The Z, Y', X'' rotation scheme, intrinsic or extrinsic, sounds like an application of Euler (sounds like "Oiler") angles [1,2].
Tom Kunicki
Center for Integrated Data Analytics
U.S. Geological Survey
8505 Research Way
Middleton, WI 53562
[1]
http://en.wikipedia.org/wiki/Euler_angles
[2]
http://mathworld.wolfram.com/EulerAngles.html
On Mar 6, 2013, at 2:17 PM, Seth McGinnis <mcginnis at ucar.edu> wrote:
> So there are an infinite number of ways to decompose a 3-D orientation into
> three orthogonal rotations. Specifying the projection in terms of the final
> location of the north pole and an optional rotation around that point means
> that CF is actually agnostic about exactly how you get there.
>
> That said, given that it's being specified in that way, the decomposition that
> maps most straightforwardly to those parameters is a Z-Y-X rotation sequence
> using an extrinsic (space-fixed) coordinate system:
>
> First rotate by angle phi around the north pole in space-fixed coordinates.
> Then rotate by angle theta around 0N 90E in space-fixed coordinates.
> Finally, rotate by angle psi around 0N 0E in space-fixed coordinates.
>
> The location of the grid's north pole in the original lat-lon coordinate system
> is now at longitude = phi, colatitude = minus theta.
>
> (I'm not sure how psi relates to north_pole_grid_longitude mathematically; it
> defaults to zero and I've never seen it in use, so I think it's included mostly
> for completeness. I agree that we may well want to change it to something more
> obvious like an azimuth angle.)
>
> Switching to the GRIB2 convention where you use the south pole instead of the
> north should be just a matter of adding 180 degrees to phi and negating theta.
>
> Cheers,
>
> --Seth
>
>
> On Wed, 6 Mar 2013 18:29:47 +0000
> David Hassell <d.c.hassell at reading.ac.uk> wrote:
>> Hello,
>>
>> It is a bit of mess! As I understand it, the full rotation described
>> is a sequence of rotations about three different axes:
>>
>> Z = 90S - 90N
>> Y' = 90W' - 90E'
>> X" = 0E" - 180E"
>>
>> where it is understood that the definitions of the N-S and W-E axes
>> change after each rotation (hence the primes and double
>> primes). Therefore the order in which they are done matters.
>>
>> I suspect that the usual and assumed order is Z, Y', X"?
>>
>>> From the GRIB-2 stuff John posted the north_pole_grid_longitude gives
>> the rotation about the X" axis.
>>
>> Something like John's "Angle of rotation" seems right to me.
>>
>> The apparent lack of consistency between the parameter names irks
>> me. Perhaps one solution could be to:
>>
>> i) Add some text to the conventions state the order (Z, Y', X", say)
>> and direction of rotations.
>>
>> ii) add three new, consistent, self-describing parameters (e.g.)
>>
>> angle_of_rotation_z
>> angle_of_rotation_y
>> angle_of_rotation_x
>>
>> iii) allow aliases for backwards compatibility
>>
>> grid_north_pole_longitude <=> angle_of_rotation_z
>> grid_north_pole_latitude <=> angle_of_rotation_y
>> north_pole_grid_longitude <=> angle_of_rotation_x
>>
>> Or would that just obfuscate things even more?
>>
>> All the best,
>>
>> David
>>
>> P.S. If you have a copy available, there are some nice descriptions in
>> "Coordinate Systems and Map Projections" by D. H. Maling
>>
>> --
>> David Hassell
>> National Centre for Atmospheric Science (NCAS)
>> Department of Meteorology, University of Reading,
>> Earley Gate, PO Box 243,
>> Reading RG6 6BB, U.K.
>>
>> Tel : 0118 3785613
>> Fax : 0118 3788316
>> E-mail: d.c.hassell at reading.ac.uk
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Received on Sat Mar 09 2013 - 18:53:16 GMT
