Li-Yong Shen

Curve fitting and optimal interpolation for CNC machining under confined error using quadratic B-splines

In CNC machining, fitting the polyline machining tool path with parametric curves can be used for smooth tool path generation and data compression. In this paper, an optimization problem is solved to find a quadratic B-spline curve whose Hausdorff distance to the given polyline tool path is within a given precision. Furthermore, adopting time parameter for the fitting curve, we combine the usual two stages of tool path generation and optimal velocity planning to derive a one-step solution for the CNC optimal interpolation problem of polyline tool paths. Compared with the traditional decoupled model of curve fitting and velocity planning, experimental results show that our method generates a smoother path with minimal machining time. The explicit Hausdorff distance of a line segment and a quadratic curve is given.G01 codes can be fitted by quadratic B-splines with confined error.We combine the tool path generating and optimal velocity planning in one step.We simulate the manufacture process with our proposed method.

Characterization of rational ruled surfaces

The algebraic ruled surface is a typical modeling surface in computer aided geometric design. In this paper, we present algorithms to determine whether a given implicit or parametric algebraic surface is a rational ruled surface, and in the affirmative case, to compute a standard parametric representation for the surface.

Certified approximation of parametric space curves with cubic B-spline curves

Approximating complex curves with simple parametric curves is widely used in CAGD, CG, and CNC. This paper presents an algorithm to compute a certified approximation to a given parametric space curve with cubic B-spline curves. By certified, we mean that the approximation can approximate the given curve to any given precision and preserve the geometric features of the given curve such as the topology, singular points, etc. The approximated curve is divided into segments called quasi-cubic Bezier curve segments which have properties similar to a cubic rational Bezier curve. And the approximate curve is naturally constructed as the associated cubic rational Bezier curve of the control tetrahedron of a quasi-cubic curve. A novel optimization method is proposed to select proper weights in the cubic rational Bezier curve to approximate the given curve. The error of the approximation is controlled by the size of its tetrahedron, which converges to zero by subdividing the curve segments. As an application, approximate implicit equations of the approximated curves can be computed. Experiments show that the method can approximate space curves of high degrees with high precision and very few cubic Bezier curve segments.

Inherently improper surface parametric supports

We identify a class of monomial supports that are inherently improper because any surface rational parametrization defined on them is improper. A surface support is inherently improper if and only if the gcd of the normalized areas of the triangular sub-supports is non-unity. The constructive proof of this claim can be used to detect all and correct almost all improper surface parametrizations due to improper supports.

Collision and intersection detection of two ruled surfaces using bracket method

Collision and intersection detection of surfaces is an important problem in computer graphics and robotic engineering. A key idea of our paper is to use the bracket method to derive the necessary and sufficient conditions for the collision of two ruled surfaces. Then the numerical intersection curve can be characterized. The cases for two bounded ruled surfaces are also discussed.

Proper reparametrization of rational ruled surface

In this paper, we present a proper reparametrization algorithm for rational ruled surfaces. That is, for an improper rational parametrization of a ruled surface, we construct a proper rational parametrization for the same surface. The algorithm consists of three steps. We first reparametrize the improper rational parametrization caused by improper supports. Then the improper rational parametrization is transformed to a new one which is proper in one of the parameters. Finally, the problem is reduced to the proper reparametrization of planar rational algebraic curves.

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We investigate conditions under which the resultant of a

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-Bases for Rational Tensor Product Surfaces and Extraneous Factors Associated to Bad Base Points and Anomalies at Infinity

-basis for a rational tensor product surface is the implicit equation of the surface without any extraneous factors. In this case, we also derive a formula for the implicit degree of the rational surface based only on the bidegree of the rational parametrization and the bidegrees of the elements of the

Implicitizing Rational Tensor Product Surfaces Using the Resultant of Three Moving Planes

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