In this study, we propose a novel equivalent-waveguide concept to design the completely adiabatic tapered waveguides. The optimal combination of taper shape and refractive-index distribution in such ideal structures is obtained by applying a series of conformal mappings to transform a single-mode straight waveguide into an equivalent tapered configuration. In an ideal taper the curved phase-front effect verifies the constant V-number concept is insufficient for designing a tapered waveguide. Also, the beam propagation method combined with the conformal mapping method is used to analyze the characteristics of light propagation in ideal structures. Simulation results predict that in the ideal taper a perfect mode-size conversion can be achieved adiabatically.