• Title, Summary, Keyword: Rational curves

### 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.

### 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.

### 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.

### 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.

### 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.

### RATIONAL CURVES ARE NOT UNIT SPEED IN THE GENERAL EUCLIDEAN SPACE

• Lee, Sun-Hong
• East Asian mathematical journal
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• v.26 no.1
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• pp.69-73
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• 2010
• We invoke the characterization of Pythagorean-hodograph polynomial curves and prove that it is impossible to parameterize any real curves, other than a straight line, by rational functions of its arc length.

### ON THE EQUATIONS DEFINING SOME CURVES OF MAXIMAL REGULARITY IN ℙ4

• LEE, Wanseok;Jang, Wooyoung
• East Asian mathematical journal
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• v.35 no.1
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• pp.51-58
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• 2019
• For a nondegenerate irreducible projective variety, it is a classical problem to describe its defining equations. In this paper we precisely determine the defining equations of some rational curves of maximal regularity in ${\mathbb{P}}^4$ according to their rational parameterizations.

### IMPLICITIZATION OF RATIONAL CURVES AND POLYNOMIAL SURFACES

• Yu, Jian-Ping;Sun, Yong-Li
• Bulletin of the Korean Mathematical Society
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• v.44 no.1
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• pp.13-29
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• 2007
• In this paper, we first present a method for finding the implicit equation of the curve given by rational parametric equations. The method is based on the computation of $Gr\"{o}bner$ bases. Then, another method for implicitization of curve and surface is given. In the case of rational curves, the method proceeds via giving the implicit polynomial f with indeterminate coefficients, substituting the rational expressions for the given curve and surface into the implicit polynomial to yield a rational expression $\frac{g}{h}$ in the parameters. Equating coefficients of g in terms of parameters to 0 to get a system of linear equations in the indeterminate coefficients of polynomial f, and finally solving the linear system, we get all the coefficients of f, and thus we obtain the corresponding implicit equation. In the case of polynomial surfaces, we can similarly as in the case of rational curves obtain its implicit equation. This method is based on characteristic set theory. Some examples will show that our methods are efficient.

### DEFINING EQUATIONS OF RATIONAL CURVES IN SMOOTH QUADRIC SURFACE

• LEE, Wanseok;Yang, Shuailing
• East Asian mathematical journal
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• v.34 no.1
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• pp.21-28
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• 2018
• For a nondegenerate irreducible projective variety, it is a classical problem to study the defining equations of a variety with respect to the given embedding. In this paper we precisely determine the defining equations of certain types of rational curves in ${\mathbb{P}}^3$.

### LOCI OF RATIONAL CURVES OF SMALL DEGREE ON THE MODULI SPACE OF VECTOR BUNDLES

• Choe, In-Song
• Bulletin of the Korean Mathematical Society
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• v.48 no.2
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• pp.377-386
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• 2011
• For a smooth algebraic curve C of genus g $\geq$ 4, let $SU_C$(r, d) be the moduli space of semistable bundles of rank r $\geq$ 2 over C with fixed determinant of degree d. When (r,d) = 1, it is known that $SU_C$(r, d) is a smooth Fano variety of Picard number 1, whose rational curves passing through a general point have degree $\geq$ r with respect to the ampl generator of Pic($SU_C$(r, d)). In this paper, we study the locus swept out by the rational curves on $SU_C$(r, d) of degree < r. As a by-product, we present another proof of Torelli theorem on $SU_C$(r, d).