THE CRITERION FOR TRANSFERABLE SELF-CONSISTENTLY TRANSLATIONALITY OF COORDINATE TRANSFORM OPERATORS AND REFERENCE FRAMES IN UNIVERSAL KINEMATICS

From an intuitive point of view universal kinematics are collections (sets) of changing objects, which evolve, being in a certain spatial-geometric environment, and evolution of whi- ch can be observed from many different frames of reference. Moreover, the definition of uni- versal kinematics impose the existence of some (preassigned) universal coordinate transform between every two reference frames of such kinematics. Transferable self-consistently translati- onal reference frames (in vector universal kinematics) are interesting because for such reference frames it is possible to give a clear and unambiguous definition of displacement of a moving reference frame relative to a fixed one, which does not depend on the choice of a fixed point in the moving frame of reference. In the present paper it is shown that an arbitrary reference frame m is transferable self-consistently translational relatively to a reference frame l (in some vector uni- versal kinematics F) if and only if the coordinate transform operator from the reference frame m to the reference frame l is transferable self-consistently translational. Therefore transferable self-consistently translational coordinate transform operators describe the conversion of coordi- nates from the moving and transferable self-consistently translational frame of reference to the (given) fixed frame in vector universal kinematics. Also in the paper it is described the structure of transferable self-consistently translational coordinate transform operators (this is the main result of the article). Using this result it have been obtained the necessary and sufficient conditi- on for transferable self-consistently translationality of one reference frame relatively to another in vector universal kinematics.

Development of Quick Three-Dimensional Shape Measurement Projection Mapping System Using a Whole-Space Tabulation Method

A grating projection method is often used as a highly accurate 3D shape measurement method. A real-time 3D shape measurement system can also be applied to measure a wide and smooth curved surface, such as in sheet metal processing. In this case, operators take much effort to recognize the positions of some problem areas on an object from a measured result displayed on a monitor. This study develops a projection mapping system projecting an evaluation image, such as height, displacement, gradient, curvature factor, and area of defect, onto an object. These evaluation results are obtained from the measured 3D shape. The evaluation image should be deformed according to the 3D shape of the object because the camera and projector positions are different. Therefore, this study proposes a method to quickly produce a deformed evaluation image using a whole-space tabulation method. A coordinate transform table allowed the conversion of a camera pixel coordinate into a projector pixel coordinate by using reference planes to apply deformation to the evaluation image according to the measured 3D shape. The quick coordinate transformation from a camera pixel coordinate into a projector pixel coordinate was realized using the coordinate transform table. This is a key idea of this study. It was confirmed that the coordinate transformation from the camera pixel coordinate to the projector coordinate could be performed in 4.5 ms using the coordinate transform table. In addition, 3D shape measurement projection mapping was applied to a curved sheet metal with small deformation, and the deformation part was clearly shown by projecting the height distribution. The architecture and the experimental results are shown herein.