• International Journal of CAD/CAM
• /
• v.6 no.1
• /
• pp.65-71
• /
• 2006

Based on the smoothness criterion of minimum curvature variation of the curve, tangent angle constraints guaranteeing an optimized geometric Hermite (OGH) curve both mathematically and geometrically smooth is given, and new methods for constructing composite optimized geometric Hermite (COH) curves are presented in this paper. The comparison of the new methods with Yong and Cheng's methods based on strain energy minimization is included.

• Communications of the Korean Mathematical Society
• /
• v.10 no.2
• /
• pp.483-490
• /
• 1995

A parametric cubic spline interpolating at fixed number of nodes is constructed by formulating a parametric cubic $g^2$ B-splines $S_3(t)$ with not equally spaced parametric knots. Since the fact that each component is in $C^2$ class is not enough to provide the geometric smoothness of parametric curves, the existence of $S_3(t)$ oriented toward the modified second-order geometric continuity is focalized in our work.

• Kyungpook Mathematical Journal
• /
• v.62 no.2
• /
• pp.271-287
• /
• 2022

In this paper, starting with the geometric constants that can characterize Hilbert spaces, combined with the isosceles orthogonality of Banach spaces, the orthogonal geometric constant Ω_X(α) is defined, and some theorems on the geometric properties of Banach spaces are derived. Firstly, this paper reviews the research progress of orthogonal geometric constants in recent years. Then, this paper explores the basic properties of the new geometric constants and their relationship with conventional geometric constants, and deduces the identity of Ω_X(α) and γ_X(α). Finally, according to the identities, the relationship between these the new orthogonal geometric constant and the geometric properties of Banach Spaces (such as uniformly non-squareness, smoothness, convexity, normal structure, etc.) is studied, and some necessary and sufficient conditions are obtained.

• Journal of the Korean Mathematical Society
• /
• v.43 no.2
• /
• pp.425-443
• /
• 2006

The aim of this paper is to obtain several results in approximation by Jackson-type generalizations of complex Picard, Poisson-Cauchy and Gauss-Weierstrass singular integrals in terms of higher order moduli of smoothness. In addition, these generalized integrals preserve some sufficient conditions for starlikeness and univalence of analytic functions. Also approximation results for vector-valued functions defined on the unit disk are given.

• International Journal of CAD/CAM
• /
• v.9 no.1
• /
• pp.55-60
• /
• 2010

Quadratic B$\acute{e}$zier curves are important geometric entities in many applications. However, it was often ignored by the literature the fact that a single segment of a quadratic B$\acute{e}$zier curve may fail to fit arbitrary endpoint unit tangent vectors. The purpose of this paper is to provide a solution to this problem, i.e., constructing $G^1$ quadratic B$\acute{e}$zier curves satisfying given endpoint (positions and arbitrary unit tangent vectors) conditions. Examples are given to illustrate the new solution and to perform comparison between the $G^1$ quadratic B$\acute{e}$zier cures and other curve schemes such as the composite geometric Hermite curves and the biarcs.

• Proceedings of the Computational Structural Engineering Institute Conference
• /
• 1989.04a
• /
• pp.33-37
• /
• 1989

According to the Rayleigh-Ritz approximation method, a solution can be represented as a finite series consisting of space-dependent functions, which satisfy all the geometric boundary conditions of the problem and appropriate smoothness requirement in the interior of the domain. In this paper, an efficient formulation for solving structural dynamics systems in frequency domain is presented. A general procedure called Ritz modes (or vectors) generation algorithm is used to generate the admissible functions, i.e. Ritz modes, Then, the use of direct superposition of the Ritz modes is utilized to reduce the size of the problem in spatial dimension via geometric coordinates projection. For the reduced system, the frequency domain approach is applied. Finally, a numerical example is presented to illustrate the effectiveness of the proposed method.

• International Journal of Precision Engineering and Manufacturing
• /
• v.8 no.3
• /
• pp.14-19
• /
• 2007

Rapid prototyping (RP) technology can fabricate any 3D physical model regardless of geometric complexity using the layered manufacturing (LM) process. Stereolithography (SL) is the best-known example of RP technology. In general, the surface quality of a raw SL-generated part is unsatisfactory for industrial purposes due to the step artefact created by the LM process. Despite of the increased number of applications for SL parts, this side effect limits their uses. In order to improve their surface quality, additional post-machining finishing, such as traditional grinding, is required, but post-machining is time consuming and can reduce the geometric accuracy of a part. Therefore, this study proposes a post-machining technology combining coating and grinding processes to improve the surface quality of SL parts. Paraffin wax and pulp are used as the coating and grinding materials. By grinding the coating wax only up to the boundary of the part, the surface smoothness can be improved without damaging the surface. Finally, moulding and casting experiments were performed to confirm the suitability of the SL parts finished using the proposed process with rapid tooling (RT) techniques.

• International Journal of CAD/CAM
• /
• v.12 no.1
• /
• pp.29-37
• /
• 2012

A new algorithm has been developed to construct surface from the contours having branches and the final smooth surface is obtained by the reversible Catmull-Clark subdivision. In branching, a particular layer has more than one contour that correspond with at least one contour at the adjacent layer. In the next step, three-dimensional composite curve is constructed from contours of a layer having correspondence with at least one contour at the adjacent layer by inserting points between them and joining the contours. The points are inserted in such a way that the geometric center of the contours should merge at the center of the contours at the adjacent layer. This process is repeated for all layers having branching problems. Polyhedra are constructed in the next step with the help of composite curves and the contours at adjacent layer. The required smooth surface is obtained in the proposed work by providing the level of smoothness.

• Journal of the Computational Structural Engineering Institute of Korea
• /
• v.27 no.3
• /
• pp.155-162
• /
• 2014

Using an isogeometric approach, a continuum-based shape design optimization method is developed for steady state power flow problems at high frequencies. In case the isogeometric method is employed to the shape design optimization, the NURBS basis functions used in CAD geometric modeling are directly utilized to embed the exact geometry into the computational framework so that the design parameterization for shape optimization is much easier than that in the finite element method and consequently provides the enhanced smoothness of design perturbations. Thus, exact geometric models can be used in both the response and the shape sensitivity analyses, where normal vector and curvature are continuous over the whole design space so that enhanced shape sensitivity can be expected. Through numerical examples, the developed isogeometric sensitivity is compared with finite difference one to provide excellent agreement. Also, it turns out that the proposed method works very well in the shape optimization problems.