Functional composition algorithms via blossoming

DeRose, Tony; Goldman, Ron; Hagen, Hans; Mann, Stephen

doi:10.1145/151280.151290

Cited by 79 publications

(57 citation statements)

References 7 publications

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“…By choosing suitable degrees and the knot sequences, we can compute the control points exactly, using formulas for the composition and multiplication of B-spline functions [3,13]. In general, however, the resulting B-spline surface has high degrees and a large number of control points.…”

Section: Generating the Blade Surfacementioning

confidence: 99%

Medial design of blades for hydroelectric turbines and ship propellers

Rossgatterer¹,

Jüttler²,

Kapl³

et al. 2012

Computers & Graphics

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We present a method for constructing blades of hydroelectric turbines and ship propellers based on design parameters that possess a clear hydraulic meaning. The design process corresponds to the classical construction of a blade using the medial surface of the blade and profile curves attached to it. The main new contribution of the paper consists in realizing this construction using B-spline techniques. In particular, it is shown how to obtain blade boundary surfaces (which describe the pressure and the suction side of the blade) which are joined with C 1 -smoothness along the leading edge. Moreover, special attention is paid to the construction of propeller blades with a well-defined tangent plane at the tip, which is a singular point of the blade boundary surfaces. In order to guarantee these smoothness properties, we generate and analyze singularly parameterized medial surfaces. We contribute novel shape modeling techniques that are based on singular parameterizations and demonstrate their potential for applications in industry. Finally, it is shown how to represent the blades as B-spline surfaces with a relatively small number of control points.

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Section: Generating the Blade Surfacementioning

confidence: 99%

Medial design of blades for hydroelectric turbines and ship propellers

Rossgatterer¹,

Jüttler²,

Kapl³

et al. 2012

Computers & Graphics

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“…We do so by extending the results for bivariate polynomial surfaces [18] to multivariate polynomials. Before proceeding, we mention that the problem of computing the Bézier control net can be formulated as a problem of changing from the monomial basis to the Bézier basis, which can be solved using the algorithms proposed in [15,22]. These algorithms also make use of the blossoming principle.…”

Section: Theorem 2 (Blossoming Principle) For Any Polynomialmentioning

confidence: 99%

Approximate Reachability Computation for Polynomial Systems

Dang

2006

Hybrid Systems: Computation and Control

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Abstract. In this paper we propose an algorithm for approximating the reachable sets of systems defined by polynomial differential equations. Such systems can be used to model a variety of physical phenomena. We first derive an integration scheme that approximates the state reachable in one time step by applying some polynomial map to the current state. In order to use this scheme to compute all the states reachable by the system starting from some initial set, we then consider the problem of computing the image of a set by a multivariate polynomial. We propose a method to do so using the Bézier control net of the polynomial map and the blossoming technique to compute this control net. We also prove that our overall method is of order 2. In addition, we have successfully applied our reachability algorithm to two models of a biological system.

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“…The composition of B-spline and NURBS freeform is not much more complex conceptually as one can always subdivide the piecewise polynomial/rational into its polynomial/rational pieces, compute the composition and reconstruct the final composed shape. More on composition can be found in [4], [6].…”

Section: The Composition Computation and Alternativesmentioning

confidence: 99%

C¹Continuous Toolpath Generation Toward 5-axis High Speed Machining

Elber

Cohen

Drake

2006

Computer-Aided Design and Applications

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We present a scheme to construct C 1 continuous toolpath for 5-axis pocketing of high speed machining. A continuous toolpath synthesis is proposed, that forms out circular segments of maximally inscribed sizes and linear segments connecting these circular segments. Extending earlier work that employed the Medial Axis Transform to compute these maximally inscribed circles for 2D planar pockets, this work examines general pockets in ℜ 3 represented as NURBS surfaces, and in 5-axis machining setup. Results are demonstrated by employing t he proposed scheme toward the machining of impellers. Possible extensions toward a C 2 continuous toolpath is discussed as well.

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Functional composition algorithms via blossoming

Cited by 79 publications

References 7 publications

Medial design of blades for hydroelectric turbines and ship propellers

Medial design of blades for hydroelectric turbines and ship propellers

Approximate Reachability Computation for Polynomial Systems

C¹Continuous Toolpath Generation Toward 5-axis High Speed Machining

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Functional composition algorithms via blossoming

Cited by 79 publications

References 7 publications

Medial design of blades for hydroelectric turbines and ship propellers

Medial design of blades for hydroelectric turbines and ship propellers

Approximate Reachability Computation for Polynomial Systems

C1Continuous Toolpath Generation Toward 5-axis High Speed Machining

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Product

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About

C¹Continuous Toolpath Generation Toward 5-axis High Speed Machining