In this paper, we propose a frequency independent approach, the numerical steepest descent path method, for computing the physical optics scattered electromagnetic field on the quadratic parabolic and saddle surfaces. Due to the highly oscillatory nature of the physical optics integral in the high frequency regime, the proposed method relies on deforming the integration path of the integral into the numerical steepest descent path on the complex plane. Furthermore, critical-point contributions which contain the stationary phase point, boundary resonance points, and vertex points, are comprehensively studied in terms of the numerical steepest descent path method. To illustrate the efficiency of the proposed method, some extensive numerical results for the physical optics integral defined on arbitrary lines, triangles and polygonal domains are demonstrated. Finally, numerical results on these quadratic surfaces illustrate that the proposed numerical steepest descent path method is frequency independent in computational cost and error controllable in accuracy.