Three-dimensional Coordinate Space Research Articles

Solutions of the nonlinear 3-wave equations in three spatial dimensions

Recently Ablowitz and Haberman have shown that, in three spatial dimensions, the nonlinear 3-wave evolution equation results from the compatibility condition between two well-defined first-order linear differential 3×3 systems having common solutions. We construct inversionlike integral equations (I.E.) associated with both these two linear differential systems so that the solutions of the I.E. embody their compatibility conditions. The scalar kernels of these I.E. depend upon three independent variables in such a way that there exist degenerate kernels confined in the three-dimensional coordinate space. Consequently we exhibit, for the nonlinear 3-wave evolution equations, an infinite number of solutions which, at fixed time, are confined in the three-dimensional coordinate space.

Patellar tracking patterns measurement by analytical X-ray photogrammetry

Variants of normal patellar tracking patterns have been implicated in the pathomechanics of patellar chondromalacia. Since these patterns are dependent upon the dynamic interaction of the quadriceps mechanism and knee geometry, a method for measuring three-dimensional patellar displacement during knee motion is needed before patellar pathomechanics can be fully understood. Once this information is available, a more rational basis for adjusting abnormal tracking patterns can be formulated. A method for accurately measuring patellar displacements using X-ray photogrammetry has been developed. Stainless steel 1 mm balls are implanted percutaneously into the patella and distal femur to provide a radiopaque landmark system which represents the two structures. In order to provide optimal measurement geometry, an analytical photogrammetry system is used. This means that the perspective centers of the X-ray anodes must be calculated before the three-dimensional space coordinates of each landmark can be determined. The displacements between the patella and femoral condyles are derived by computing the relative motion between their respective ball systems. Anterior-posterior, medial-lateral and tilting displacements of the patella were determined for the valgus osteotomy patients shown in this research. The positional accuracy of the patella can be determined by comparing distances between two fixed points (steel balls) in one position with the same distances as computed independently from the other positions. The mean square error is a measure of the patellar position accuracy. Since the implanted markers remain in place permanently, the repeatability of this reference system for future measurements is assured. This method is applicable to a variety of situations requiring the determination of small, three-dimensional displacements between two rigid structures.

Theory of Three-Axis Stable Systems Employing External Information

I (l) and astronomical (2) systems may be used to compute x, y, z (coordinates in three-dimensional space) and x, y, z (the corresponding velocities). An inertial system can give the velocities reliably, but the coordinates are subject to major errors because of drift in the gyros and instability in the system generally. An astronomical system can give the coordinates accurately, but the velocities, which are deduced b} differentiation, are subject to very large errors. A combined (astro-inertial) system can give coordinates and velocities with the desired accuracy. Previous work (1-3) forms the basis of the present treatment of a three-axis astro-inertial system; we assume that the astronomical information is used not merely to correct the coordinate readings but also to control the response of the inertial system (to suppress instability, to cause damping, and to reduce the response time).