I want to calculate the movement of points in polar coordinates.

Consider the following issues:

On the xy plane, first move the length r_1 along the x-axis from the origin O (where angle 1_1=0).
Then, the length r_2 is moved at angle __2, and the length r_3 is moved at angle __3, and
After that, I will continue to do the same.
The blue line in the figure below shows the movement above.

For example,
The length of the red dotted line in the figure, s_3, s_4,
The angles __3 and __4 between the red dotted line and the x-axis are
How can I calculate using r and を?

Also, what general formula can s_n and __n be used to calculate using r and を?

Ultimately, we would like to generalize in a three-dimensional polar coordinate system.

Thank you for your cooperation.

mathematics

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