• Korean Journal of Computational Design and Engineering
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• v.11 no.4
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• pp.246-255
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• 2006

Ribs and fans are interesting geometric entities that are derived from an ordinary $B\'{e}zier$ curve or surface. A rib itself is a $B\'{e}zier$ curve or surface with a lower degree than the given curve or surface. A fan is a vector field whose degree is also lower than its origin. First, we present methods to transform the control points of a $B\'{e}zier$ curve or surface into the control points and vectors of its ribs and fans. Then, we show that a $B\'{e}zier$ curve of degree n is decomposed into a rib of degree (n-1), a fan of degree (n-2), and a scalar function of degree 2. We also show that a $B\'{e}zier$ surface of degree (m, n) is decomposed into a rib of degree (m-1, n-1) and three fans of degrees (m-1, n-2), (m-2, n-1), and (m-2, n-2), respectively. In addition, the lengths of the fans are further controlled by scalar functions of degree 2 and (2, 2). We present relevant notations and definitions, introduce theories, and present some of design examples.

• 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 Ship and Ocean Technology
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• v.10 no.2
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• pp.24-36
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• 2006

Ship hullform is usually designed with a curve network, and smooth hullform surfaces are supposed to be generated by filling in (or interpolating) the curve network with appropriate surface patches. Tensor-product surfaces such as B-spline and $B\'{e}zier$ patches are typical representations to this interpolating problem. However, they have difficulties in representing the surfaces of irregular topological type which are frequently appeared in the fore- and after-body of ship hullform curve network. In this paper, we proposed a method that can automatically generate discrete $G^1$ continuous B-spline surfaces interpolating given curve network of ship hullform. This method consists of three steps. In the first step, given curve network is reorganized to be of two types: boundary curves and reference curves of surface patches. Especially, the boundary curves are specified for their surface patches to be rectangular or triangular topological type that can be represented with tensor-product (or degenerate) B-spline surface patches. In the second step, surface fitting points and cross boundary derivatives are estimated by constructing virtual iso-parametric curves at discrete parameters. In the last step, discrete $G^1$ continuous B-spline surfaces are generated by surface fitting algorithm. Finally, several examples of resulting smooth hullform surfaces generated from the curve network data of actual ship hullform are included to demonstrate the quality of the proposed method.

• 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.