This paper formulates a class-E synchronous RF rectifier from a new viewpoint. The key point is to introduce a matrix and convolute the DC terms into RF matrices. The explicit expression of input impedance is demonstrated in plane geometry. We find out their input impedance exhibits a geodesic arc in hyperbolic geometry under ZVS operation, where the theoretical RF-DC conversion efficiency results in 100%. We verify the developed theory both numerically (circuit simulation) and experimentally (6.78MHz, 100W). We confirm that the input impedance becomes a geodesic arc for a wide range of DC load resistance. The presented theory is quite elegant since it is based on a matrix-based formulation and plane-geometrical expression.