Suppose I am given two unit vectors starting from origin in three dimensions, $\vec{v} = (v_x, v_y, v_z)$ and $\vec{u}=(\sin\theta\cos\phi, \sin\theta\sin\phi, \cos\theta) $, which is just the spherical representation of the end point of the vector $\vec{u}$. Given the knowledge about the two vectors, I know that the angle between them is $$\cos\theta_0 = v_x\sin\theta\cos\phi + v_y\sin\theta\sin\phi + v_z\cos\theta~.$$ However, suppose I want to know all vectors that make angle $\theta_0$ with $\vec{v}$, i.e I want to know all the $(\theta, \phi)$. In this case, would Rodrigue's rotation formula apply?

  • $\begingroup$ \begin{eqnarray*} P_{n}(z)=\frac{1}{2^n n!} \frac{d^{n}}{dz^n} (z^2-1)^n. \end{eqnarray*} What is Rodrigues's Formula ? $\ddot \smile$ $\endgroup$ – Donald Splutterwit May 10 '18 at 19:58
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    $\begingroup$ Sorry, have a look at the edited link now in the question. $\endgroup$ – konstant May 10 '18 at 20:00
  • $\begingroup$ Yes, you could certainly use the rotation formula as part of the parameterization of the cone. $\endgroup$ – amd May 10 '18 at 22:54

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