ANNALS OF THE UNIVERSITY OF ORADEA FASCICLE OF TEXTILES, LEATHERWORK A STUDY FOR THE MATHEMATIC MODELING OF 2D IRREGULAR SHAPES FOR FOOTWEAR CAD SYSTEM
For using a specialized footwear CAD system it's imperative to know the analytical expression of the outlines of the footwear patterns. This brings us to the field of mathematical modeling. Mathematic modeling is based on the equation of the function defining the outline of the model contour. Shapes, contours cannot be identified, in designing, by simple function of the form y=f(x), because most of them have irregular forms, with many concavities and convexities, wich explains why their form is
... intrisically dependent on the coordinates system. For example, if we want to plot a curve, it is absolutely necesary that we choose the right set of contour points in a system of coordinates, but the important factor in determining the form of the object is the relation between these points, not that between the points and the randomly chosen coordinates system.Furthe more, the contour forms may have vertical tangents. If the shape were represented by a function y=f(x), the vertical tangents would be an incovenient in designing, which might be avoided by an aproximation of analytic function (e.g. of polynomials) For all these reasons, the dominant representation of shapes in CAD is not possible a function y=f(x) but a set of function wich can be obtained on various partions. This paper presents a study regarding the interpolation of the footwear components and outlines contours and the graphic visualization, using the following methods: Lagrange, B-Spline, Bezier. 1. MATHEMATIC MODELLING OF COMPONENTS. REQUIEREMENTS Geometrical shapes commonly used in Computer Aided Design (CAD) systems can be defined by several points obtained through the digitizing process. The coordinates of the points situated between two nodes can be approximated through both analytic and graphic methods, with interpolation curves. Thus, the analytic expression of the curve that approximates the points will be a interpolation function. The graphical form will be represented by a curve that crosses all the coordinates of the digitized points, without bringing any mutations of the initial curve. The analytic functions, approximating the contour to be designed, may be obtained by extrapolation if and only if there has been made a numeric coding of the geometric body, wich should furnish all necesary data. As, however, many contours have irregular forms, the mathematic models are aproximative. Taking into account the advantage of computation technigue for these last years, we can asseses that highly performant programmes lead to approximations with minimum of errors. Mention has to be of the processing centers with numerical command existent in highly developed countries, where, on the basis the coorditates of a set of points, the model to be designed in physically made up with the desired accuracy. On a local scale, although the description and modelling of bodies, wich is an initial stage of data input, is slow (form keyboard) or very expensive (with specialized equipment) all efforts will be warranted by the spectacular results obtained.