• 대한수학회논문집
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• 제36권3호
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• pp.609-622
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• 2021

In the Galilean space G₃, we study a special kind of tube surfaces, called tube-like surfaces. They are defined by sweeping a space curve along another central space curve. In this setting, we investigate some equations in terms of Gaussian and mean curvatures, showing some relevant theorems. Our theoretical results are illustrated with some plotted examples.

• Kyungpook Mathematical Journal
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• 제58권1호
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• pp.117-135
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• 2018

The orthogonal trajectories of the first tangents of the curve are called the involutes of x. The hyperspheres which have higher order contact with a curve x are known osculating hyperspheres of x. The centers of osculating hyperspheres form a curve which is called generalized evolute of the given curve x in n-dimensional Euclidean space ${\mathbb{E}}^n$. In the present study, we give a characterization of involute curves of order k (resp. evolute curves) of the given curve x in n-dimensional Euclidean space ${\mathbb{E}}^n$. Further, we obtain some results on these type of curves in ${\mathbb{E}}^3$ and ${\mathbb{E}}^4$, respectively.

• 천문학회보
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• 제45권1호
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• pp.48.4-49
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• 2020

We reconstruct the expansion history of the universe using type Ia supernovae (SN Ia) in a manner independent of any cosmological model assumptions. To do so, we implement a nonparametric iterative smoothing method on the Joint Light-curve Analysis (JLA) data while exploring the SN Ia light-curve hyperparameter space by Markov Chain Monte Carlo (MCMC) sampling. We test to see how the posteriors of these hyperparameters depend on cosmology, whether using different dark energy models or reconstructions shift these posteriors. Our constraints on the SN Ia light-curve hyperparameters from our model-independent analysis are very consistent with the constraints from using different parameterizations of the equation of state of dark energy, namely the flat ΛCDM cosmology, the Chevallier-Polarski-Linder model, and the Phenomenologically Emergent Dark Energy (PEDE) model. This implies that the distance moduli constructed from the JLA data are mostly independent of the cosmological models. We also studied that the possibility the light-curve parameters evolve with redshift and our results show consistency with no evolution. The reconstructed expansion history of the universe and dark energy properties also seem to be in good agreement with the expectations of the standard ΛCDM model. However, our results also indicate that the data still allow for considerable flexibility in the expansion history of the universe. This work is published in ApJ.

• Kyungpook Mathematical Journal
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• 제60권2호
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• pp.255-277
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• 2020

Divide knots and links, defined by A'Campo in the singularity theory of complex curves, is a method to present knots or links by real plane curves. The present paper is a sequel of the author's previous result that every knot in the major subfamilies of Berge's lens space surgery (i.e., knots yielding a lens space by Dehn surgery) is presented by an L-shaped curve as a divide knot. In the present paper, L-shaped curves are generalized and it is shown that every knot in the minor subfamilies, called sporadic examples of Berge's lens space surgery, is presented by a generalized L-shaped curve as a divide knot. A formula on the surgery coefficients and the presentation is also considered.

• ETRI Journal
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• 제39권4호
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• pp.493-501
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• 2017

This paper introduces a new motion-synthesis technique for animating multiple characters. At a high level, we introduce a hub-sub-control-point scheme that automatically generates many different spline curves from a user scribble. Then, each spline curve becomes a trajectory along which a 3D character moves. Based on the given curves, our algorithm synthesizes motions using a cyclic motion. In this process, space-time warp curves, which are time-warp curves, are embedded in the 3D environment to control the speed of the motions. Since the space-time warp curve represents a trajectory over the time domain, it enables us to verify whether the trajectory causes any collisions between characters by simply checking whether two space-time warp curves intersect. In addition, it is possible to edit space-time warp curves at run time to change the speed of the characters. We use several experiments to demonstrate that the proposed algorithm can efficiently synthesize a group of character motions. Our method creates collision-avoiding trajectories ten times faster than those created manually.

• 대한전기학회:학술대회논문집
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• 대한전기학회 1991년도 추계학술대회 논문집 학회본부
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• pp.67-70
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• 1991

When analyzing field or optimizing the shape of electrode in three dimensional space by using the surface charge method, we need to divide finely the surface of electrode into surface element like triangle or rectangle. In this case, there exist any variables in field analysis or field optimization. In particular, smoothness on the surface of optimized shape is not good. Recently, A paper is published where introducing NURB curve to field analysis and field optimization about two dimensional space model and axial symmetric three dimensional space model results in reduced variables, enhenced accuracy and improved smoothness. NURB curve has some useful properties like continuity, controllability and locality. Therefore in this paper, in order to improve the demerits of the established optimization method for three dimensional space models, the NURB surface that has same properties in common with NURB curve is used to analyze and optimize simple model.

• 대한수학회논문집
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• 제35권3호
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• pp.967-977
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• 2020

In this article, we find the necessary and sufficient condition for a proper biharmonic Frenet curve in the Lorentzian Sasakian space forms 𝓜³₁(H) except the case constant curvature -1. Next, we find that for a slant curve in a 3-dimensional Sasakian Lorentzian manifold, its ratio of "geodesic curvature" and "geodesic torsion -1" is a constant. We show that a proper biharmonic Frenet curve is a slant pseudo-helix with 𝜅² - 𝜏² = -1 + 𝜀₁(H + 1)𝜂(B)² in the Lorentzian Sasakian space forms x1D4DC³₁(H) except the case constant curvature -1. As example, we classify proper biharmonic Frenet curves in 3-dimensional Lorentzian Heisenberg space, that is a slant pseudo-helix.

• 한국의류학회지
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• 제39권6호
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• pp.826-837
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• 2015

This study analyzes the shapes of the adjustment of a Napoleon Collar which combines a stand collar with an upper collar. It established experimental conditions for fixing the conditions of Napoleon Collar components (lapel width, stand collar size and upper collar size) as well as varied the shape of the neckline, the length of the curve of a stand collar and the size of the drawing space at the center back. It produced 22 test clothes of muslin, which were dressed on dress form No. 8. The results indicate that: 1. Neckline shape determines the amount of stand and fall. Less curved neckline stands higher against the neck and a more curved neckline is laid lower onto the body. 2. A shorter curve length of a stand collar allows a longer roll line to fall farther away from the neck with more space between the neck and collar. However, the longer the depth of curve of a stand collar creates a shorter roll line that stands high against neck and closer to neck without any space between the neck and collar with a collar line matching the neck of the dress form. 3. The smaller the drawing space at the center back creates a shorter the style line of the upper collar. However, a narrower back width of the collar creates a bigger drawing space at the center back with a longer the style line and a more naturally placed back width of the collar. 4. A Napoleon Collar creates a longer depth of curve for a stand collar and a smaller drawing space at the center back that is tightly and stably stuck to the neck.

• 호남수학학술지
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• 제40권3호
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• pp.561-576
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• 2018

In this paper, we define null Cartan and pseudo null Bertrand curves in Minkowski space ${\mathbb{E}}^3_1$ according to their Bishop frames. We obtain the necessary and sufficient conditions for pseudo null curves to be Bertand B-curves in terms of their Bishop curvatures. We prove that there are no null Cartan curves in Minkowski 3-space which are Bertrand B-curves, by considering the cases when their Bertrand B-mate curves are spacelike, timelike, null Cartan and pseudo null curves. Finally, we give some examples of pseudo null Bertrand B-curve pairs.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제12권4호
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• pp.275-287
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• 2005

Geometric invariants are basic tools for geometric processing and computer vision. In this paper, we give a linear approximation for the differentiation of the binormal vector field of a space curve by using the forward and backward differences of discrete binormal vectors. Two kind of discrete torsion, say, back-ward torsion $T_b$ and forward torsion $T_f$ can be defined by the dot product of the (backward and forward) discrete differentiation of binormal vectors that are linear approximations of torsion. Using Frenet formula and Taylor series expansion, we give error estimations for the discrete torsions. We also give numerical tests for a curve. Notably the average of $T_b$ and $T_f$ looks more stable in errors.