The Rational Approximation Of Planar Bézier Offset Curve

Offset curves, also called parallel curves, are defined as locus of the points which are at constant distant d along the normal from the generator curves. Offsets are widely used in various engineering applications, which is one of the most important geometric operations in CAD/CAM systems. In general, an offset is functionally more complex than its progenitor because of the square root involved in the expression of the unit normal vector. If the progenitor is a rational curve, then its offset is usually not a rational curve except for special cases. Special cases for curves include straight lines and circles while planes, spheres, circular cylinders, circular cones, toruses and cyclides are the special cases for surfaces. Many methods approximating the offset curves should be used to make CAD/CAM systems operated effectively. At present, the main offset approximation methods are: (1) control polygon-based methods;(2) interpolation methods;(3) circle approximation methods. The first and second methods are simple and look more intuitionistic, but have lower accuracy and larger storage. The third method has a good accuracy, but the disadvantage is a high degree.In this thesis, the algebraic rational approximation algorithm of the planar Bezier offset curve is given, which is based on the approximation of the norm of parametric speed. This method can not only preserve the offset curve moving along the normal direction, which is convenient for the alternate operation, but also decrease the computational complexity and the space of storage by using the recursion property of the continued fraction. At the same time, the method leads to significant improvements in accuracy and efficiency. A numerical example is given to show the effectiveness of the new method.