Fast conical surfaces evaluation with null-screens and randomized algorithms

A method for the evaluation of the shape of fast conical surfaces based on the null-screen method is presented. The proposed method is based on randomized algorithms, where instead of integration as is usually performed, the shape of the surface is recovered in a direct and random way. This is done by proposing a new test surface; the surface parameters are randomly generated and the calculated null-screen is compared against the originally designed one. The coefficients that generate the null-screen closest to the reference null-screen are taken as the final values that describe the surface under test. The algorithm validation was done in the theoretical and experimental way. For the theoretical way, a convex parabolic surface with 21.36 mm radius of curvature and f/0.178 was simulated and analyzed; in the experimental way, a convex spherical surface f/0.33 and a concave ellipsoidal mirror f/0.2381 were evaluated. The results obtained with our algorithm in the evaluation of the ellipsoidal surface were compared against the results obtained with traditional methods. We found that the radius of curvature, conic constant, and the decentering coefficients that were used to generate each surface were retrieved in a good manner with our algorithm. The results show that these parameters can be recovered with a percent error smaller than 1% in the case of simulations, and smaller than 2% in the two experimental cases.