as I move around the parameters. (The goal is eventually to plot it wrt one of the variables inside.) However, Mathematica complains when I set V[1,T] as a manipulated variable: Manipulate[Evaluate[myExpr], {A[0], 0, 1}, {V[1, T], 0, 1}] (*Manipulate::vsform: Manipulate argument {V[1,T],0,1} does not have the correct form for a variable specification. >> *) As a workaround, if I get rid of the symbol T inside the argument, it works fine: Manipulate[ Evaluate[myExpr /. T -> 15], {A[0], 0, 1}, {V[1, 15], 0, 1}] Why this behavior? Can anyone point me to the documentation that says what counts as a valid variable? And is there a way to get Manpiulate to accept an expression with a symbolic argument as a variable? Investigations I've done so far: I tried using variableQ from this answer , but it says V[1...