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As it's well known, assuming the earth fixed and non rotating, the trajectory of a falling body with initial speed $v_0 = \{v_{0x},v_{0y},{v_{0z}}\}$ is contained in a plane. Assuming an observer in the origin of a fixed reference frame, he will measure three coordinates of the falling body: $P=\{\rho,\theta,\phi\}$. We can suppose to be $\rho$,distance, $\theta$, elevation and $\phi$, azimuth. How is it possible to find a reference frame in which the azimuth $\phi$ is zero? Thanks in advance.

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The observer has to be in the plane of the motion. The surfaces where $\phi$ is constant are planes; the observer just has to orient her axes appropriately for one of these planes to be the plane of motion.

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    The observer isn't in the plane of the motion. The question is: what is the transormation matrix from the observer reference frame to the frame in which the azimuth doesn't change in time?2012-10-19
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    @Riccardo: If by "transformation matrix" you mean only a rotation without translation, the answer is that there is no such transformation. The surfaces of constant azimuth are planes through the origin; thus, for the azimuth to remain constant, the motion has to occur in a plane through the origin, that is, the origin has to be in the plane of motion; and "observer" is just a confusing synonym for "origin".2012-10-19
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    I mean rotation and translation.2012-10-19
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    Obviously there are infinite reference frames in which the condition: $\phi=0$ is satisfyed. One of this, for example, could be the reference frame of the impact point.2012-10-19
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    @Riccardo: Then the problem is underspecified -- you need some point on the trajectory.2012-10-19
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    I agree. We suppose we can have all the measurements we need for the calculation2012-10-19
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    @Riccardo: Yes, the impact point or any other point on the trajectory. If you already know that, I don't understand what the question is. A translation isn't described by a transformation matrix.2012-10-19
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    Ok. If, for example the $(x,z)$ plane is coincident with the $(x',z')$, I need only a translation. But, in general, if I put the $x$ toward the north star, I will need also a rotation.2012-10-19
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    @Riccardo: Could you please state exactly what you're looking for? It seems you know all the concepts required for the solution. The only concrete thing that you've asked for is a transformation matrix, and as I said the problem cannot be solved with only a transformation matrix. What is given, and what do you want to calculate?2012-10-19
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    OK. Suppose I measure three points of the trajectory in my reference frame: $P_1$, $P_2$ and $P_3$. The equation of the plane defined by this points is: $a(x-x_1)+b(y-y_1)+c(z-z_1)$ where $a$,$b$ and $c$ are calculated from the three points coordinates. Assuming I know the impact point of the particle on the trajectory belonging to this plane. In this case I can put the origin of the new reference frame in this point and put, for example, the $(x,z)$ plane coincident with the trajectory plane. What kind of rotation and translation have I do?2012-10-19