Offset surfaces are widely used in industrial fields, such as tool path geometry generation and robot path planning in NC machining, surface plate representation in solid modeling, etc. Geodesic, line of curvature and asymptotic curve are three kinds of important characteristic curves which play a decisive role in the properties and shapes of surfaces. The construction of offset surface pencils preserving common characteristic curves is studied, that is, when constructing a surface pencil P（s，t） with a given curve r（s） as a common characteristic curve, the corresponding curve r（s） on the offset surface pencil P（s，t） is the same type of characteristic curve. Firstly, P（s，t）is expressed as the linear combination of the Frenet frame of r（s）, and the coefficients of the linear combination are called the scale functions. Then, P（s，t） is taken as the base surface to construct its offset surface pencil P（s，t）. According to the properties of the characteristic curves, whether r（s） can be guaranteed to be the same type of characteristic curve as r（s） on P（s，t） completely depends on the given curve and has nothing to do with the scale functions. Finally, the cylindrical helix and circle are taken as examples to verify the correctness of the above conclusions.