A simple and efficient method is developed for generating oftset curves interactively using a set ot control knots for a uniform cubic B-spline.
geometry, curves, oEsets
The construction of offset curves has many applications in industry. For example, it can be used in the numerical control of sewing machines in the textile and shoe industry or in the numerical control of milling machines in the automobile industry. The problems of generating offset curves have been dealt with by a few authors ~-3.
Klass 2 has generated a curve and its offset as segments of cubic splines. A cubic spline segment is constructed from two given end points and their tangent vectors. The offset of these end points and the corresponding tangent vectors will then be computed using Newton' s method for solving systems of nonlinear equations. Another cubic spline segment is constructed that gives a good approximation to the offset line of the original curve if the curvatures at the end points of the segment do not differ too much. If the result is not good enough, the segment will be divided into two parts and the whole process will start again.
The main disadvantage of using cubic splines is the requirement that the tangent vectors need to be known in advance. Furthermore, since the offset of end derivatives is too obscure to be visualized, it is not easy to manipulate the curve nor to predict the modification, hence the method is not suitable for an interactive curve design environment.
Cubic B-splines, on the other hand, offer flexibility in the modification of curves since their control polygons replicate the curves and the effects of perturbation are Iocalised. Furthermore, their computation is more efficient.
Tiller and Hanson ~ have implemented an algorithm to create an offset curve using the offset of the control vertices of the original curve.
In this paper, a simple and fast method is discussed that allows the generation of an offset curve using a set of control knots for a uniform cubic B-spline. As the control knots lie on the curve, it is easier to visualize and modify the curve in the interactive designing stage.