• Title/Summary/Keyword: rational curves

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The Closed Form of Hodograph of Rational Bezier curves and Surfaces (유리 B$\acute{e}$zier 곡선과 곡면의 호도그래프)

  • 김덕수;장태범;조영송
    • 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.

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A Brief History of Study on the Bound for Derivative of Rational Curves in CAGD (CAGD에서 유리 곡선의 미분과 그 상한에 관한 연구의 흐름)

  • Park, Yunbeom
    • Journal for History of Mathematics
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    • v.27 no.5
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    • pp.329-345
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    • 2014
  • CAGD(Computer Aided Geometric Design) is a branch of applied mathematics concerned with algorithms for the design of smooth curves and surfaces and for their efficient mathematical representation. The representation is used for the computation of the curves and surfaces, as well as geometrical quantities of importance such as curvatures, intersection curves between two surfaces and offset surfaces. The $B\acute{e}zier$ curves, B-spline, rational $B\acute{e}zier$ curves and NURBS(Non-Uniform Rational B-Spline) are basically and widely used in CAGD. The definitions and properties of these curves are presented in this paper. And a brief history of study on the bound for derivative of rational curves in CAGD is also presented.

APPLICATION OF DEGREE REDUCTION OF POLYNOMIAL BEZIER CURVES TO RATIONAL CASE

  • PARK YUNBEOM;LEE NAMYONG
    • Journal of applied mathematics & informatics
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    • v.18 no.1_2
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    • pp.159-169
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    • 2005
  • An algorithmic approach to degree reduction of rational Bezier curves is presented. The algorithms are based on the degree reduction of polynomial Bezier curves. The method is introduced with the following steps: (a) convert the rational Bezier curve to polynomial Bezier curve by using homogenous coordinates, (b) reduce the degree of polynomial Bezier curve, (c) determine weights of degree reduced curve, (d) convert the Bezier curve obtained through step (b) to rational Bezier curve with weights in step (c).

Approximate Conversion of Rational Bézier Curves

  • Lee, Byung-Gook;Park, Yunbeom
    • 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.

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The Detection of Inflection Points on Planar Rational $B\'{e}zier$ Curves (평면 유리 $B\'{e}zier$곡선상의 변곡점 계산법)

  • 김덕수;이형주;장태범
    • 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.

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Exceptional bundles of higher rank and rational curves

  • Kim, Hoil
    • Bulletin of the Korean Mathematical Society
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    • v.35 no.1
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    • pp.149-156
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    • 1998
  • We relate the existence of rational curves with the existence of rigid bundles of any even rank on Enriques surfaces and compare with the case of K3 surfaces.

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SOME RATIONAL CURVES OF MAXIMAL GENUS IN ℙ3

  • Wanseok LEE;Shuailing Yang
    • East Asian mathematical journal
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    • v.40 no.1
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    • pp.75-83
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    • 2024
  • For a reduced, irreducible and nondegenerate curve C ⊂ ℙr of degree d, it was shown that the arithmetic genus g of C has an upper bound π0(d, r) by G. Castelnuovo. And he also classified the curves that attain the extremal value. These curves are arithmetically Cohen-Macaulay and contained in a surface of minimal degree. In this paper, we investigate the arithmetic genus of curves lie on a surface of minimal degree - the Veronese surface, smooth rational normal surface scrolls and singular rational normal surface scrolls. We also provide a construction of curves on singular rational normal surface scroll S(0, 2) ⊂ ℙ3 which attain the maximal arithmetic genus.

HIGHER DIMENSIONAL MINKOWSKI PYTHAGOREAN HODOGRAPH CURVES

  • Kim, Gwang-Il;Lee, Sun-Hong
    • 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.

ALGEBRAIC CHARACTERIZATION OF GENERIC STRONGLY SEMI-REGULAR RATIONAL PH PLANE CURVES

  • KIM GWANG-IL
    • Journal of applied mathematics & informatics
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    • v.19 no.1_2
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    • pp.241-251
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    • 2005
  • In this paper, we introduce a new algebraic method to characterize rational PH plane curves. And using this method, we study the algebraic characterization of generic strongly regular rational plane PH curves expressed in the complex formalism which is introduced by R.T. Farouki. We prove that generic strongly semi-regular rational PH plane curves are completely characterized by solving a simple functional equation H(f, g) = $h^2$ where h is a complex polynomial and H is a bi-linear operator defined by H(f, g) = f'g - fg' for complex polynomials f,g.