Abstract: In order to analyze the relationship between the minimum position (lwmin, lwmax) of the minor and major axes of the isophotes and the mutation point (lR1,lR2) of the radius of curvature R1, R2 of the generalized Gaussian beam, which never studied before, and find some new and special nature of the generalized Gaussian beam from their relationship, at first, using the abc parameter method to carry out theoretical derivation, a simple analytical solution expression between lwmin,lwmax, lR1,lR2 was obtained. Secondly, after numerical simulation, the relationship figures between lwmin,lwmax, lR1,lR2 and wmin,0,wmax,0,R1,0,R2,0, φR1 were obtained. The results show that lwmin,lwmax is always between lR1 and lR2,and lwmin,lwmax, lR1,lR2 is distributed near l=0.lwmin has serious non-continuous points other than the other three positions. This conclusion adapts to the generalized Gaussian beam in actual and general situations.