• Korean Journal of Computational Design and Engineering
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• v.4 no.4
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• pp.350-354
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• 1999

The problem of the computation of derivatives arises in various applications of rational Bezier curves. These applications sometimes require the computation of derivative on numerous points. Therefore, many researches have dealt with the representation for the computation of derivatives with the small computation error. This paper compares the performances of the representations for the derivative of rational Bezier curves in the performances. The performance is measured as computation requirements at the pre-processing stage and at the computation stage based on the theoretical derivation of computational bound as well as the experimental verification. Based on this measurement, this paper discusses which representation is preferable in different situations.

• 한국전산유체공학회:학술대회논문집
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• 2002.05a
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• pp.94-101
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• 2002

The steam turbine rotor blade is designed using the Turbine Rotor Design Package developed by the authors. It can quickly accomplish blade shape design in the power plant industry. The quasi-3d code is employed for analysis of passage flow in the blade sections. Iterative change of each blade shape is made by moving position of control points in the Bezier curve under GUI(graphic user interface) environment. The full 3-D blade shape is obtained by stacking of the section blades.

• International Journal of Control, Automation, and Systems
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• v.2 no.1
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• pp.107-117
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• 2004

A new algorithm for planning a collision-free path based on an algebraic curve as well as the concept of path space is developed. Robot path planning has so far been concerned with generating a single collision-free path connecting two specified points in a given robot workspace with appropriate constraints. In this paper, a novel concept of path space (PS) is introduced. A PS is a set of points that represent a connection between two points in Euclidean metric space. A geometry mapping (GM) for the systematic construction of path space is also developed. A GM based on the 2$^{nd}$ order base curve, specifically Bezier curve of order two is investigated for the construction of PS and for collision-free path planning. The Bezier curve of order two consists of three vertices that are the start, S, the goal, G, and the middle vertex. The middle vertex is used to control the shape of the curve, and the origin of the local coordinate (p, $\theta$) is set at the centre of S and G. The extreme locus of the base curve should cover the entire area of actual workspace (AWS). The area defined by the extreme locus of the path is defined as quadratic workspace (QWS). The interference of the path with obstacles creates images in the PS. The clear areas of the PS that are not mapped by obstacle images identify collision-free paths. Hence, the PS approach converts path planning in Euclidean space into a point selection problem in path space. This also makes it possible to impose additional constraints such as determining the shortest path or the safest path in the search of the collision-free path. The QWS GM algorithm is implemented on various computer systems. Simulations are carried out to measure performance of the algorithm and show the execution time in the range of 0.0008 ~ 0.0014 sec.

• Journal of the Korean Society for Aeronautical & Space Sciences
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• v.39 no.5
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• pp.424-432
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• 2011

Shape parameters and design variables for a centrifugal compressor impeller were investigated for optimizing a centrifugal compressor. In order to compare the performance of an optimized impeller with the performance of the original impeller, an already tested impeller was chosen and design variables for optimization were selected. The meridional shapes at the shroud and at the hub were re-designed using the Bezier curve. The camber-lines of the impeller blade at the hub and at the tip were also expressed by the Bezier curve. The shape curves for impeller could be expressed using 6-8 control points. Among them, eight control points which have strong effect to the shape can be selected as design variables for optimization. Therefore, any impeller which is expressed by data points for its shape can be optimized using few design variables.

• The Transactions of the Korean Institute of Electrical Engineers B
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• v.53 no.8
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• pp.469-476
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• 2004

In this paper, a 3D shape optimization algorithm which guarantees a smooth optimal shape is presented using parameterized sensitivity analysis. The design surface is parameterized using Bezier spline and B-spline, and the control points of the spline are taken as the design variables. The parameterized sensitivity for the control points are found from that for nodal points. The design sensitivity and adjoint variable formulae are also derived for the 3D non-linear problems. Through an application to the shape optimization of 3D electromagnet to get a uniform magnetic field, the effectiveness of the proposed algorithm is shown.

• Proceedings of the KIEE Conference
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• 2003.07b
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• pp.915-917
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• 2003

In this paper, a 3D shape optimization algorithm which guarantees a smooth optimal shape is presented using parameterized sensitivity analysis. The design surface is parameterized using Bezier spline and the control points of the spline are taken as the design variables. The parameterized sensitivity for the control points are found from that for nodal Points. The design sensitivity and adjoint variable formulae are also derived for the 3D non-linear problems. Through an application to the shape optimization of 3D electromagnet to get a uniform magnetic field, the effectiveness of the proposed algorithm is shown.

• Korean Journal of Computational Design and Engineering
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• v.3 no.3
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• pp.183-191
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• 1998

There are many available algorithms based on the different approaches to solve the intersection problems between two curves. Among them, the implicitization method is frequently used since it computes precise solutions fast and is robust in lower degrees. However, once the degrees of curves to be intersected are higher than cubics, its computation time increases rapidly and the numerical stability gets worse. From this observation, it is natural to transform the original problem into a set of easier ones. Therefore, curves are subdivided appropriately depending on their geometric behavior and approximated by a set of rational quadratic Bezier cures. Then, the implicitization method is applied to compute the intersections between approximated ones. Since the solutions of the implicitization method are intersections between approximated curves, a numerical process such as Newton-Raphson iteration should be employed to find true intersection points. As the seeds of numerical process are close to a true solution through the mix-and-match process, the experimental results illustrates that the proposed algorithm is superior to other algorithms.

• Journal of the Korean Society for Precision Engineering
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• v.15 no.8
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• pp.46-59
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• 1998

This paper presents shape design, surface construction, and cutting path generation for the surface of marine ship propeller blades. A propeller blade should be designed to satisfy performance constraints that include operational speed which impacts rotations per minutes, stresses related to deliverable horst power, and the major length of the marine ship which impacts the blade size and shape characteristics. Primary decision variables that affect efficiency in the design of a marine ship propeller blade are the blade diameter and the expanded area ratio. The blade design resulting from these performance constraints typically consists of sculptured surfaces requiring four or five axis contoured machining. In this approach a standard blade geometry description consisting of blade sections with offset nominal points recorded in an offset table is used. From this table the composite Bezier surface geometry of the blade is created. The control vertices of the Hazier surface patches are determined using a chord length fitting procedure from tile offset table data. Cutter contact points and path intervals are calculated to minimize travel distance and production time while maintaining a cusp height within tolerance limits. Long path intervals typically generate short tool paths at the expense of increased however cusp height. Likewise, a minimal tool path results in a shorter production time. Cutting errors including gouging and under-cut, which are common errors in machining sculptured surfaces, are also identified for both convex and concave surfaces. Propeller blade geometry is conducive to gouging. The result is a minimal error free cutting path for machining propeller blades for marine ships.

• Journal of Korean Institute of Industrial Engineers
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• v.21 no.4
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• pp.529-540
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• 1995

The paper deals with a procedure for constructing a composite surface interpolating a polygonal curve mesh defined from 3D scattered points. The procedure consists of a poly-angulation, construction of a curve net, and interpolation of the curve net. The poly-angulation contains a stage that changes a triangular edge net obtained from a triangulation into a poly-angular edge net. A curve net is constructed by replacing edges on the edge net with cubic Bezier curves. Finally, inside of an n-sided polygon is interpolated by n subdivided triangular subpatches. The method interpolates given point data with relatively few triangular subpatches. For an n-sided polygon, our method constructs an interpolant with n subdivided triangular subpatches while the existing triangular surface modeling needs 3(n-2) subpatches. The obtained surface is composed of quartic triangular patches which are $G^1$-continuous to adjacent patches.

• Journal of Korean Institute of Industrial Engineers
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• v.19 no.3
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• pp.123-137
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• 1993

In this paper, an integrated approach is proposed to generate gouging-free Cartesian tool paths for machining sculptured surfaces from 3D measurement data. The integrated CAD/CAM system consists of two modules : offset surface module an Carteian tool path module. The offset surface module generates an offset surface of an object from its 3D measurement data, using an offsetting method and a surface fitting method. The offsetting is based on the idea that the envelope of an inversed tool generates an offset surface without self-intersection as the center of the inversed tool moves along on the surface of an object. The surface-fitting is the process of constructing a compact representation to model the surface of an object based on a fairly large number of data points. The resulting offset surtace is a composite Bezier surface without self-intersection. When an appropriate tool-approach direction is selected, the tool path module generates the Cartesian tool paths while the deviation of the tool paths from the surface stays within the user-specified tolerance. The tool path module is a two-step process. The first step adaptively subdivides the offset surface into subpatches until the thickness of each subpatch is small enough to satisfy the user-defined tolerance. The second step generates the Cartesian tool paths by calculating the intersection of the slicing planes and the adaptively subdivided subpatches. This tool path generation approach generates the gouging-free Cartesian CL tool paths, and optimizes the cutter movements by minimizing the number of interpolated points.