Calculation of a length using Involute curve

[am1234567]

As shown in the attached figure, a circle with its involute curve has an initial position at 0 degree, and have a length of 'a' which was measured from the y-intercept to the center of the circle. After that, the circle was rotate for 45 degree and now have the length of 'b'. Then it was again rotate for 90 degree and has a length of c. Based on the information provided, is there any possible way for me to calculate the length 'b' and 'c'. Is there any relation I can use to calculate the length for any degree of rotation ?

 

[IRstuff]

https://en.wikipedia.org/wiki/Involute#Involutes_of_a_circle

The paradigm of unwrapping the circumference will help solve this problem

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