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Content available Thickness analysis of a saddle
EN
The study analyzes the thickness of the solid-surface lobe saddle formed by two geometric surface parabolic hyperboloids as a result of the translation. The obtained results demonstrate the dependence between the thickness of the saddle and the ratio of height and base edge length of the base cuboid. With some values of that ratio the thickness of the saddle fit in acceptable deviations from the initial value.
PL
W pracy dokonano analizy grubości plata-bryły powierzchni siodłowej powstałego w wyniku przesunięcia geometrycznej powierzchni hiperboloidy parabolicznej o dany wektor. Otrzymane rezultaty wskazują na istnienie zależności grubości od proporcji wysokości i długości krawędzi podstawy prostopadłościanu bazowego. Przy pewnych proporcjach, grubości te mieszczą się w dopuszczalnych granicach odchyleń od założonej grubości płata.
EN
Purpose: The purpose is to present a mathematical model of the function of the thumb carpometacarpal joint (TCMCJ) based on measurements of human joints. In the TCMCJ both articulating surfaces are saddle-shaped. The aim was to geometrically survey the shapes of the articulating surfaces using precise replicas of 28 TCMCJs. Methods: None of these 56 articulating surfaces did mathematically extend the differential geometrical neighbourhood around the main saddle point so that each surface could be characterised by three main parameters: the two extreme radii of curvature in the main saddle point and the angle between the saddles’ asymptotics (straight lines). Results: The articulating surfaces, when contacting at the respective main saddle points, are incongruent. Hence, the TCMCJ has functionally five kinematical degrees of freedom (DOF); two DOF belong to flexion/extension, two to ab-/adduction. These four DOF are controlled by the muscular apparatus. The fifth DOF, axial rotation, cannot be adjusted but stabilized by the muscular apparatus so that physiologically under compressive load axial rotation does not exceed an angle of approximately ±3°. Conclusions: The TCMCJ can be stimulated by the muscular apparatus to circumduct. The mechanisms are traced back to the curvature incongruity of the saddle surfaces. Hence we mathematically proved that none of the individual saddle surfaces can be described by a quadratic saddle surface as is often assumed in literature. We derived an algebraic formula with which the articulating surfaces in the TCMCJ can be quantitatively described. This formula can be used to shape the articulating surfaces in physiologically equivalent TCMCJ-prostheses.
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