The µ-bases of rational curves/surfaces are newly developed tools which play an important role in connecting parametric forms and implicit forms of the rational curves/surfaces. They provide efficient algorithms to implicitize rational curves/surfaces as well as algorithms to compute singular points of rational curves and to reparametrize rational ruled surfaces. In this paper, we present an efficient algorithm to compute the µ-basis of a rational curve/surface by using polynomial matrix factorization followed by a technique similar to Gaussian elimination. The algorithm is shown superior than previous algorithms to compute the µ-basis of a rational curve, and it is the only known algorithm that can rigorously compute the µ-basis of a general rational surface. We present some examples to illustrate the algorithm.
The optimal feedrate planning problem for the five-axis parametric tool path remains challenging due to the nonlinear relationships between the joint space and the Cartesian space. We present a novel and complete optimal feedrate planning method for a five-axis parametric tool path by constraining the velocity, acceleration, jerk in the joint space and the chord error in the Cartesian space. Our method formulates the problem as an optimal control problem, and we propose an iteration control vector parametrization (ICVP) method to compute the optimal solution. Compared with the new development five-axis feedrate planning methods, our solution satisfies ”bang-bang” optimal control, and each constraint is strictly under the limits globally. Examples and comparisons with two other methods are demonstrated to show the effectiveness of the algorithm.
Combining computer-aided design and computer numerical control (CNC) with global technical connections have become interesting topics in the manufacturing industry. A framework was implemented that includes point clouds to workpieces and consists of a mesh generation from geometric data, optimal surface segmentation for CNC, and tool path planning with a certified scallop height. The latest methods were introduced into the mesh generation with implicit geometric regularization and total generalized variation. Once the mesh model was obtained, a fast and robust optimal surface segmentation method is provided by establishing a weighted graph and searching for the minimum spanning tree of the graph for extraordinary points. This method is easy to implement, and the number of segmented patches can be controlled while preserving the sharp features of the workpiece. Finally, a contour parallel tool-path with a confined scallop height is generated on each patch based on B-spline fitting. Experimental results show that the proposed framework is effective and robust.
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