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"In general, if we restrict ourselves to rotations about one of the Cartesian axes, three successive rotations are required to transform the fixed frame into the body frame"

The origin of our fixed frame and the body frame are common. The axes of the body frame points in the direction of the principal axes of rotation of the body.

I do not understand, how three rotations are all what is needed to transform the two frames.

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  • $\begingroup$ There are three degrees of freedom rotationally needed to achieve any orientation possible. $\endgroup$ Mar 24, 2014 at 16:31

3 Answers 3

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The body frame is defined by the principal axes of rotation, which are three unit vectors in $\mathbb{R}^3$. If restricted to rotations around the $x$, $y$ and $z$ axes, it takes (in general) one rotation around each axis to align the original three axes to the principal axes of rotation. For example, one can first choose the $x$ axis and use two rotations to align it with one of the principal axes of rotation; then one more is needed to align the $y$ and $z$ axes.

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The OP has never accepted any of the answers. I'll give a constructive proof using an intrinsic rotation sequence.

I do not understand how three rotations are all what is needed to transform the two frames.

I'll denote the original set of axes as $x,y,z$ and the final set as $X,Y,Z$, both with the same handedness. Suppose the $xy$ and $XY$ planes do not coincide. Both planes contain the origin and thus there is a line $N$ that passes through the origin and lies on both the $xy$ and $XY$ planes.

Rotate about the $\hat z$ axis such that the $\hat x$ axis now points along the line $N$. Next, rotate about the once-rotated $\hat x$ axis so that $\hat z$ points along $\hat Z$. Finally, rotate about the twice-rotated $\hat z$ axis such that the thrice rotated $\hat x$ and $\hat y$ unit vectors align with $\hat X$ and $\hat Y$.

The above assumed that the $xy$ and $XY$ planes do not coincide. Suppose they do coincide. This means that either $\hat Z = \hat z$ or that $\hat Z = -\hat z$. In the first case ($\hat Z = \hat z$), this devolves to a plane rotation, so that only one rotation about the $\hat z = \hat Z$ axis is needed.

Two rotations are needed when $\hat Z = -\hat z$, one about $\hat z$ to align $\hat x$ with $\hat X$, and the second a 180 the second case, or that $\hat Z = -\hat z$, in which case two rotations are needed, one about about $\hat z$ to align $\hat x$ with $\hat X$, and a 180° rotation about the once-rotated $\hat x$ to align $\hat y$ with $\hat Y$ and $\hat z$ with $\hat Z$.


This sequence of up to three rotations, first about the $z$ axis, then about the once rotated $x$ axis, and a third about the twice-rotated $z$ axis, is the Euler rotation sequence, as original developed by Leonard Euler.

There are a number of alternative approaches that use rotations about primary axes only. For example, there's nothing special about the use of the $x$ axis as the middle rotation in the above sequence. The $y$ axis works just as well. The same goes for the $z$ axis as the first and last axes. There are six Euler-like sequences. Alternatively, one could rotate about the $z$ axis, then about the once-rotated $y$ axis, and finally about the twice-rotated $x$ axis. This results in another six set of rotation sequences. Collectively these form the twelve intrinsic rotation sequences. Another set of sequences results from rotating about the fixed $x$, $y$, and $z$ axes. Just as there are twelve intrinsic sequences, there are twelve of these extrinsic sequences. All 24 of these rotation sequences require at most three rotations to attain any arbitrary orientation.

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A general orientation can be defined by three mutually perpendicular unit vectors. Actually, only two are needed, as the third is derived by a cross product. Two vectors have 6 parameters in general, and to make them unit vectors, you need 4 parameters. Since the vectors have to be perpendicular this reduces the independent parameters to 3.

Euler angles is one of many ways 3 parameters can define the full rotational orientation. Another can be a single rotation axis (unit vector, 2 parameters) and a rotation angle (1 parameter), for a total of 3.

I guess I do not fully understand what you are asking about, but I hope this helps.

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