In a scientific context, our world does not adhere to a true Cartesian framework: plumb lines deviate rather than run parallel, and a potential surface—defined by a water body—fails to conform to a geometric plane.

Although this fact is known, the distances and angles measured by a total station are transformed into Cartesian coordinates to establish local topocentric networks. This method is applicable for project areas of up to several hundred meters. However, for larger project areas, traditional geodesy employs corrections to the measurements: heights determined by trigonometric methods include a correction term for Earth's curvature (in addition to refraction), and horizontal distances measured at varying elevations are adjusted to a reference surface.

The procedures previously described are typically viewed as adjustments to measurements; nonetheless, these refined measurements establish a novel quasi-Cartesian coordinate system. Within this framework, horizontal coordinates are determined on the reference surface while height is measured relative to this curved surface. Although coordinate lines remain orthogonal in this new curvilinear coordinate system, the horizontal lines are not straight, and the vertical lines are no longer parallel.

Another effect of the non-Cartesian world poses unique challenges for aligning terrestrial scans. If, on the one hand, the vertical axes of the scanners are assumed parallel, the point clouds scanned from various positions do not align precisely. On the other hand, if the point clouds are aligned, the scanner axes become tilted.

The author advises implementing the aforementioned corrections on each terrestrial scan prior to merging the point clouds. By applying the Earth's curvature correction to the true Cartesian vertical coordinates and performing reduction to the reference level on the true Cartesian horizontal coordinates of individual scans, transforming them into this quasi-Cartesian curvilinear coordinate system. These transformed scans can be seamlessly co-registered and tranformed to map projection coordinates through a 2D Helmert transformation.