Bilinear-inverse-mapper: Analytical solution and algorithm for inverse mapping of bilinear interpolation of quadrilaterals

Abstract

The challenge of finding parametric coordinates of bilinear interpolation of a point with respect to a quadrilateral in 2D or 3D frequently arises as a subproblem in various applications, e.g. finite element methods, computational geometry, and computer graphics. The accuracy and efficiency of inverse mapping in such cases are critical, as the accumulation of errors can significantly affect the quality of the overall solution to the broader problem. This mapping is nonlinear and typically solved with Newton’s iterative method, which is not only prone to convergence issues but also incurs high computational cost. This paper presents an analytical solution to this inverse mapping, along with a comprehensive geometric analysis covering all possible quadrilateral configurations. It describes the invertibility of all points and extends the discussion to 3D and concave quadrilaterals. The proposed algorithm is robust, free from failure due to convergence issues or oscillations in iterative methods, and achieves approximately $2.4\times$ higher computational speed compared to Newton’s method for quadrilaterals with non-parallel opposite edges. This enables an efficient calculation of shape functions or interpolation functions at all invertible spatial points. The high-accuracy, high-speed computational solution will be particularly advantageous in applications involving high spatial or temporal discretisation (i.e. fine mesh and small timesteps) where iterative methods will be computationally expensive. The analytical solution based algorithm is available as an open-source library at https://github.com/sahu-indrajeet/Bilinear-Inverse-Mapper.