Creting a versor from any two elements

I know that is possible to compute a versor from the ratio of two lines V^2 = L_2L_1^{-1} that represents the transformation from L_1 to L_2 where L_k = p_k\wedge\mathbf{n}_k\wedge \infty, and that V\hat{B}V^{-1} is a general rigid body motion.

Given two circles in their dual form \kappa_i = \sigma_i\wedge\pi_i, \ \ i=1,2. Can I compute a versor that does the transformation from \kappa_1 to \kappa_2 as V^2=\kappa_2\kappa_1^{-1}?
What I want is a V that satisfies V\kappa_1V^{-1} = \kappa_2.

Is this possible in general? Given two similar objects A and B is it possible to determine the versor that transforms one to the other. Does the application of the following versor V = \text{exp}(\frac{1}{2}\text{log}(AB^{-1})) transforms object A into object B as VAV^{-1} = B?