• Journal of applied mathematics & informatics
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• v.24 no.1_2
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• pp.521-528
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• 2007

We present a new proof of the characterization theorem for Minkowski Pythagorean-hodograph curves in the Minkowski spaces $\mathbf{R}^{n+1,m}$. For an polynomial curves $\mathbf{s}(t)=(x_1(t),...,\;x_{n+m}(t))$, we also find Minkowski Pythagorean-hodograph curves $\mathbf{r}(t)=(x_0(t),\;x_1(t),...,\;x_{n+m}(t))$. In case m=0, Minkowski Pythagorean-hodograph curves become Pythagorean-hodograph curves in the Euclidean spaces $\mathbf{R}^{n+1}$ and Theorems in this paper hold for these Pythagorean-hodograph curves.

• East Asian mathematical journal
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• v.28 no.1
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• pp.13-23
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• 2012

In this paper, we present the geometric Hermite interpolation for planar Pythagorean-hodograph cubics for some general Hermite data.

• East Asian mathematical journal
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• v.29 no.1
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• pp.53-68
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• 2013

We solve the geometric Hermite interpolation problem with planar Pythagorean-hodograph cubics. For every Hermite data, we determine the exact number of the geometric Hermite interpolants and represent the interpolants explicitly. We also present a simple criterion for determining whether the interpolants have a loop or not.

• 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.

• Korean Journal of Computational Design and Engineering
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• v.4 no.4
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• pp.312-317
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• 1999

An inflection point on a curve is a point where the curvature vanishes. An inflection point is useful for various geometric operations such as the approximation of curves and intersection points between curves or curve approximations. An inflection point on planar Bezier curves can be easily detected using a hodograph and a derivative of hodograph, since the closed from of hodograph is known. In the case of rational Bezier curves, for the detection of inflection point, it is needed to use the first and the second derivatives have higher degree and are more complex than those of non-rational Bezier curvet. This paper presents three methods to detect inflection points of rational Bezier curves. Since the algorithms avoid explicit derivations of the first and the second derivatives of rational Bezier curve to generate polynomial of relatively lower degree, they turn out to be rather efficient. Presented also in this paper is the theoretical analysis of the performances of the algorithms as well as the experimental result.

• Journal of applied mathematics & informatics
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• v.23 no.1_2
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• pp.73-86
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• 2007

The general stereographic projection which maps a point on a sphere with arbitrary radius to a point on a plane stereographically and its inverse projection have the pythagorean-hodograph (PH) preserving property in the sense that they map a PH curve to another PH curve. Upon this fact, for given spatial $C^1$ Hermite data, we construct a spatial PH curve on a sphere that is a $C^1$ Hermite interpolant of the given data as follows: First, we solve $C^1$ Hermite interpolation problem for the stereographically projected planar data of the given data in $\mathbb{R}^3$ with planar PH curves expressed in the complex representation. Second, we construct spherical PH curves which are interpolants for the given data in $\mathbb{R}^3$ using the inverse general stereographic projection.

• Journal of the Korea Computer Graphics Society
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• v.7 no.1
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• pp.19-25
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• 2001

In this paper, we study the special geometric reparametization of the (plane polynomial) Pythagorean Hodograph curves in the view point of their roots. The PH curves are completely determined by the roots of their hodographs. we show that the loci of roots of the PH curves satisfy some interesting geometric properties.

• Journal of applied mathematics & informatics
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• v.14 no.1_2
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• pp.405-413
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• 2004

Rational parameterization of curves and surfaces is one of the main topics in computer-aided geometric design because of their computational advantages. Pythagorean hodograph (PH) curves and Minkowski Pythagorean hodograph (MPH) curves have attracted many researcher's interest because they provide for rational representation of their offset curves in Euclidean space and Minkowski space, respectively. In [10], Kim presented the characterization of the PH curves in the Euclidean space and analyzed the relation between the class of PH curves and the class of rational curves. In this paper, we extend the characterization of PH curves in [10] into that of MPH curves in the general Minkowski space and consider some generalized MPH curves satisfying this characterization.

• Journal of applied mathematics & informatics
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• v.30 no.3_4
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• pp.381-393
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• 2012

In this paper, we show two ways of the time reparametrization of piecewise Pythagorean-hodograph $C^1$ Hermite interpolants. One is the time reparametrization with no shape change, and the other is that with shape change. We show that the first reparametrization does not depend on the boundary data and that it is uniquely determined by the size of parameter domain, up to the general cases. We empirically show that the second parametrization can cause the change of the shape of interpolant.

• Journal of applied mathematics & informatics
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• v.26 no.1_2
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• pp.121-133
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• 2008

In this article, we define Minkowski Pythagorean-hodograph (MPH) curves in the Minkowski plane $\mathbb{R}^{1,1}$ and obtain $C^1$ Hermite interpolations for MPH quintics in the Minkowski plane $\mathbb{R}^{1,1}$. We also have the envelope curves of MPH curves, and make surfaces of revolution with exact rational offsets. In addition, we present an example of $C^1$ Hermite interpolations for MPH rational curves in $\mathbb{R}^{2,1}$ from those in $\mathbb{R}^{1,1}$ and a suitable MPH preserving mapping.