• Journal of Integrative Natural Science
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• v.6 no.1
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• pp.46-52
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• 2013

In this thesis, we consider isoparametric curves of quadratic F-B$\acute{e}$zier curves. F-B$\acute{e}$zier curves unify C-B$\acute{e}$zier curves whose basis is {sint, cos t, t, 1} and H-B$\acute{e}$zier curves whose basis is {sinht, cosh t, t,1}. Thus F-B$\acute{e}$zier curves are more useful in Geometric Modeling or CAGD(Computer Aided Geometric Design). We derive the relation between the quadratic F-B$\acute{e}$zier curves and the quadratic rational B$\acute{e}$zier curves. We also obtain the geometric properties of isoparametric curve of the quadratic F-B$\acute{e}$zier curves at both end points and prove the continuity of the isoparametric curve.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.2 no.1
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• pp.88-93
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• 1998

It is frequently important to approximate a rational B$\acute{e}$zier curve by an integral, i.e., polynomial one. This need will arise when a rational B$\acute{e}$zier curve is produced in one CAD system and is to be imported into another system, which can only handle polynomial curves. The objective of this paper is to present an algorithm to approximate rational B$\acute{e}$zier curves with polynomial curves of higher degree.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.15 no.2
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• pp.123-135
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• 2011

In this paper, we present arc-length estimations for quadratic rational B$\acute{e}$zier curves using the length of polygon and distance between both end points. Our arc-length estimations coincide with the arc-length of the quadratic rational B$\acute{e}$zier curve exactly when the weight ${\omega}$ is 0, 1 and ${\infty}$. We show that for all ${\omega}$ > 0 our estimations are strictly increasing with respect to ${\omega}$. Moreover, we find the parameter ${\mu}^*$ which makes our estimation coincide with the arc-length of the quadratic rational B$\acute{e}$zier curve when it is a circular arc too. We also show that ${\mu}^*$ has a special limit, which is used for optimal estimation. We present some numerical examples, and the numerical results illustrates that the estimation with the limit value of ${\mu}^*$ is an optimal estimation.

• The Pure and Applied Mathematics
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• v.22 no.1
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• pp.25-34
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• 2015

In this paper we characterize the isogonal and isotomic conjugates of conic. Every conic can be expressed by a quadratic rational B$\acute{e}$zier curve having control polygon $b_0b_1b_2$ with weight w > 0. We show that the isotomic conjugate of parabola and hyperbola with respect to ${\Delta}b_0b_1b_2$ is ellipse, and that the isotomic conjugate of ellipse with the weight $w={\frac{1}{2}}$ is identical. We also find all cases of the isogonal conjugate of conic with respect to ${\Delta}b_0b_1b_2$. Our characterizations are derived easily due to the expression of conic by the quadratic rational B$\acute{e}$ezier curve in standard form.

• Journal of Integrative Natural Science
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• v.7 no.2
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• pp.124-129
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• 2014

In this paper, we construct an logarithmic spiral-like curve using curvature-continuous quadratic spline and quadratic rational spline. The quadratic (rational) spline has self-similarity. We present some properties of the quadratic spline. Also using this $G^2$ quadratic spline, an approximation of logarithmic spiral is proposed and error analysis is obtained.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.13 no.4
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• pp.257-265
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• 2009

In this paper we find an explicit form of upper bound of Hausdorff distance between given cubic spline curve and its quadratic spline approximation. As an application the approximation of offset curve of cubic spline curve is presented using our explicit error analysis. The offset curve of quadratic spline curve is exact rational spline curve of degree six, which is also an approximation of the offset curve of cubic spline curve.

• 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.

• 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.

• Proceedings of the Korean Operations and Management Science Society Conference
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• 2005.05a
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• pp.842-847
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• 2005

Although there are many applications of Euclidean Voronoi diagram for spheres in a 3D space in various disciplines from sciences and engineering, it has not been studied as much as it deserves. In this paper, we present an edge-tracing algorithm to compute the Euclidean Voronoi diagram of 3-dimensional spheres in O(mn) in the worst-case, where m is the number of edges of Voronoi diagram and n is the number of spheres. After building blocks for the algorithm, we show an example of Voronoi diagram for atoms using actual protein data and discuss its applications for protein analysis.