Helmert Geometry

The simplest version of the problem is described by two r by s matrices X and Y. Their columns contain the r-dimensional coordinates of s points that have been observed in two different coordinate systems with random observational errors. The problem is to estimate in some least-squares sense a transformation matrix H (rotations and scale changes) and an r-dimensional vector t (translation between the two origins); the translational matrix is teT where the s-dimensional vector e has mere ones as components: Y =H??X - t?eT. We generalize by introducing weights p, and q. All coordinates in X are observed with weight p and the Y coordinates with weight q. There is another type of generalization: LW wants to transform between more than two point sets X and Y. The networks also can be perceived as repeatedly measured versions of one and the same network and the problem changes into one of interpolation between different coordinate sets for the same point set. The problem admits a geometrical theory which in the opinion of the authors is interesting and even beautiful, and, what is perhaps more important, it suggests a method for computing the parameters. (Kai Borre and Torben Krarup, Horsens)