• Title, Summary, Keyword: Bezier Surface

### 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.

### Modeling of Blend Surfaces by Bezier Surface Patches (비지어곡면에 의한 블렌드곡면의 모델링)

• 주상윤
• Korean Journal of Computational Design and Engineering
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• v.2 no.2
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• pp.122-129
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• 1997
• Ball rolling blending is a popular technique for blending between parametric surfaces. The ball rolling blend surface is conceptually a trajectory of a ball rolling between two base sufaces. It is constructed by sweeping a circular arc along a ball contact curve pair. Since a ball rolling blend surfaces does not have a polynomial form like a Bezier surface patch, it is impossible to apply this method directly to a commercial CAD/CAM system. In this paper an algorithm is developed to approximate a ball rolling blend surface into Bezier surface patches. Least square method is applied to obtain proper Bezier surface patches under a given tolerance. The Bezier surface patches have degree three or more and guarantee VC1-continuity.

### Bezier Control Points for the Image of a Domain Curve on a Bezier Surface (베지어 곡면의 도메인 곡선의 이미지 곡선에 대한 베지어 조정점의 계산)

• 신하용
• Korean Journal of Computational Design and Engineering
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• v.1 no.2
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• pp.158-162
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• 1996
• Algorithms to find the Bezier control points of the image of a Bezier domain curve on a Bezier surface are described. The diagonal image curve is analysed and the general linear case is transformed to the diagonal case. This proposed algorithm gives the closed form solution to find the control points of the image curve of a linear domain curve. If the domain curve is not linear, the image curve can be obtained by solving the system of linear equations.

### A Unified Surface Modeling Technique Using a Bezier Curve Model (de Casteljau Algorithm) (베지에 곡선모델 (드 카스텔죠 알고리듬) 을 이용한 곡면 통합 모델링 기법)

• Rhim, Joong-Hyun;Lee, Kyu-Yeul
• Journal of the Society of Naval Architects of Korea
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• v.34 no.4
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• pp.127-138
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• 1997
• In this study, a new technique is presented, by which one can define ship hull form with full fairness from the input data of lines. For curve modeling, the de Casteljau Algorithm and Bezier control points are used to express free curves and to establish the unified curve modeling technique which enables one to convert non-uniform B-spline (NUB) curve or cubic spline curve into composite Bezier curves. For surface modeling, the mesh curve net which is required to define surface of ship hull form is interpolated by the method of the unified curve modeling, and the boundary curve segments of Gregory surface patches are generated by remeshing(rearranging) the given mesh curve net. From these boundary information, composite Gregory surfaces of good quality in fairness can be formulated.

### Development of an Effective Walking System for a Hexapod Robot on Uneven Terrain (오프로드 환경에서 효율적인 6족 로봇 보행 시스템 개발)

• Kim, Jun Woo;Lee, Gi Won;Lee, Suk Gyu
• Journal of Institute of Control, Robotics and Systems
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• v.19 no.12
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• pp.1152-1159
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• 2013
• This paper proposes an effective walking system for a hexapod robot on uneven terrain. To overcome the deficiencies of two-pair walking systems, which are effective on even terrain, the use of only three legs changes the steps required for movement. The proposed system receives feedback data from switches attached to the bottom of the legs and gyro sensor to carry out stable walking using the Bezier curve algorithm. From the coordinates of the Bezier curve, which guarantees the circular motion of legs, the motor's angle value can be obtained using inverse kinematics. The angle values are sent to each motor though RS-485 communication. If a switch is pushed by the surface during navigation in the Bezier curve pattern, the robot is designed to change its circular course. Through the changed course, each leg can be located on an optimal surface and the wobble phenomenon is reduced by using a normal vector algorithm. The simulation and experiment results show the efficiency of the proposed algorithm.

### 3D Surface Approximation to Serial 2D Cross Sections (단면정보로부터 3차원 근사곡면의 생성)

• 박형준;김광수
• Proceedings of the Korean Society of Precision Engineering Conference
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• pp.719-724
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• 1994
• This paper describes a hybrid surface-based method for smooth 3D surface approximation to a sequence of 2D cross sections. The resulting surface is a hybrid G $^{1}$ surface represented by a mesh of triangular and rectangular Bezier patches defined on skinning, branching, or capping regions. Each skinning region is approximated with a closed B_spline surface, which is transformed into a mesh of Bezier patches. Triangular G $^{1}$ surfaces are constructed over brabching and capping regions such that the transitions between each capping regions such that the transitions between each triangular surface and its neighboring skinning surfaces are G $^{1}$ continuous. Since each skinning region is represented by an approximated rectangular C $^{2}$ suface instead of an interpolated trctangular G $^{[-1000]}$ surface, the proposed method can provide more smooth surfaces and realize more efficient data reduction than triangular surfacebased method.

### A DESIGN OPTIMIZATION STUDY OF BLUNT NOSE HYPERSONIC FLIGHT VEHICLE MINIMIZING SURFACE HEAT-TRANSFER RATE AND DRAG (표면 열전달율과 항력을 최소화한 극초음속 비행체 선두부 형상 최적설계)

• Lim S.;Seo J. I.;Kim S. D.;Song D. J.
• Journal of computational fluids engineering
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• v.10 no.3
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• pp.27-35
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• 2005
• A design optimization of hypersonic flight vehicle has been studied by using upwind Navier-Stokes method and numerical optimization method. CFD method is linked to numerical optimization method by using a Bezier curve and a design optimization of blunt nose hypersonic flight vehicle has been studied. Heat transfer coefficient and drag coefficient are selected as objective functions or design constraints. The Bezier curve-based shape function was applied to blunt body shape.

### Variational surface design under normal field guidance

• Wu, Weidong;Yang, Xunnian
• Journal of Computational Design and Engineering
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• v.2 no.3
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• pp.129-136
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• 2015
• This paper proposes a novel method for shape design of a Bezier surface with given boundary curves. The surface is defined as the minimizer of an extended membrane functional or an extended thin plate functional under the guidance of a specified normal field together with an initial prescribed surface. For given boundary curves and the guiding normal field, the free coefficients of a Bezier surface are obtained by solving a linear system. Unlike previous PDE based surface modeling techniques which construct surfaces just from boundaries, our proposed method can be used to generate smooth and fair surfaces that even follow a specified normal field. Several interesting examples are given to demonstrate the applications of the proposed method in geometric modeling.

### Composite $G^{1}$ surface construction from 2D cross-sections (2차원 단면 데이터로부터 복합 $G^{1}$ 자유곡면 생성)

• Park, Hyung-Jun;Na, Sang-Wook;Bae, Chae-Yeol
• Proceedings of the Korean Operations and Management Science Society Conference
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• pp.44-47
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• 2004
• This paper proposes an approach for composite surface reconstruction from 2D serial cross-sections, where the number of contours varies from section to section. In a triangular surface-based approach taken in most reconstruction methods, a triangular $G^{1}$ surface is constructed by stitching triangular patches over a triangular net generated from the compiled contours. In the proposed approach, the resulting surface is a composite $G^{1}$ surface consisting of three kinds of surfaces: skinned, surface is first represented by a B-spline surface approximating the serial contours of the skinned region and then serial contours of the skinned region and then transformed into a mesh of rectangular Bezier patches. On branched and capped regions, triangular $G^{1}$ surfaces are constructed so that the connections between the triangular surfaces and their neighboring surfaces are $G^{1}$ continuous. Since each skinned region is represented by an approximated rectangular $G^{2}$ surface instead of an interpolated triangular $G^{1}$ surface, the proposed approach can provide more visually pleasing surfaces and realize more efficient data reduction than the triangular surface-based approach. Some experimental results demonstrate its usefulness and quality.

### A Design Optimization Study of Blunt Nose Hypersonic Flight Vehicle Using Surface Heat-transfer and Drag Minimization (표면열전달과 항력을 고려한 극초음속 비행체 선두부 최적형상설계)

• Lim S.;Seo J. I.;Song D. J.
• 한국전산유체공학회:학술대회논문집
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• pp.197-201
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• 2004
• A design optimization of Sphere-Cone blunt nose hypersonic flight vehicle has been studied by using upwind Navier-Stokes method and numerical optimization method. Heat transfer coefficient and drag coefficient are selected as objective function or design constraint. Control points of Bezier curve are considered as design variable.