This two-part paper aims at providing the basic mathematical theory of omnidirectional wireless power transfer (WPT). Based on current vector control, the magnetic field vector can be generated and pointed at any direction, thus achieving genuine omnidirectional wireless power flow. Part-I of this paper contains the mathematical analysis of the two-dimensional (2-D) system. The 2-D analysis shows that the total input power and the system energy efficiency of the WPT system can be expressed as functions of the angle of the input magnetic field vector. These functions follow the Lemniscates of Bernoulli on the 2-D plane. The directional nature of such curves provides the crucial information of the locations of the receivers so that the wireless power can theoretically be directed to the loads efficiently. The 2-D theory is presented here with the support of experimental verification. Understanding the 2-D omnidirectional WPT theory would make it relatively easy to appreciate the three dimensional one reported in Part II.