Coordinate Transformations
Jason Rupert
Hello All,
I have crusied the web, read the Mathematica help, and looked at many of the
posted email questions but none answer my question. Maybe ya'll can help. I
am working on a project that involves several different coordinate systems.
These coordinate systems are arbitrarily set-up by me to easyily describe
that part of the system. I will have to be able to write the position,
velocity, acceleration, etc. vectors in terms of these several different
coordinates. All the unit vectors are orthogonal and right-handed. I also
have all the angles involved in the rotation from one coordinate system to
another. Is their any way to set-up a rotation matrices so that this can
easily be done? Does any one have an example that I can follow? Any help you
can offer will be greatly appreciated. Can you refrence me to any sources to
examine. Are there any Mathematica user groups where daily postings are
made?
Thanks again,
Jason Rupert
UAH Huntsville, AL
Reply to: rup...@email.uah.edu
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Jens-Peer Kuska
I assume you have a set of second order differential equations deqn
depending
on the coordinates q[1][t], q[2][t], q[3][t], ...
and wish to transform to the new coordinates Q[1][t], Q[2][t], ...
You can express q[_][t] as functions of Q[_}[t] by the relations
q[k][t]=t[k][Q[1][t],Q[2][t],..] In Mathematica this ist written as
trules={q[1][t]->t[1][Q[1][t],..],..}
since Mathematica does not replace the derivatives you must generate the
rules for the
first and scond derivative by
dtrules=Join @@ ({#,D[#,t],D[#,{t,2}]} & /@ trules)
now you can transform your differential equations with
deqnQ=deqn /. dtrules
for the next coordinate system you can repeat the steps and nest all the
transformations.
Please notice that this not assume that the transformations or the
differential
equations are linear, it works for any set of variable transformations.
For a linear transformation your q=T.Q (q and Q are vectors T is a
matrix)
you get some simple linear combinations.
Hope that
helps
Jens
The future steps depend on your differential equations, because it is
not
John Doty
the other's coordinates, it's easy: the vectors are just the rows (or
columns, depending on which direction you want to make the
transformation) of the rotation matrix (in the obvious order).
As a major application area for this math is spacecraft attitude
control, I suggest you ask some rocket scientist to lend you a copy of
Wertz (shouldn't be hard to find a rocket scientist in Huntsville :-).
Jason Rupert wrote:
>
> Please Reply to: rup...@email.uah.edu
> Hello All,
> I have crusied the web, read the Mathematica help, and looked at many of the
> posted email questions but none answer my question. Maybe ya'll can help. I
> am working on a project that involves several different coordinate systems.
> These coordinate systems are arbitrarily set-up by me to easyily describe
> that part of the system. I will have to be able to write the position,
> velocity, acceleration, etc. vectors in terms of these several different
> coordinates. All the unit vectors are orthogonal and right-handed. I also
> have all the angles involved in the rotation from one coordinate system to
> another. Is their any way to set-up a rotation matrices so that this can
> easily be done? Does any one have an example that I can follow? Any help you
> can offer will be greatly appreciated. Can you refrence me to any sources to
> examine. Are there any Mathematica user groups where daily postings are
> made?
>
> Thanks again,
> Jason Rupert
> UAH Huntsville, AL
>
> Reply to: rup...@email.uah.edu
>
> ______________________________________________________
> Get Your Private, Free Email at http://www.hotmail.com
--
John Doty "You can't confuse me, that's my job."
Home: j...@w-d.org
Work: j...@space.mit.edu
Dave Grosvenor
Hi Jason
This is my solution for the problem I think you have.
I will post some mathematica code to help you but you still need to do
some work.
Problem
--------
Calculate the rotation transformation between two right-handed
coordinate systems.
The appraoch is in two stages
i) Pick on one pair of corresponding axes, determine a transform which
will align the z-axes and the x-y planes. This is done by rotating about
the cross product of the corresponding z-axes, by the angle between the
two vectors defining the z-axes. Call this the intermediate coordinate
system where the z-axis are aligned but the x-y axes may be an arbitrary
rotation different.
ii) Now align the x-y planes of the two cordinate systems. This is done
by rotating about the z-axis (in the target and intermediate coordinate
system), by the angle between the intermediate x -axis and the target
x-axis. This is where the assumption that both coordinate systems are
right handed gets used.
QED
There is loads of scope for going wrong (signs of angles,cross products
handedness of coordinate system,etc..) but it should be obvious when you
get it wrong. So you have plenty to do!
So making a general routine is a pain.
Rotation about an axis and point
----------------------------------
Approach is to define a 3D coordinate transformation by specifying axis
of rotation (a point {0,0,0} and a vector), plus an angle of rotation.
This is standard (see Graphic Gems I "Rotation Tools", Michael E.
Pique.p466.)
see yuz
Dave Grosvenor
Mathematica code
=================
The two stages for the approach are something like below:-
i) Mat =
rotatePointVector[{0,0,0},angleBetweenVectors[{ax,ay,az},{bx,by,bz}],
Cross[{ax,ay,az},{bx,by,bz}]]
where a and b are the vector of the corresponding axes
ii) let {icx,icy,icz} = deHomogenise[Mat . homogenise[{cx,cy,cz]]
where homogenise[{x_,y_,z_}]:= {x,y,z,1}
deHomogenise[{x_,y_,z_,w_}]:= {x/w,y/w,z/w}
so now we rotate about
Mat1 =
rotatePointVector[{0,0,0},angleBetweenVectors[{icx,icy,icz},{dx,dy,dz}],
Cross[{icx,icy,icz},{dx,dy,dz}]]
{dx,dy,dz} is the corresponding axis in the target
Angle between Vector
---------------------
These expressions are derived by dot and cross product for vectors.
magnitude[v_]:= Sqrt[v.v]
sineBetweenVectors[u_,v_]:=
magnitude[Cross[u,v]]/(magnitude[u] magnitude[v])
cosineBetweenVectors[u_,v_]:= Dot[u,v]/(magnitude[u] magnitude[v])
angleBetweenVectors[u_,v_]:=With[{acos=cosineBetweenVectors[u,v],
asin=sineBetweenVectors[u,v]},
ArcTan[acos,asin]]
Rotation 4x4 matrix routine
-----------------------------
rotate routine:- from graphics gems, this routine is more general than
required (it allows a translate to an arbitrary point)--but is slow as
it does not pre-multiply the translation.
Rotate3x3[{x_,y_,z_},\[Theta]_]:=
With[{c=Cos[\[Theta]],s=Sin[\[Theta]],t=1-Cos[\[Theta]]},
{{t x^2 + c,t x y + s z, t x z - s y},
{t x y - s z, t y^2 + c,t y z + s x},
{t x z + s y, t y z - s x, t z^2 + c}}]
Now a function to uplift the rotate matrix to a 4x4 transformation
uplift[{{a00_,a01_,a02_},{a10_,a11_,a12_},{a20_,a21_,a22_}}]:=
{{a00,a01,a02,0},{a10,a11,a12,0},{a20,a21,a22,0},{0,0,0,1}}
In[29]:=
MatrixForm[uplift[Rotate3x3[{x,y,z},theta]]]
Out[29]//MatrixForm=
\!\(\*
TagBox[
RowBox[{"(", GridBox[{
{\(x\^2\ \((1 - Cos[theta])\) + Cos[theta]\),
\(x\ y\ \((1 - Cos[theta])\) + z\ Sin[theta]\),
\(x\ z\ \((1 - Cos[theta])\) - y\ Sin[theta]\), "0"},
{\(x\ y\ \((1 - Cos[theta])\) - z\ Sin[theta]\),
\(y\^2\ \((1 - Cos[theta])\) + Cos[theta]\),
\(y\ z\ \((1 - Cos[theta])\) + x\ Sin[theta]\), "0"},
{\(x\ z\ \((1 - Cos[theta])\) + y\ Sin[theta]\),
\(y\ z\ \((1 - Cos[theta])\) - x\ Sin[theta]\),
\(z\^2\ \((1 - Cos[theta])\) + Cos[theta]\), "0"},
{"0", "0", "0", "1"}
}], ")"}],
(MatrixForm[ #]&)]\)
Now we define a function to define a translate transformation as a 4x4
matrix
In[231]:=
translate[{tx_,ty_,tz_}] := {{1,0,0,tx},{0,1,0,ty},{0,0,1,tz},{0,0,0,1}}
In[232]:=
translate[{ax,ay,az}]
Out[232]=
{{1,0,0,ax},{0,1,0,ay},{0,0,1,az},{0,0,0,1}}
Conceptually we apply a transformation to move the origin to the desired
point and then perform the rotate and then we change the coordinate
system
back to undo the translate. We calculate and simplify this 4x4
transformation
In[32]:=
FullSimplify[translate[{tx,ty,tz}] . uplift[Rotate3x3[{x,y,z},theta]] .
translate[{-tx,-ty,-tz}]]
Out[32]=
\!\({{x\^2 + Cos[theta] - x\^2\ Cos[theta],
x\ y - x\ y\ Cos[theta] + z\ Sin[theta],
x\ z - x\ z\ Cos[theta] - y\ Sin[theta],
\((tx\ \((\(-1\) + x\^2)\) + x\ \((ty\ y + tz\ z)\))\)\
\((\(-1\) + Cos[theta])\) + \((tz\ y - ty\ z)\)\ Sin[theta]},
{
x\ y - x\ y\ Cos[theta] - z\ Sin[theta],
y\^2 + Cos[theta] - y\^2\ Cos[theta],
y\ z - y\ z\ Cos[theta] + x\ Sin[theta],
\((tx\ x\ y + ty\ \((\(-1\) + y\^2)\) + tz\ y\ z)\)\
\((\(-1\) + Cos[theta])\) + \((\(-tz\)\ x + tx\ z)\)\
Sin[theta]}, {
x\ z - x\ z\ Cos[theta] + y\ Sin[theta],
y\ z - y\ z\ Cos[theta] - x\ Sin[theta],
z\^2 + Cos[theta] - z\^2\ Cos[theta],
\((\((tx\ x + ty\ y)\)\ z + tz\ \((\(-1\) + z\^2)\))\)\
\((\(-1\) + Cos[theta])\) + \((ty\ x - tx\ y)\)\ Sin[theta]},
{0,
0, 0, 1}}\)
In[33]:=
MatrixForm[%%]
Out[33]//MatrixForm=
\!\(\*
TagBox[
RowBox[{"(", GridBox[{
{"1", "0", "0", "ax"},
{"0", "1", "0", "ay"},
{"0", "0", "1", "az"},
{"0", "0", "0", "1"}
}], ")"}],
(MatrixForm[ #]&)]\)
Then (lazilly) just define the desired routine as the un-simplified
matrix multiplications -- the assumption is that the vector defining the
axis of rotation is a unit vector. .
Clear[rotatePointVector]
rotatePointVector[{tx_,ty_,tz_},theta_,{x0_,y0_,z0_}]:=
With[{unit = unitVector[{x0,y0,z0}]},
With[{x=unit[[1]],y=unit[[2]],z=unit[[3]]},
translate[{tx,ty,tz}] . uplift[Rotate3x3[{x,y,z},theta]] .
translate[{-tx,-ty,-tz}]]]