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I have this problem to solve. I want to compute the inclination of a plane $\theta(t)$ at every frame of a simulation given the following rule for its angular speed of rotation $\omega(t)$

$$ \omega(t) = - \frac{\mathbf{V}\cdot\hat{\mathbf{n}}}{(||\mathbf{P}-\mathbf{P}'||)}\frac{1}{\tan(\theta(t))} $$

given that the starting angle $\theta(t=0)=45°$

The quantity $$\frac{\mathbf{V}\cdot\hat{\mathbf{n}}}{(||\mathbf{P}-\mathbf{P}'||)}$$ is the ratio between the velocity of the observer along his line of sight ($\mathbf{V}\cdot\hat{\mathbf{n}}$) and the distance of the observer $\mathbf{P}$ from a point $\mathbf{P}'$ and is computed externally at every frame.

I'm unable to figure out how to solve this equation in Mathematica, is it possible analytically?

If this is not possible, how can I implement a iterative method that can deal with the imprecision of computation of the $\frac{1}{\tan(\theta(t))} $ term?

In a first implementation I choose:

double omegaz= -speedAlongLineOfSight/((cyclopeanEye-projPoint).norm() *tan( toRadians(theta) ));
theta = theta - (omegaz)*(deltaT);

but I think this continuosly sums errors, creating instability of the solution.

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