• 한국CDE학회논문집
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• 제3권2호
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• pp.135-139
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• 1998

The hodograph, which are usually defined as the derivative of parametric curve or surface, is useful far various geometric operations. It is known that the hodographs of Bezier curves and surfaces can be represented in the closed form. However, the counterparts of rational Bezier curves and surface have not been discussed yet. In this paper, the equations are derived, which are the closed form of rational Bezier curves and surfaces. The hodograph of rational Bezier curves of degree n can be represented in another rational Bezier curve of degree 2n. The hodograph of a rational Hazier surface of degree m×n with respect to a parameter can be also represented in rational Bezier surface of degree 2m×2n. The control points and corresponding weight of the hodographs are directly computed using the control points and weights of the given rational curves or surfaces.

• Journal of applied mathematics & informatics
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• 제18권1_2호
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• pp.159-169
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• 2005

An algorithmic approach to degree reduction of rational Bezier curves is presented. The algorithms are based on the degree reduction of polynomial Bezier curves. The method is introduced with the following steps: (a) convert the rational Bezier curve to polynomial Bezier curve by using homogenous coordinates, (b) reduce the degree of polynomial Bezier curve, (c) determine weights of degree reduced curve, (d) convert the Bezier curve obtained through step (b) to rational Bezier curve with weights in step (c).

• 한국CDE학회논문집
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• 제4권4호
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• pp.312-317
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• 1999

An inflection point on a curve is a point where the curvature vanishes. An inflection point is useful for various geometric operations such as the approximation of curves and intersection points between curves or curve approximations. An inflection point on planar Bezier curves can be easily detected using a hodograph and a derivative of hodograph, since the closed from of hodograph is known. In the case of rational Bezier curves, for the detection of inflection point, it is needed to use the first and the second derivatives have higher degree and are more complex than those of non-rational Bezier curvet. This paper presents three methods to detect inflection points of rational Bezier curves. Since the algorithms avoid explicit derivations of the first and the second derivatives of rational Bezier curve to generate polynomial of relatively lower degree, they turn out to be rather efficient. Presented also in this paper is the theoretical analysis of the performances of the algorithms as well as the experimental result.

• Journal of applied mathematics & informatics
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• 제9권1호
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• pp.151-165
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• 2002

An a1gorithmic approach to degree elevation of NURBS curves is presented. The new algorithms are based on the weighted blossoming process and its matrix representation. The elevation method is introduced that consists of the following steps: (1) decompose the NURBS curve into piecewise rational Bezier curves, (b) elevate the degree of each rational Bezier piece, and (c) compose the piecewise rational Bezier curves into NURBS curve.

• 대한전자공학회:학술대회논문집
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• 대한전자공학회 2002년도 하계종합학술대회 논문집(3)
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• pp.17-20
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• 2002

This paper proposes a new 75 fuzzy model approximation method which reduces error in nonlinear fuzzy model approximation over the V-type decision rules. Employing rational Bezier curves used in computer graphics to represent curves or surfaces, the proposed method approximates the decision rule by constructing a tractable linear equation in the highly non-linear fuzzy rule interval. This algorithm is applied to the self-adjusting air cushion for spinal cord injury patients to automatically distribute the patient's weight evenly and balanced to prevent decubitus. The simulation results indicate that the performance of the proposed method is bettor than that of the conventional TS Fuzzy model in terms of error and stability.

• 한국CDE학회논문집
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• 제4권4호
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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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• 한국전산유체공학회 2001년도 춘계 학술대회논문집
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• pp.72-77
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• 2001

The computational efficiency of an shape optimization procedure is highly dependent upon the proper selection of shape representation methods and design variables. In this study, shape functions, Bezier and NURBS(non-uniform rational B-splines) curves are selected as configuration generation methods and their efficiencies on the nose shape design of high-speed air vehicles, are compared. The effects of the number of control points, weighting factors and the optimization methods when utilizing the NURBS curves, are investigated. By implementing Bezier and NURBS curves, shapes having lower drag than the optimization case utilizing the shape functions, were obtained, hence it was demonstrated that these curves have better capability in representing the configuration. Efforts will be given to improve the convergence behavior when utilizing the NURBS, hence to reduce the number of Navier-Stokes analysis calculations.

• 한국CDE학회논문집
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• 제3권3호
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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.

• 한국정보처리학회:학술대회논문집
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• 한국정보처리학회 2002년도 춘계학술발표논문집 (상)
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• pp.625-628
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• 2002

디지틸워터마킹은 디지털콘텐츠의 저작권보호 솔루션으로서 현재 주로 이미지, 오디오, 비디오, 텍스트 등을 대상으로 연구되고 있다. 컴퓨터 하드웨어, 네트워크, 그리고, 응용 소프트웨어의 빠른 발전과 함께 국가 차원의 초고속 통신망의 인프라 구축에 힘입어, 3차원 폴리곤과 곡선 및 곡면에 대한 디지털워터마킹에 관심이 높아지고 있다. 본 논문에서는 유리 베지에곡선에 대한 디지털워터마킹에 대한 하나의 방법을 제시한다. 기존의 베지에곡선의 차수를 증가시키는 일반적인 방법이 아닌 유리항의 분모와 분자에 공통의 다항식을 곱하여 차수를 증가시킨다. 이때 공통으로 칠하는 다항식의 관들의 복비(cross ratio) 값에 우리가 숨기고자하는 마크를 삽입하고, 추출해내는 방법을 제시한다. 본 논문에서 제시된 알고리듬은 워터마크를 삽입하는 과정에서 곡선의 형태를 전혀 변화시키지 않는 형태 유지성(shape preserving property)을 갖는다. 또한. 본 알고리듬의 다른 중요한 특징은 곡선이 CAD 시스템에 의해 이용되는 과정에서 흔히 일어나는 재매개화 방법 중 뫼비우스 변환을 이용한 재매개화에 저항성이 있는 알고리듬이라는 것이다. 마지막으로 본 연구에서 제시한 방범에 의한 예제의 결과를 보여준다.