Construction of freeform surface with adjacent orthogonal tangent vectors for point light source

Abstract

Illumination design problem for light source with zero-étendue can be transformed into optimal mass transport problem based on ray mapping. In this paper, an improved numerical method for constructing freeform surface that best fits the normal field is proposed. First, the normal vectors at each point on freeform surface are constructed by two adjacent orthogonal tangent vectors. Then the nonlinear equations corresponding to the surface coordinates can be constructed according to the geometric relationship and Snell's law. Finally, a continuous and accurate freeform surface can be acquired by solving these nonlinear equations. Compared with the previous work, this method combines the coordinates of the initial boundary and the internal ones of the surface to construct a set of nonlinear equations, which effectively eliminates the bad points generated in the equation solving process caused by the arbitrariness of the initial boundary, and improves the integrability and accuracy. The simulation results show that for point light source illumination, freeform surface constructed by traditional methods have large distortion and relative standard deviation (RSD) values in the target area, while using this method can effectively reduce the distortion and RSD. Furthermore, the proposed method can provide lower normal deviation than the traditional methods, especially for the point-source lighting condition. In addition, the simulation found that when the illuminated area increases the normal deviation of both methods decrease, the RSD of the traditional method decrease, while the RSD of proposed method will increase slowly