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EternityZA

Eular angle question

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Hi.

I have a very specific problem. Explaining exactly what Im doing is most likely redundent so Im just going to try and explain what I need as best as possible using a simple example.

Im applying a eular angle to a vector in 3D space by first rotating the vector around the x-axes, then y, then z.
Its a givin that for this particular eular angle the same end result can be achived by only rotating around the z axis by a certian amount. For this givin Eualr angle how would I calculate the neccesary rotation around the Z axis.

Thnx in Advance!

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I could be wrong but it seems like what you are looking for is a quaternion. You can't represent any rotated vector by only rotating it around the Z-Axis. A quaternion is simply a vector in space that can represent your rotation and then an amount of rotation around that axis. But constraining the rotation to only rotating around the Z-Axis will limit your degrees of freedom. Hopefully this is what you were trying to explain. If not, correct me and hopefully I can help.

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Yes rotating around only 1 axis will limit the degrees of freedom but thats fine. As i tried to explain the input eular angle will always fall within the allowed range as a givin. Heres some actual numbers

rotating a point in 3D space with eular angle


x: 180
y: 180
z: 50

in the order of x,y and then z will have the same effect on the point as just rotating around the z-axis by 130 degrees

so i want to know how i can get from

x: 180
y: 180
z: 50

to

x: 0
y: 0
z: 130

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When using column vector notation, the Euler rotation is formulated as
[b]R[/b][sub]z[/sub] * [b]R[/b][sub]y[/sub] * [b]R[/b][sub]x[/sub] =: [b]R[/b][sub]E[/sub]
which you want to equal a single z axis rotation, so that
[b]R[/b][sub]E[/sub] = [b]R[/b][sub]z[/sub]'

This implies 9 equations for a single variable (namely the argument angle to [b]R[/b][sub]z[/sub]'), what obviously can be solved in special situations only. When such a situation is actually given, then you have 4 equations like
cos( z' ) = a
cos( z' ) = b
sin( z' ) = c
-sin( z' ) = d
where the numerical values of a and b as well as those of c and -d are obviously the same; further a and c need to differ in a phase shift of 90 degree.

When you use the arccos or arcsin function for solving, then you get a result that covers a half circle only. So you need to take 2 of the above functions into account. One way to do this is to choose e.g.
tan( z' ) = sin( z' ) / cos( z' ) = a / c
and hence
z' = arctan( a / c )
in principle. However, this formula still has problems, because it (1.) is vulnerable against c==0 and (2.) only 2 of the 4 quadrants can be identified (i.e. the same problem as those above).

Fortunately, most programming languages provide a variant of arctan that internally deals with this problem. This function is usually named atan2. So try
z' = atan2( c, a )
where a and c are picked from the combined Euler rotation matrix.


FYI: The above does something like looking at the problem in the 2D space given by the x/y plane. It sees a vector (a c)[sup]t[/sup] and computes the angle that is between the principal x axis and the vector. You can find several discussions of this in this forum if you search for e.g. "atan2".

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I once wrote some examples about generic Euler angle conversions:

[url="http://khayyam.kaplinski.com/2011/02/converting-rotation-matrix-to.html"]http://khayyam.kapli...-matrix-to.html[/url]
[url="http://khayyam.kaplinski.com/2011/06/more-about-matrix-to-euler-angles.html"]http://khayyam.kapli...ler-angles.html[/url]

These allow you:[list]
[*]Convert rotation matrix to arbitrarily ordered Euler angles (useful for animations)
[*]Pick the smallest set of angles from alternate rotation variants (that is what you needed)
[*]Pick the set of angles from alternate rotation variants that is closest to some previous set (useful for animations)
[*]Should be gimbal-lock safe
[/list]

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