• Communications of the Korean Mathematical Society
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• v.22 no.4
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• pp.641-648
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• 2007

In this paper, we present the exact Hausdorff distance between the offset curve of quadratic $B\'{e}zier$ curve and its quadratic $GC^1$ approximation. To illustrate the formula for the Hausdorff distance, we give an example of the quadratic $GC^1$ approximation of the offset curve of a quadratic $B\'{e}zier$ curve.

• 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.

• 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.

• Honam Mathematical Journal
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• v.29 no.3
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• pp.475-480
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• 2007

In this paper we find the point at which the rational B$\'{e}$zier curve has the given tangent direction. We also analyze the geometric properties of the point of quadratic rational B$\'{e}$zier curve.

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

• Journal of KIISE:Computer Systems and Theory
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• v.36 no.1
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• pp.21-25
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• 2009

In this paper, we study the problem to fit a piecewise-quadratic polynomial curve to points in the plane. The curve consists of quadratic polynomial segments and two points are connected by a segment. But it passes through a subset of points, and for the points not to be passed, the error between the curve and the points is estimated in $L^{\infty}$ metric. We consider two optimization problems for the above problem. One is to reduce the number of segments of the curve, given the allowed error, and the other is to reduce the error between the curve and the points, while the curve has the number of segments less than or equal to the given integer. For the number n of given points, we propose $O(n^2)$ algorithm for the former problem and $O(n^3)$ algorithm for the latter.

• 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.

• Communications of the Korean Mathematical Society
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• v.20 no.4
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• pp.861-873
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• 2005

In this paper we approximate a cylindrical helix by bi-conic and bi-quadratic Bezier curves. Each approximation method is $G^1$ end-points interpolation of the helix. We present a sharp upper bound of the Hausdorff distance between the helix and each approximation curve. We also show that the error bound has the approximation order three and monotone increases as the length of the helix increases. As an illustration we give some numerical examples.

• 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.