| Re: 3D vector rotation to the Z axis [message #73974] |
Tue, 14 December 2010 12:28  |
David Fanning
Messages: 11724
Registered: August 2001
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Senior Member |
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MartyL writes:
> Any help appreciated.
I don't have any suggestions. You obviously know more
about this than I do. Just an observation. I know any
time I try to translate matrix operations out of a book
into IDL my head swells up to about three times its
normal size trying to keep columns and rows straight.
That said, using the # operator, rather than the ## operator
always throws me back from discovering the solution at LEAST
two days, sometimes more. :-(
Cheers,
David
--
David Fanning, Ph.D.
Fanning Software Consulting, Inc.
Coyote's Guide to IDL Programming: http://www.dfanning.com/
Sepore ma de ni thui. ("Perhaps thou speakest truth.")
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| Re: 3D vector rotation to the Z axis [message #73975 is a reply to message #73974] |
Tue, 14 December 2010 12:20  |
MartyL
Messages: 3
Registered: December 2010
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Junior Member |
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On Dec 13, 7:23 pm, Paulo Penteado <pp.pente...@gmail.com> wrote:
> On Dec 14, 1:11 am, MartyL <mlandsf...@yahoo.com> wrote:
>
>> I would like to rotate and arbitrary 3D vector to be aligned with the
>> Z axis. I can translate and scale the vector but I can not find a
>> formula to create the rotation matrix to perform the rotation. It
>> sounds simple but I can't seem to find one.
>> Thanks
>
> Do you mean something like
>
> IDL> t3d,identity(4),matrix=matrix,rotate=[45d0,0d0,0d0]
> IDL> print,matrix
> 1.0000000 0.0000000 0.0000000 0.0000000
> 0.0000000 0.70710678 -0.70710678 0.0000000
> 0.0000000 0.70710678 0.70710678 0.0000000
> 0.0000000 0.0000000 0.0000000 1.0000000
Yes, I am trying to get the transformation matrix for an arbitrary 3D
vector. So the problem is that I need to find the degrees which to
rotate. If I have the vector:
pts = [[5.1,20.4,-12.2],[15.0,55.5,30.4]]
I can translate that to the origin
pts_trans = [[0.0, 0.0, 0.0],[9.9,35.1,42.6]]
and normalize it to a unit vector
pts_norm = [[0.0, 0.0, 0.0],[0.176,0.625,0.759]]
but then I need to determine the rotation angles to rotate it to [0.0,
0.0, 1.0] or the Z axis.
I have found this solution in another news group but it did not
produce the correct results.
--------------------------------
Let V1 = [ x1, y1, z1 ], V2 = [ x2, y2, z2 ]. Assume V1 and V2 are
already normalized.
V3 = normalize(cross(V1, V2)). (the normalization here is
mandatory.)
V4 = cross(V3, V1).
[ V1 ]
M1 = [ V4 ]
[ V3 ]
cos = dot(V2, V1), sin = dot(V2, V4)
[ cos sin 0 ]
M2 = [ -sin cos 0 ]
[ 0 0 1 ]
The sought transformation matrix is just M1^-1 * M2 * M1. This might
well be a standard-text solution.
--------------------------------
I used [0.176,0.625,0.759], from pts_norm, as V1 and [0.0, 0.0, 1.0],
the Z axis, as V2.
My implementation is this:
v1 = [0.176,0.625,0.759D]
v2 = [0.0, 0.0, 1.0D]
v = CROSSP(v1, v2)
v3 = v/NORM(v)
v4 = CROSSP(v3, v1)
m1 = DBLARR(3,3)
m1[*,0] = [v1]
m1[*,1] = [v4]
m1[*,2] = [v3]
rot_cos = TRANSPOSE(v1)#v2
rot_sin = TRANSPOSE(v2)#v4
m2 = DBLARR(3,3)
m2[*,0] = [rot_cos, rot_sin, 0.0]
m2[*,1] = [-rot_sin, rot_cos, 0.0]
m2[*,2] = [0.0, 0.0, 1.0]
m = MATRIX_MULTIPLY(INVERT(m1), m2)
m4 = MATRIX_MULTIPLY(m, m1)
v1_trans = MATRIX_MULTIPLY(v1, m4)
print, v1_trans
IDL output is:
-0.0875152
0.933033
0.341724
which should have been:
0.0
0.0
1.0
I am wondering if I have the order of vectors and matrices wrong in
MATRIX_MULTIPLY.
Any help appreciated.
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| Re: 3D vector rotation to the Z axis [message #74027 is a reply to message #73974] |
Thu, 16 December 2010 11:48 |
James[2]
Messages: 44
Registered: November 2009
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Member |
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Do you have to use a rotation matrix? Quaternions (http://
en.wikipedia.org/wiki/Quaternion) are more numerically stable, and
there is an easy-to-use library by Craig Markwardt at
http://www.physics.wisc.edu/~craigm/idl/math.html. I was just using
them to draw some rotating polyhedra in IDL, and they work great!
Your code would look like this:
;define the axis of rotation
rotaxis = crossp (input, [0,0,1])
;find the angle of rotation
rotangle = transpose(input) # [0,0,1]
;make the quaternion
q = qtcompose(rotaxis, rotangle)
;do it
rotated = qtvrot(input, q)
;there is even a routine to create a rotation matrix from the
quaternion:
rmatx = qtmat(q)
by the way, I like the # operator. It lets you treat 1D arrays as
column vectors; then defining a matrix is just concatenating a group
of column vectors across the second dimension. Matrix-by-vector
multiplication works like you expect.
- James
On Dec 14, 12:28 pm, David Fanning <n...@dfanning.com> wrote:
> MartyL writes:
>> Any help appreciated.
>
> I don't have any suggestions. You obviously know more
> about this than I do. Just an observation. I know any
> time I try to translate matrix operations out of a book
> into IDL my head swells up to about three times its
> normal size trying to keep columns and rows straight.
>
> That said, using the # operator, rather than the ## operator
> always throws me back from discovering the solution at LEAST
> two days, sometimes more. :-(
>
> Cheers,
>
> David
>
> --
> David Fanning, Ph.D.
> Fanning Software Consulting, Inc.
> Coyote's Guide to IDL Programming:http://www.dfanning.com/
> Sepore ma de ni thui. ("Perhaps thou speakest truth.")
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| Re: 3D vector rotation to the Z axis [message #74178 is a reply to message #74027] |
Mon, 03 January 2011 17:17 |
MartyL
Messages: 3
Registered: December 2010
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Junior Member |
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On Dec 16 2010, 11:48 am, James <donje...@gmail.com> wrote:
> Do you have to use a rotation matrix? Quaternions (http://
> en.wikipedia.org/wiki/Quaternion) are more numerically stable, and
> there is an easy-to-use library by Craig Markwardt athttp://www.physics.wisc.edu/~craigm/idl/math.html. I was just using
> them to draw some rotating polyhedra in IDL, and they work great!
> Your code would look like this:
>
> ;define the axis of rotation
> rotaxis = crossp (input, [0,0,1])
>
> ;find the angle of rotation
> rotangle = transpose(input) # [0,0,1]
>
> ;make the quaternion
> q = qtcompose(rotaxis, rotangle)
>
> ;do it
> rotated = qtvrot(input, q)
>
> ;there is even a routine to create a rotation matrix from the
> quaternion:
> rmatx = qtmat(q)
>
> by the way, I like the # operator. It lets you treat 1D arrays as
> column vectors; then defining a matrix is just concatenating a group
> of column vectors across the second dimension. Matrix-by-vector
> multiplication works like you expect.
>
> - James
>
James,
I do not need to use a rotation matrix per se, I just need to rotate
the vectors to the Z axis.
I copied your code and made a quick test procedure and I am not
getting the results I would expect. Here is the code I used:
PRO Rotate_to_Z_axis, input
print, 'input = ', input
;define the axis of rotation
rotaxis = crossp (input, [0,0,1])
;find the angle of rotation
rotangle = transpose(input) # [0,0,1]
;make the quaternion
q = qtcompose(rotaxis, rotangle)
;do it
rotated = qtvrot(input, q)
print, 'rotated = ', rotated
end
I then input the following at the command prompt:
rotate_to_z_axis, [cos(!dtor*30.0), sin(!dtor*30.0), 0.0]
and the results are:
input = 0.866025 0.500000 0.00000
rotated = 0.866025 0.500000 0.00000
Shouldn't the rotated vector be [0,0,1] if it is being rotated to the
Z axis? That is what I am trying to do at least.
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