• Title, Summary, Keyword: Derivative of rational curves

### A Brief History of Study on the Bound for Derivative of Rational Curves in CAGD (CAGD에서 유리 곡선의 미분과 그 상한에 관한 연구의 흐름)

• Park, Yunbeom
• Journal for History of Mathematics
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• v.27 no.5
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• pp.329-345
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• 2014
• CAGD(Computer Aided Geometric Design) is a branch of applied mathematics concerned with algorithms for the design of smooth curves and surfaces and for their efficient mathematical representation. The representation is used for the computation of the curves and surfaces, as well as geometrical quantities of importance such as curvatures, intersection curves between two surfaces and offset surfaces. The $B\acute{e}zier$ curves, B-spline, rational $B\acute{e}zier$ curves and NURBS(Non-Uniform Rational B-Spline) are basically and widely used in CAGD. The definitions and properties of these curves are presented in this paper. And a brief history of study on the bound for derivative of rational curves in CAGD is also presented.

### The Closed Form of Hodograph of Rational Bezier curves and Surfaces (유리 B$\acute{e}$zier 곡선과 곡면의 호도그래프)

• 김덕수;장태범;조영송
• Korean Journal of Computational Design and Engineering
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• v.3 no.2
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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.

### Evaluations of Representations for the Derivative of Rational $B\{e}zier$ Curve (유리 $B\{e}zier$ 곡선의 미분계산방법의 평가)

• 김덕수;장태범
• 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.

### The Detection of Inflection Points on Planar Rational $B\'{e}zier$ Curves (평면 유리 $B\'{e}zier$곡선상의 변곡점 계산법)

• 김덕수;이형주;장태범
• Korean Journal of Computational Design and Engineering
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• v.4 no.4
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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.

### Braking performance of working rail-mounted cranes under wind load

• Jin, Hui;Chen, Da
• Wind and Structures
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• v.19 no.1
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• pp.1-14
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• 2014
• Rail-mounted cranes can be easily damaged by a sudden gust of wind while working at a running speed, due to the large mass and high barycenter positions. In current designs, working rail-mounted cranes mainly depend on wheel braking torques to resist large wind load. Regular brakes, however, cannot satisfactorily stop the crane, which induces safety issues of cranes and hence leads to frequent crane accidents, especially in sudden gusts of wind. Therefore, it is necessary and important to study the braking performance of working rail mounted cranes under wind load. In this study, a simplified mechanical model was built to simulate the working rail mounted gantry crane, and dynamic analysis of the model was carried out to deduce braking performance equations that reflect the qualitative relations among braking time, braking distance, wind load, and braking torque. It was shown that, under constant braking torque, there existed inflection points on the curves of braking time and distance versus windforce. Both the braking time and the distance increased sharply when wind load exceeded the inflection point value, referred to as the threshold windforce. The braking performance of a 300 ton shipbuilding gantry crane was modeled and analyzed using multibody dynamics software ADAMS. The simulation results were fitted by quadratic curves to show the changes of braking time and distance versus windforce under various mount of braking torques. The threshold windforce could be obtained theoretically by taking derivative of fitted curves. Based on the fitted functional relationship between threshold windforce and braking torque, theoretical basis are provided to ensure a safe and rational design for crane wind-resistant braking systems.