New Algorithms For Rational Approximation Of Offset Curves

Offset curves, also called parallel curves, and defined as locus of the points which are at constant distant d along the normal from the generator curves. Offset curves are widely used in various engineering applications, which is one of the most important geometric operations in CAD/CAM systems. In general, the offset curve of a rational curve is not a rational curve except for special cases. In order to compatible with CAD/CAM system, many methods approximating the offset curves should be used. In the thesis, two new methods for rational approximation of offset curve are presented, which are based on the approximation of the norm of parametric speed and circle approximation. At first, both the Tchebyshev approximation and the Tchebyshev-Padéapproximation of parametric speed of Said-Bézier curves are presented, and two rational approximation functions of the offset curves are also obtained. Then using Tchebyshev polynomial to approximate circular arcs, the arbitrary order Bézier polynomial approximation of circle is obtained by transformation of Tchebyshev-Bernstein polynomial basis, and we also use this circular arcs approximation method to obtain an offset approximation curve which has the same degree and the same form with the base curve. Finally, some examples are given to show the effectiveness of these methods, and the result is compared with other methods.