I'm doing my homework on flight mechanics and the first lesson is about non-inertial reference frames. I've learnt there are three basic reference frames to be account in the study of flight mechanics. Body (b), wind (w) and local horizon (h).

  • Body-local horizon orientation. With three angles: $\psi$ (yaw), $\theta$ (pitch), $\phi$ (roll).
  • Wind-local horizon orientation. With three angles: $\chi$ (wind yaw), $\gamma$ (wind pitch), $\mu$ (wind roll).
  • Body-wind orientation. With two angles: $\alpha$ (angle of attack), $\beta$ (slip angle).

I have determined the matrix of conversion to convert any vector from one to another frame of reference by using Euler angles, which is:

$$L_{fi}=$$ $$\begin{pmatrix} \cos \delta_2 \cdot \cos \delta_3 & \cos \delta_2\cdot \sin\delta_3 & -\sin\delta_2 \\ \sin\delta_1\cdot \sin\delta_2\cdot \cos\delta_3 - \cos\delta_1\cdot \sin\delta_3 & \sin\delta_1\cdot \sin\delta_2\cdot \sin\delta_3 + \cos\delta_1\cdot \cos\delta_3 & \sin\delta_1\cdot \cos\delta_2\\ \cos\delta_1\cdot \sin\delta_2\cdot \cos\delta_3 + \sin\delta_1\cdot \sin\delta_3 & \cos\delta_1\cdot \sin\delta_2\cdot \sin\delta_3 - \sin\delta_1\cdot \cos\delta_3 & \cos\delta_1\cdot \cos\delta_2 \end{pmatrix}$$

Where the subindex ($fi$) indicates the final reference frame (f) and the initial reference frame (i). And the angles $\delta_1,\delta_2,\delta_3$ are to be replaced for the corresponding angles depending on our conversion.

In such a way, that if we have a vector $\textbf{A}_w$ and we want to get $\textbf{A}_h$ we'd do: $$\textbf{A}_h=L_{hw} \cdot \textbf{A}_w$$

But I have found a problem, and here it's when my question comes, that asks you to write the following:

$$\gamma = f(\theta,\phi,\alpha,\beta)$$

That is, writing the angle $\gamma$ as function of the angles $\theta, \phi, \alpha, \beta$.

How can I do this? I have been thinking of how to do it, but how can I combine two reference frames to express the angle of another reference frame? Should I use the matrix of transformation or should I use another method?

Note: I'm not asking to get my homework done, just obtain a hint on how to do it since I've been stuck for a good deal of time.


closed as off-topic by ACuriousMind, Kyle Kanos, user81619, John Rennie, Ryan Unger Sep 28 '15 at 13:03

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" – ACuriousMind, Kyle Kanos, Community, John Rennie, Ryan Unger
If this question can be reworded to fit the rules in the help center, please edit the question.


One approach that uses the matrices you've already derived is to set $L_{bh}=L_{bw}L_{wh}$ and then solve for $\gamma$.

  • $\begingroup$ Thank you, I have performed the product $L_{bw}L_{wh}$ and I have equated with $L_{bh}$. In order to solve for $\gamma$, can I choose whatever row/column? I mean, will the results be equivalent? $\endgroup$ – user3780731 Sep 27 '15 at 12:02
  • $\begingroup$ They should be. If not, then the rotation matrices are incorrect. $\endgroup$ – Rick Sep 27 '15 at 13:21

Not the answer you're looking for? Browse other questions tagged or ask your own question.