Consider the geometric construction shown in the figure.

The point P is constrained to slide along the vertical line v$.$

The OP segment, of variable lenght r$,$ rotates around O by an angle ${}_0-,$ with $$ ranging from $0$ to $90$ degrees; since ${}_0$ is $45$ degrees, the angle ${}_0-$ ranges from $45$ to $-45$ degrees.

In addition, the distance d between the vertical line v and the center of rotation O is know, which, of course, is a parameter that remains constant as the angle of rotation $$ changes.

Other parameters that remain constant are the angle OQP $($the angle between the segments PQ and OQ$)$ that is equal to $90$ degrees and the angle PAO $($the angle between the segment PA and OA$)$ that is equal to $90$ degrees.

Instead, other parameters that change during rotation are: the lenght of the segment PA $($c$),$ the lenght of the segment PQ $($t$),$ the lenght of the segment OQ $($b$)$ and the angle OPQ $($the angle between the segment OP and QP$),$ that in the figure is called $.$

Specifically, the beta angle initially takes on $($in the configuration in the figure, that is$,$ when $=0)$ a certain value that can be called ${}_0.$

I need to calculate the expression by which the beta angle varies as a function of the aforementioned know parameters, i$.$e$.$ the distance d and the angle of rotation $.$