Abstract: In solving the problems about reference frame transformation in the theory of relativity，it is often required

to have a long and complicated calculation. Two-inertial-system mapping is often used to transform the computational

problem into a geometric problem. This method is quick and intuitive but hard for beginners to grasp. To

express the dynamic process of Lorentz transformation and transfer physical pursuit invariance values，a new geometric

method named hyperbolic rotation was put forward. Based on the two inertial system mapping，the hyperbolic

rotation was carried out based on time-interval invariance and the final geometric results were determined by the restriction

of causality. There was an example to explain the application of two geometric methods in detail.

Key words: geometric, two inertial system mapping, hyperbolic rotation, causal relationship