Damping Ratio Line on a root locus plot

[kenambo]

Hi all

I come across damping ratio line in a root locus method, in which i can't able to understand How a damping ratio is drawn in a cartesian graph.

For example damping ratio = 0.456 means they draw a line of 117.13 deg on the root locus in norman Nise book of control engineering.


How a damping ratio of 0.456 converts into a line of 117.13 deg.?

Thanks

 

[erikl]

0.456 = -cos(117.13deg)

 

[kenambo]

But May I know why we need to take cos(damping_ratio)

Is there any specific reason ..?

Thanks