I need help with this specific problem.

There is a spiral expressed as r=aθ^m(a and m are positive constant、α≦θ≦β（positive)). The curvature of the spiral changes uniformly and a rotational angle φ is given (the spiral total length is always constant). I would like to know how will the spiral equation change. I guess that the spiral after rotation is expressed as the following equation.

r=bθ^m(b positive constant、α'≦θ≦β'（positive）). The radius at which the spiral begins is the same before and after the rotation（aα^m=bα'^m, β'-α'=β-α+φ）.

In conclusion, if we give a curvature change uniformly, how will the spiral equation change?