• 제목/요약/키워드: rational curves

검색결과 98건 처리시간 0.027초

ON THE EQUATIONS DEFINING SOME CURVES OF MAXIMAL REGULARITY IN ℙ4

  • LEE, Wanseok;Jang, Wooyoung
    • East Asian mathematical journal
    • /
    • 제35권1호
    • /
    • pp.51-58
    • /
    • 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.

DEGREE ELEVATION OF NURBS CURVES BY WEIGHTED BLOSSOM

  • Lee, Byung-Gook;Park, Yun-Beom
    • Journal of applied mathematics & informatics
    • /
    • 제9권1호
    • /
    • pp.151-165
    • /
    • 2002
  • An a1gorithmic approach to degree elevation of NURBS curves is presented. The new algorithms are based on the weighted blossoming process and its matrix representation. The elevation method is introduced that consists of the following steps: (1) decompose the NURBS curve into piecewise rational Bezier curves, (b) elevate the degree of each rational Bezier piece, and (c) compose the piecewise rational Bezier curves into NURBS curve.

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

  • Choe, In-Song
    • 대한수학회보
    • /
    • 제48권2호
    • /
    • pp.377-386
    • /
    • 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).

Convexity preserving piecewise rational interpolation for planar curves

  • Sarfraz, Muhammad
    • 대한수학회보
    • /
    • 제29권2호
    • /
    • pp.193-200
    • /
    • 1992
  • This paper uses a piecewise ratonal cubic interpolant to solve the problem of shape preserving interpolation for plane curves; scalar curves are also considered as a special case. The results derived here are actually the extensions of the convexity preserving results of Delbourgo and Gregory [Delbourgo and Gregory'85] who developed a $C^{1}$ shape preserving interpolation scheme for scalar curves using the same piecewise rational function. They derived the ocnstraints, on the shape parameters occuring in the rational function under discussion, to make the interpolant preserve the convex shape of the data. This paper begins with some preliminaries about the rational cubic interpolant. The constraints consistent with convex data, are derived in Sections 3. These constraints are dependent on the tangent vectors. The description of the tangent vectors, which are consistent and dependent on the given data, is made in Section 4. the convexity preserving results are explained with examples in Section 5.

  • PDF

IMPLICITIZATION OF RATIONAL CURVES AND POLYNOMIAL SURFACES

  • Yu, Jian-Ping;Sun, Yong-Li
    • 대한수학회보
    • /
    • 제44권1호
    • /
    • pp.13-29
    • /
    • 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.

Isoparametric Curve of Quadratic F-Bézier Curve

  • Park, Hae Yeon;Ahn, Young Joon
    • 통합자연과학논문집
    • /
    • 제6권1호
    • /
    • pp.46-52
    • /
    • 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.

ON THE EQUATIONS DEFINING SOME RATIONAL CURVES OF MAXIMAL GENUS IN ℙ3

  • Wanseok LEE;Shuailing Yang
    • East Asian mathematical journal
    • /
    • 제40권3호
    • /
    • pp.287-293
    • /
    • 2024
  • For a nondegenerate irreducible projective variety, it is a classical problem to describe its defining equations and the syzygies among them. In this paper, we precisely determine a minimal generating set and the minimal free resolution of defining ideals of some rational curves of maximal genus in ℙ3.

PYTHAGOREAN-HODOGRAPH CURVES IN THE MINKOWSKI PLANE AND SURFACES OF REVOLUTION

  • Kim, Gwang-Il;Lee, Sun-Hong
    • Journal of applied mathematics & informatics
    • /
    • 제26권1_2호
    • /
    • pp.121-133
    • /
    • 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.

  • PDF

Weight Control and Knot Placement for Rational B-spline Curve Interpolation

  • Kim, Tae-Wan;Lee, Kunwoo
    • Journal of Mechanical Science and Technology
    • /
    • 제15권2호
    • /
    • pp.192-198
    • /
    • 2001
  • We consider an interpolation problem with nonuniform rational B-spline curves given ordered data points. The existing approaches assume that weight for each point is available. But, it is not the case in practical applications. Schneider suggested a method which interpolates data points by automatically determining the weight of each control point. However, a drawback of Schneiders approach is that there is no guarantee of avoiding undesired poles; avoiding negative weights. Based on a quadratic programming technique, we use the weights of the control points for interpolating additional data. The weights are restricted to appropriate intervals; this guarantees the regularity of the interpolating curve. In a addition, a knot placement is proposed for pleasing interpolation. In comparison with integral B-spline interpolation, the proposed scheme leads to B-spline curves with fewer control points.

  • PDF