I have a distribution of vector $\textbf{x}=\langle \sin{\phi_x}\cos{\theta_x}, \sin{\phi_x}\sin{\theta_x}, \cos{\phi_x} \rangle$ on the unit sphere (von Mises-Fisher):

\begin{align}f(\phi_x,\theta_x;\kappa)=C(\kappa)e^{\kappa \cos{\phi_x}},\end{align}

where $\phi_x$ and $\theta_x$ are the spherical coordinate angles ($\phi_x$: angle between $\textbf{x}$ and the $z$-axis; $\theta_x$: angle around the $z$-axis to the $x$-axis), $\kappa \geq 0$ is a scaling factor, and $C(\kappa)$ is a normalization factor. This distribution is radially symmetric.

Another vector $\textbf{v}=\langle \sin{\phi_v}, 0, \cos{\phi_v} \rangle$ is set to a fixed point on the sphere and on the $xz$-plane, where the angle between $\textbf{v}$ and $\textbf{x}$,

\begin{align}\psi(\textbf{v}, \textbf{x}) = \cos^{-1}{(\textbf{v}\cdot \textbf{x})}=\cos^{-1}{(\sin{\phi_v}\sin{\phi_x}\cos{\theta_x} + \cos{\phi_v}\cos{\phi_x})}.\end{align}

I am interested in the PDF $g(\psi)$, with a fixed $\textbf{v}$, which should only be a function of $\phi_v$ and $\kappa$. Unfortunately, $\psi(\textbf{v},\textbf{x})$is not a 1-to-1 function. How can I go about finding this PDF $g(\psi)$?

I have a distribution of vector $\textbf{x}=\langle \sin{\phi_x}\cos{\theta_x}, \sin{\phi_x}\sin{\theta_x}, \cos{\phi_x} \rangle$ on the unit sphere (von Mises-Fisher):

\begin{align}f(\phi_x,\theta_x;\kappa)=C(\kappa)e^{\kappa \cos{\phi_x}}\sin{\phi_x},\end{align}

where $\phi_x$ and $\theta_x$ are the spherical coordinate angles ($\phi_x$: angle between $\textbf{x}$ and the $z$-axis; $\theta_x$: angle around the $z$-axis to the $x$-axis), $\kappa \geq 0$ is a scaling factor, and $C(\kappa)$ is a normalization factor. This distribution is radially symmetric.

Another vector $\textbf{v}=\langle \sin{\phi_v}, 0, \cos{\phi_v} \rangle$ is set to a fixed point on the sphere and on the $xz$-plane, where the angle between $\textbf{v}$ and $\textbf{x}$,

\begin{align}\psi(\textbf{v}, \textbf{x}) = \cos^{-1}{(\textbf{v}\cdot \textbf{x})}=\cos^{-1}{(\sin{\phi_v}\sin{\phi_x}\cos{\theta_x} + \cos{\phi_v}\cos{\phi_x})}.\end{align}

I am interested in the PDF $g(\psi)$, with a fixed $\textbf{v}$, which should only be a function of $\phi_v$ and $\kappa$. Unfortunately, $\psi(\textbf{v},\textbf{x})$is not a 1-to-1 function. How can I go about finding this PDF $g(\psi)$?

Edit: Added spherical area element.

I have a distribution of vector $\textbf{x}=\langle \sin{\phi_x}\cos{\theta_x}, \sin{\phi_x}\sin{\theta_x}, \cos{\phi_x} \rangle$ on the unit sphere (von Mises-Fisher):

\begin{align}f(\phi_x,\theta_x;\kappa)=C(\kappa)e^{\kappa \cos{\phi_x}},\end{align}

where $\phi_x$ and $\theta_x$ are the spherical coordinate angles ($\phi_x$: angle between $\textbf{x}$ and the $z$-axis; $\theta_x$: angle around the $z$-axis to the $x$-axis), $\kappa \geq 0$ is a scaling factor, and $C(\kappa)$ is a normalization factor. This distribution is radially symmetric.

Another vector $\textbf{v}=\langle \sin{\phi_v}, 0, \cos{\phi_v} \rangle$ is set to a fixed point on the sphere and on the $xz$-plane, where the angle between $\textbf{v}$ and $\textbf{x}$,

\begin{align}\psi(\textbf{v}, \textbf{x}) = \cos^{-1}{(\textbf{v}\cdot \textbf{x})}=\cos^{-1}{(\sin{\phi_v}sin{\phi_x}\cos{\theta_x} + \cos{\phi_v}\cos{\phi_x})}.\end{align}

I am interested in the PDF $g(\psi)$, with a fixed $\textbf{v}$, which should only be a function of $\phi_v$ and $\kappa$. Unfortunately, $\psi(\textbf{v},\textbf{x})$is not a 1-to-1 function. How can I go about finding this PDF $g(\psi)$?

I have a distribution of vector $\textbf{x}=\langle \sin{\phi_x}\cos{\theta_x}, \sin{\phi_x}\sin{\theta_x}, \cos{\phi_x} \rangle$ on the unit sphere (von Mises-Fisher):

\begin{align}f(\phi_x,\theta_x;\kappa)=C(\kappa)e^{\kappa \cos{\phi_x}},\end{align}

where $\phi_x$ and $\theta_x$ are the spherical coordinate angles ($\phi_x$: angle between $\textbf{x}$ and the $z$-axis; $\theta_x$: angle around the $z$-axis to the $x$-axis), $\kappa \geq 0$ is a scaling factor, and $C(\kappa)$ is a normalization factor. This distribution is radially symmetric.

Another vector $\textbf{v}=\langle \sin{\phi_v}, 0, \cos{\phi_v} \rangle$ is set to a fixed point on the sphere and on the $xz$-plane, where the angle between $\textbf{v}$ and $\textbf{x}$,

\begin{align}\psi(\textbf{v}, \textbf{x}) = \cos^{-1}{(\textbf{v}\cdot \textbf{x})}=\cos^{-1}{(\sin{\phi_v}\sin{\phi_x}\cos{\theta_x} + \cos{\phi_v}\cos{\phi_x})}.\end{align}

I am interested in the PDF $g(\psi)$, with a fixed $\textbf{v}$, which should only be a function of $\phi_v$ and $\kappa$. Unfortunately, $\psi(\textbf{v},\textbf{x})$is not a 1-to-1 function. How can I go about finding this PDF $g(\psi)$?