Minimal Configurations in 3D Similarity Transformation: Selection Criteria for Stable Solutions

Abstract

We revisit minimal seven-equation selections for the 7-parameter 3D similarity (Helmert) transformation using three common points. From the 9 available scalar equations there are 36 possible seven-row subsets. Rather than claiming that a fixed subset of these is always stable, we show that stability is geometry-dependent: the same selection can be wellor ill-behaved depending on the spatial arrangement of the three points. We provide two simple, geometry-aware selection rules that perform reliably across typical geodetic and photogrammetric scenarios: (A) choose two full 3D points (six rows) and add one component from the third point whose direction is not parallel to the twist axis defined by the two full points; and (B) choose one full 3D point and, from the other two points, take the same 2D pair that includes the local normal (e.g., YZ or XZ in near-planar setups). These rules are motivated by a first-order sensitivity analysis of the transformation equations and are validated by synthetic experiments with non-unit scale, non-zero rotations, translations, and measurement noise. For each geometry we rank all 36 selections using a normalized, unit-invariant conditioning measure derived from the scaled Jacobian. Results show that the number of acceptable selections may be less than, equal or greater than nine, depending on point configuration, and the proposed rules consistently identify the well-behaved cases while flagging the risky ones. The paper offers practical guidance for choosing seven-row subsets that avoid hidden instabilities when working with minimal threepoint data.