A novel approach for computing C-2-continuous offset of NURBS curves

Computing offset curves is an important geometric operation in areas of CAD/CAM, robotics, cam design and many industrial applications. In this paper, an algorithm for computing offsets of NURBS curves using C-2-continuous B-spline curves is presented. The progenitor curve in database is initially approximated by a line-fitting curve, and then the exact offset of this line-fitting curve is introduced as an initial offset. Based on the initial offset and a set of selected knots, an intended C-2-continuous B-spline curve is subsequently constructed. The method uses a new error-measuring scheme, which is based on the convex hull property of Bezier curves and the idea of cumulative errors, to calculate the global error bound of offset approximation. The method obtains offset curves with C-2 continuity and guarantees that the actual error bound is precisely within the prescribed tolerance. In addition, it also allows one to selectively parametrize the offset curve

NURBS output based tool path generation for freeform pockets

A robust method is proposed to generate tool paths for NURBS-based machining of arbitrarily shaped freeform pockets with islands. Although the input and output are all of higher-degree NURBS curves, only one simple category of geometric entities, i.e., line segments, is required for initial offsetting and for detecting and removing self-intersecting loops. Furthermore, using those linear non-self-intersecting offsets as the legs of NURBS control polygons, NURBS-format tool paths can be smoothly reconstructed with G(1)-continuity, no overcutting, no cusps, and global error control. Since all operations involved in computing tool path curves are linear geometric calculations, the method is robust and simple. Examples with integrated rough and finish cutting tool paths of pockets demonstrate the usefulness and effectiveness of this method

Multipatched B-spline surfaces and automatic rough cut path generation

A method is proposed to generate rough cut machining tool paths for multipatched B-spline surfaces. For a group of B-spline surfaces, smoothly connected, separated or intersecting one another, this method can guarantee generating paths without overcutting any of the surfaces under consideration. The use of multipatched surfaces to construct complex surfaces can greatly increase the controllability and friendliness in the design of surfaces. A method for rough cut planning by convex hull boxing is proposed to generate paths guaranteeing no over-cut. since no computation for solving nonlinear equation is involved, the rough cut plan is robust and efficient

One-sided offset approximation of freeform curves for interference-free NURBS machining

Generating valid tool path Curves in NURBS form is important in realizing an efficient NURBS machining. In this paper, a method for computing one-sided offset approximations of freeform curves with NURBS format as tool paths is presented. The approach first uses line segments to approximate the progenitor curve with one-sided deviations. Based on the obtained line approximating curve and its offsets, a unilateral tolerance zone (UTZ) is constructed subsequently. Finally, a C(1)-continuous and completely interference-free NURBS offset curve is generated within the UTZ to satisfy the required tolerance globally. Since all of the geometric computations involved are linear, the proposed method is efficient and robust. Interference-free tool path generation thus can be achieved in NURBS based NC machining. (C) 2008 Elsevier Ltd. All rights reserved

Feature design on deformation of freeform surfaces using range control

This research is to develop a freeform surface design method which uses feature-based design approach to process surface deformation. Using this method called range control, a user modifies or changes the position of a characteristic control point of the control polyhedron for a B-spline surface, and regional surface deformation is then carried out through automatic displacements of a group of control points surrounding the characteristic point. The position changes of the group of control points are the interpolations of the maximum movement of the characteristic point related to a feature dimension and zero movements of some fixed points on the designated boundary. In this range control surface deformation method, three interpolating approaches are proposed to smooth out the changes between the characteristic vertex and fixed vertices. They are based on the linear interpolation under the consideration of index, distance and angle distributions. The ideas are implemented on a feature-based designing system for shoe lasts. Design examples ale given to shore that the range control method is simple, fast and practically applicable for shape deformation of freeform surfaces

One-sided arc approximation of B-spline curves for interference-free offsetting

An algorithm is presented to generate a piecewise curve with G(1) continuity using area, which approximates a B-spline curve without crossing the curve. Initially, the B-spline curve in database is decomposed into piecewise Bezier curves. Using convex hulls of the Bezier curves to protect the original curve from interference, the line segments are chosen all on the same side of the B-spline curve to give a one-sided approximating curve. Based on the obtained approximating line segments, biarc fitting and single are fitting methods are applied to construct a smooth, G(1) continuous curve which does not cross the original curve. If the resulting curve is offset to generate tool paths for pocketing boundaries with B-spline curves, the over-cutting problem can be eliminated completely, and abrupt direction changes on tool paths can be greatly reduced. This method can also be applied to generate collision free paths for robot arms. (C) 1999 Elsevier Science Ltd. All rights reserved

Reverse engineering of sculptured surfaces by four-axis non-contacting scanning

Reverse engineering can be divided into Soul steps: surface data extraction; extracted data processing; surface reconstruction; and CNC part program generation. The goal of this research is to deal with data extraction fr-om complex sculptured surfaces, redundant data elimination, and surface reconstruction problems of reverse engineering. To deal with the complex-sculptured-surface data extraction problem, an efficient four-axis non-contacting surface data scanning system, which integrates a PC-based XYZ table, a laser displacement meter (LDM), and a personal computer is developed. In this fonr-axis system, an LDM an XYZ table and a PC are used as a displacement sensor, sensor carrier and central controller respectively. In extracted data processing, we apply the medium filter technique to smooth data points, and propose a simple algorithm to discard efficiently those redundant measured points according to the required degree of accuracy. While in the surface reconstruction, we first try To convert all eliminated column or row data into spline curves. Methods for 2D spline (u, v directions) curve construction are described Non-meshed 2D spline curves are then blended to a spline surface in terms of a sparse matrix data structure. Experimental results show that the proposed four-axis surface data scanning machine can be programmed to measure most of the complex sculptured surfaces which is nor possible using a three-axis machine. The proposed extracted dataprocessing method can reduce the surface reconstruction time substantially for only the cost of a little extracting lime and minor modelling errors