Optimization scheme is attempted regarding the overall performance of ship's rudders from wide aspects of ship hydrodynamics, i. e. maneuverability, resistance and propulsion. Two different approaches, analytical and synthetic, are adopted simultaneously. In this first report, analytical procedures are developed for obtaining lift and drag characteristics of rudders with given geometry, where non-linear lifting surface theory are applied. Considering that rudders are low aspect ratio wings that operate at high angle of incidence in non-uniform flow field, a new procedure for solving non-linear lifting-surface equation which implicitly includes shedding angle of free vortex to the wing plane is developed and a general method for analysing hydrodynamical characteristics of arbitrary aspect ratio rudders is completed. Present method do not impose any limitation on aspect ratio, angle of incidence, inflow velocity distribution and rudder section profile, whereas methods so far used essentially impose limitations. Analysed forces, moments and pressure distributions agree fairly well with experimental ones. As applications of this method, it is shown that performance of rudder as a maneuvering equipment is improved with the present analysis method and that advanced design method for reaction rudders is composed.