The coordinate systems in the Hess screen test and the Harms tangent screen test for ocular motility analysis of strabismus: their metrics and relationship

In the diagnosis of strabismus due to paralysis of the eye muscles, the Hess screen test and the Harms tangent screen test are used to measure the misalignment of the eyes in different gaze directions (motility analysis). Both examination methods use different procedures to define the gaze direction and coordinate systems. Coordinate systems with scales in angular degrees are used to determine the direction of gaze and simplify reading the squint angle in the event of a misalignment (horizontal and vertical deviation). In the measurement results, however, their numerical values differ due to the different angle definitions of the direction of gaze in the Hess and Harms tests. In this paper, the angle definitions are clearly described, the coordinate systems are mathematically derived from them, their metrics are formally analysed and compared with the surface theory according to Carl Friedrich Gauss, and their transformation relationship is specified and visualised. The surface theory is used to justify the finding that a measured misalignment of the eye leads to the same motility disorder with the same spatial orientation of the direction of gaze, regardless of the choice of angle. The coordinate systems according to Harms and Hess are therefore mathematically equivalent. However, assessment of the paretic eye muscle(s) and the subsequent therapeutic measures are based on the numerical values of the squint angle measurement. Therefore, knowledge of the test used and its angle definition is important. The procedure for determining the gaze direction determines the choice of coordinate system in the test according to Harms and Hess to avoid systematic measurement deviations.