Algorithm On Approximation Of Offset Curves Based On Singular Blending Spline

Offset curves, also called parallel or potential difference curves, are defined as locus of the points which are at constant distance d along the normal from the base curves. It has been a hot topic on computer aided geometric design in recent ten years.In chapter one, we firstly review the phytogeny of the offset curves, then summarize PH curves and OR curves concerning the inherited geometric structure of offset curves.In chapter two, after discussing the possible problems encountered in the offset approximation, an overview of algorithms on the offset approximation is presented, and several representative algorithms are introduced in detail.In chapter three, a singular blending spline curve is introduced. The curve can be obtained by blending a cubic B-spline curve with a singular polygon. Because of complicated expression and difficult application, the author converts the singular blending curve to a uniform B-spline curve by reparametrization, basis transformation and matrix transformation.In chapter four, an algorithm on offset approximation based on a singular blending spline is presented, which is a key point in this paper. After getting a prior formula for subdividing the base curve, which depends on the user defined tolerance and the bound on the second derivative of the normal curve, we firstly sample the base curve and get sample points on the exact offset curve, then interpolate these points with a cubic singular blending spline curve, and finally convert it into a uniform cubic B-spline curve. The error can be strictly controlled by computing the number of subdivision layers and adjusting the blending factor. Meanwhile, the algorithm can also be generalized to the case of surfaces.