Mathematica Rotation of 3D Plot using Euler angles

[Mr. Rho]

So, I'm trying to plot a 3D "dipole" (an arrow with a small torus around it basically) in mathematica, and want to rotate it according to Euler angles...

I use this code for the rotation matrix:

rot[a, b, g] := RotationMatrix[g, {1, 0, 0}].RotationMatrix[b, {0, 1, 0}].RotationMatrix[a, {0, 0, 1}]

and I obtain the standard Euler angles rotation matrix.

For the arrow and the torus I'm using:

arrow[x_, y_, z_,a_, b_, g_, r_] := Graphics3D[{Red, Cone[{rot[a, b, g].{0, 0, 2^4 r} + {x, y, z}, rot[a, b, g].{0, 0, 2^4 r + 2^2 r} + {x, y, z}}, 3 r], Cylinder[{rot[a, b, g].{0, 0, 0} + {x, y, z}, rot[a, b, g].{0, 0, 2^4 r} + {x, y, z}}, r]}]

torus[x_, y_, z_, a_, b_, g_, r_] := ParametricPlot3D[rot[a, b, g].{(r*Cos[m] + 4 r) Cos[v], (r*Cos[m] + 4 r) Sin[v], r*Sin + z} + {x, y, z}, {m, 0, 2 Pi}, {v, 0, 2 Pi}, Mesh -> None, PlotStyle -> {Green, Opacity[.25]}]

(Note that I also introduced a boost) The next step is to show both plots using manipulate to change the angle and boost values, the code I use is simply:

Manipulate[Show[arrow[x,y,z-8,a,b,g,1], torus[x,y,z,a,b,g,1]], {x,0,1}, {y,0,1}, {z,0,1}, {a,0,2Pi}, {b,0,2Pi}, {g,0,2Pi}]

Everything is ok until here, the plot of a dipole is shown and it moves smootly. The problem is when I start introducing more and more arrows and torus. If I want, for example, 4 dipoles, the plot gets really really slow...

So, any plot optimization tips?

Thank you for your help!

 

[J Gregory Moxness]

I have seen complex combinations of plots bog down in a Manipulate. It seems doing only 4 dipoles (each with a Graphic3D and a ParametricPlot3D) shouldn't be too bad, but that may depend on the CPU/cores/memory of your system. Did you integrate each of the 4 within the one Graphic3D and one ParametricPlot3D or are you processing 8 separate delayed functions and then Show'ing that?

If so, you might also try creating a single (complex) graphic object and use a Rotate command against that (instead of doing all the ParametricPlot3D's for each).

In a worst case scenario when I have hundreds or thousands of objects floating around a 3D space (e.g. where I am doing a parallelized CPU or GPU based simulation of some early inflationary Universe N-Body gravitational model of theoretical particles), I create a 2D raster of it and simply do an Inset of that into the 3D space.

 

[Mr. Rho]

I have seen complex combinations of plots bog down in a Manipulate. It seems doing only 4 dipoles (each with a Graphic3D and a ParametricPlot3D) shouldn't be too bad, but that may depend on the CPU/cores/memory of your system. Did you integrate each of the 4 within the one Graphic3D and one ParametricPlot3D or are you processing 8 separate delayed functions and then Show'ing that?

If so, you might also try creating a single (complex) graphic object and use a Rotate command against that (instead of doing all the ParametricPlot3D's for each).

In a worst case scenario when I have hundreds or thousands of objects floating around a 3D space (e.g. where I am doing a parallelized CPU or GPU based simulation of some early inflationary Universe N-Body gravitational model of theoretical particles), I create a 2D raster of it and simply do an Inset of that into the 3D space.

Hi Gregory, thank you for your reply. The solution for me was processing the 4 separate dipoles and then showing that as you say, the problem was that I was trying to plot all the four dipoles at the same time