Obtaining xy-coordinate linear motion from angles of rotation

[sk63]

Hello,

I have a laser beam that passes through two 3 degree angled wedges, resulting in the ability to project the laser anywhere within a 6cm radius circle (each wedge alone produces a 3cm radius circle). I need to be able to tell each wedge how much to rotate in order to get the laser from one point (x1,y1) within that circle to another point (x2,y2), and can’t seem to get the formula for the rotation angles correct…

I have determined that the angles for the two wedges will have the following relationship:
ThetaWedge1= arctan(y/x) + b
ThetaWedge2 = arctan(y/x) – b

Where “b” is a function of x and y. and 2b=ThetaWedge1-ThetaWedge2

The equation for “b” is what I am having trouble calculating and have been getting results that produce correct angles of rotation for some (x,y) coordinates and incorrect ones for others.

(Note: this equation form is just for the first quadrant, for other quadrants there will be another factor to take the quadrant into account)

This will be in a Matlab loop, so it can be assumed that each (x2,y2) point will become the (x1,y1) point for the next movement/ set of rotations.

If anyone has any ideas or has done calculations similar to these in the past, you help is much appreciated! Thanks!

 

[fresh_42]

The equation for “b” is what I am having trouble calculating and have been getting results that produce correct angles of rotation for some (x,y) coordinates and incorrect ones for others.

(Note: this equation form is just for the first quadrant, for other quadrants there will be another factor to take the quadrant into account)

It would help to have a picture or at least the other formulas, too. Otherwise you can still solve for $b$ in each quadrant.