A Symbolic Solution of a 3D Affine Transformation

Abstract

We demonstrate a symbolic elimination technique to solve a nine-parameter 3D affine transformation when only three known points in both systems are given. The system of nine equations is reduced to six by subtracting the equations and eliminating the translation parameters. From these six equations, five variables are eliminated using a Grobner basis to get a quadratic univariate polynomial, from which the solution can be expressed symbolically. The main advantage of this result is that we do not need to guess initial values of the nine parameters, which is necessary in the case of the traditional solution of the nonlinear system of equations. This result can be useful in geodesy, robotics, and photogrammetry when occasionally only three known points in both systems are given or when a Gauss‐ Jacobi combinatorial solution may be required for certain reasons, for example detecting outliers by using variancecovariance matrices.