Northam conceptualizes the urbanization course with an S-shaped curve, which was divided into three stages. Although this theory has been widely validated and referred to by various researchers, the dividing points between each stage are not explicitly defined in mathematical method. This essay is to deduce the equation of the Logistic model of urbanization level according to the variable t (time series) by mathematical methods. Through qualitative and quantitative studies, it proves that the Northam curve has three characteristic points which are the inflexion point and two biggest curvature points. It is validated that the urbanization growth rate attains the maximum while the acceleration is down to zero at the inflexion point, as the urbanization growth is accelerated before reaching the point and decelerated afterwards. Developed further from the Northam's theory of three phases of urbanization, the two biggest curvature points are identified to divide the urbanization growth course into three stages. This essay also presents the mathematical formula of the three characteristic points and the duration and the mean growth rate at the second stage. By modeling the urbanization processes of Japan and other 22 countries with historical data, it is proved that the result tallies with other relevant studies and the world urbanization development trends. It further indicates that the China's urbanization reached its inflexion point in 2004. Finally, it discusses the applicability and the limitation of this method.