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Based on the theory of orthogonal frame, a general equation in form of vector function was established for scroll profile. Engagement condition of meshed profiles was derived. According to the parameter equation of Cartesian coordinate, the function of curvature radius could be expressed by integration of tangential angle. Curvature radius function based on differential equation of profile was deduced. Furthermore, demonstrated by four typical profiles, the corresponding relation among radius of curvature, general generating radius and general basic radius was determined. The differential equation for the involute of a regular triangle was demonstrated in form of function of radius of curvature. The integration profile could not only cover the existed profiles, but also be used to optimize properties of these profiles by controlling coefficients of series of tangential angles. The conclusions could be applied in flexible design of scroll fluid machine. 