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Vector products as a tool for the investigation of the isometric transformation of objects

Grabowski R. J.

Reports on Geodesy and Geoinformatics

2015

Vol. 98

60--71

The identification of isometric displacements of studied objects with utilization of the vector product is the aim of the analysis conducted in this paper. Isometric transformations involve translation and rotation. The behaviour of distances between check points on the object in the first and second measurements is a necessary condition for the determination of such displacements. For every three check points about the measured coordinate, one can determine the vector orthogonal to the two neighbouring sides of the triangle that are treated as vectors, using the definition of the vector product in three-dimensional space. If vectors for these points in the first and second measurements are parallel to the studied object has not changed its position or experienced translation. If the termini of vectors formed from vector products treated as the vectors are orthogonal to certain axis, then the object has experienced rotation. The determination of planes symmetric to these vectors allows the axis of rotation of the object and the angle of rotation to be found. The changes of the value of the angle between the normal vectors obtained from the first and second measurements, by exclusion of the isometric transformation, are connected to the size of the changes of the coordinates of check points, that is, deformation of the object. This paper focuses mainly on the description of the procedure for determining the translation and rotation. The main attention was paid to the rotation, due to the new and unusual way in which it is determined. Mean errors of the determined parameters are often treated briefly, and this subject requires separate consideration.

Обеспечение качества сборки конической передачи путём уменьшения радиального биения валов на опорах скольжения

Gabdullina A. Z.

Technologia i Automatyzacja Montażu

2013

nr 1

42--44

В статье рассматриваются технологические вопросы обеспечения точности зацепления конической передачи путeм уменьшения радиального биения базовых валов.

W artykule przedstawiono zagadnienia technologiczne zapewnienia dokładności zazębiania przekładni stożkowej przez zmniejszanie bicia promieniowego czopów oporowych.

The article deals with the technological issues of ensuring the precision of bevel gear by reducing the radial run-out of shaft.

Migration of the instantaneous axis of motion during axial rotation in lumbar segments and role of the zygapophysial joints

Wachowski M. M., Hawellek T., Hubert J., Lehmann A., Mansour M., Dumont C., Dorner J., Raab B. W., Kubein-Meesenburg D., Nägerl H.

Acta of Bioengineering and Biomechanics

2010

Vol. 12, no. 4

39-47

The biomechanical role of the zygapophysial joints was investigated for axial rotations of lumbar segments by recording the positions of the instantaneous helical axis (IHA) against the axial rotational angle and by relating these IHA-positions to anatomical landmarks. Cyclically varying pure axial moments were applied to 3 L1/L2, 7 L3/L4 and 3 L4/L5 segments. There were 800 segment positions per cycle taken by a custom-made high precision 3D-position measuring system. In intact segments IHA-migration reached from one zygapophysial joint to the other IHA-paths came up to 10–60 mm within small angular intervals (š1 deg). After removing the right joints, IHA-migration remained comparable with that of intact segments only for segment positions rotated to the right. Rotation to the left, however, approximately yielded stationary IHA-positions as found after resection of both joints. Hence, IHA-migration is determined by the joints already for small rotational angles. Each type of segment showed a typical pattern of IHA-migration.