• Journal of the Chungcheong Mathematical Society
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• v.29 no.2
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• pp.267-281
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• 2016

In this paper, we consider multi-degree reduction of $B{\acute{e}}zier$ curves with continuity of any (r, s) order with respect to $L_2$ norm. With help of matrix theory about generalized inverses we can use Lagrange multipliers to obtain the degree reduction matrix in a very simple form as well as the degree reduced control points. Also error analysis comparing with the least squares degree reduction without constraints is given. The advantage of our method is that the relationship between the optimal multi-degree reductions with and without constraints of continuity can be derived explicitly.

• Composites Research
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• v.33 no.6
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• pp.353-359
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• 2020

In this work, a contour-based section analysis method incorporating the use of Bézier curves is attempted for the construction of optimal structural design framework of composite helicopter blades. The suggested section analysis method is able to analyze composite blades with solid cores made of arbitrary materials and geometric shapes. The contour-based section analysis method is integrated into a blade structural optimization framework to confirm the efficiency of the present approach. The numerical simulation result demonstrates that the optimized blade configurations are obtained with a reduction in mass by 52%, compared to the baseline blade. For the structural optimization of composite blades with 19 subsections, it takes about one hour for the successful optimization while satisfying all the design constraints considered in this study, which reveals the efficiency of the present approach.

• 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.

• Journal of applied mathematics & informatics
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• v.3 no.2
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• pp.245-252
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• 1996

We described a relationship of the $L_2$ norm of the $L_2$norm of a Bzier curve and l2 norm of its confrol points. The use of Bezier curves finds much application in the general description of curves and surfaces and provided the mathematical basis for many computer graphics system. We define the $L_2$ norm for Bezier curves and find a upper and lower bound for many computer graphics system. We define the $L_2$ norm for Bezier curves and find a upper and lower bound for the $L_2$ norm with respect to the $L_2$ norm for its control points for easy computation.

• Journal of the Korea Society of Computer and Information
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• v.10 no.1 s.33
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• pp.35-44
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• 2005

The degree reduction of B-splines is necessary in exchanging parametric curves and surfaces of the different geometric modeling systems because some systems limit the supported maximal degree. In this paper, We provide an our experimental results in approximate conversion for B-splines apply to degree reduction. We utilize the existing Bezier degree reduction methods, and analyze the methods. Also, knot removal algorithm is used to reduce data in the degree reduction Process.

• Journal of the Korean Society for Aeronautical & Space Sciences
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• v.38 no.11
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• pp.1049-1059
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• 2010

This paper treats the aerodynamic optimization of the blade planform for helicopters. The blade shapes, which should be determined during the threedimensional aerodynamic configuration design step, are defined and are parameterized using the B$\acute{e}$zier curves. This research focuses on the design approaches generally adopted by industries and or research institutes using their own experiences and know-hows for the parameterization and for the definition of design constraints. The hover figure of merit and the equivalent lift-to-drag ratio for the forward flight are used to define the objective function. The resultant nonlinear programming (NLP) problem is solved using the sequential quadratic programming (SQP) method. The applications show the present method can design the important planform shapes such as the airfoil distribution, twist and chord variations in the efficient manner.

• Honam Mathematical Journal
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• v.43 no.1
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• pp.88-99
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• 2021

The purpose of this paper is to assign a movable frame to an arbitrary point of a rational Bézier curve on the 2-sphere S2 in Euclidean 3-space R3 to provide a better understanding of the geometry of the curve. Especially, we obtain the formula of geodesic curvature for a quadratic rational Bézier curve that allows a curve to be characterized on the surface. Moreover, we give some important results and relations for the Darboux frame and geodesic curvature of a such curve. Then, in specific case, given characterizations for the quadratic rational Bézier curve are illustrated on a unit 2-sphere.