• Journal of the Korean Mathematical Society
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• v.37 no.3
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• pp.437-454
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• 2000

Let C be a smooth k-gonal curve of genus g. We study the number of pencils of degree k on C. In case $g\geqk(k-a)/2$ we state a conjecture based on a discussion on plane models for C. From previous work it is known that if C possesses a large number of pencils then C has a special plane model. From this observation the conjectures are split up in two cases : the existence of some types of plane curves should imply the existence of curves C with a given number of pencils; the non-existence of plane curves should imply the non-existence of curves C with some given large number of pencils. The non-existence part only occurs in the range $k(k-1)/2\leqg\leqk(k-1)/2] if k\geq7$. In this range we prove the existence part of the conjecture and we also prove some non-existence statements. Those result imply the conjecture in that range for $k\leq10$. The cases $k\leq6$ are handled separately.

• Bulletin of the Korean Mathematical Society
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• v.56 no.5
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• pp.1159-1186
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• 2019

In this paper, we study plane curves of degree 6 with points whose multiplicities of the tangents are 5. We determine all the Weierstrass semigroups of ramification points on double covers of the plane curves when the genera of the covering curves are greater than 29 and the ramification points are on the points with multiplicity 5 of the tangent.

• Honam Mathematical Journal
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• v.19 no.1
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• pp.107-116
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• 1997

In this paper, we get some properties of curves of constant relative breadth in the Minkowski plane.

• 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.

• 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.

• Korean Journal of Computational Design and Engineering
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• v.8 no.4
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• pp.262-269
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• 2003

An approach to compute characteristic curves such as section curves, reflection characteristic lines, and asymptotic curves on a surface is introduced. Each problem is formulated as a surface-plane inter-section problem. A single-valued function that represents the characteristics of a problem constructs a property surface on parametric space. Using a contouring algorithm, the property surface is intersected with a horizontal plane. The solution of the intersection yields a series of points which are mapped into object space to become characteristic curves. The approach proposed in this paper eliminates the use of traditional searching methods or non-linear differential equation solvers. Since the contouring algorithm has been known to be very robust and rapid, most of the problems are solved efficiently in realtime for the purpose of visualization. This approach can be extended to any geometric problem, if used with an appropriate formulation.

• Journal of the Korean Mathematical Society
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• v.37 no.3
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• pp.411-436
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• 2000

We compute lim t 0 Ct in 3 of some family $\pi$:Clongrightarrow$\Delta$ of plane quartics whose general members are nonsingular.

• The Pure and Applied Mathematics
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• v.28 no.1
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• pp.71-89
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• 2021

We define the constant ratio curves in the isotropic plane and investigate their deflection properties.

• 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 the Korean Mathematical Society
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• v.58 no.6
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• pp.1485-1500
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• 2021

A curve is rectifying if it lies on a moving hyperplane orthogonal to its curvature vector. In this work, we extend the main result of [Chen 2017, Tamkang J. Math. 48, 209] to any space dimension: we prove that rectifying curves are geodesics on hypercones. We later use this association to characterize rectifying curves that are also slant helices in three-dimensional space as geodesics of circular cones. In addition, we consider curves that lie on a moving hyperplane normal to (i) one of the normal vector fields of the Frenet frame and to (ii) a rotation minimizing vector field along the curve. The former class is characterized in terms of the constancy of a certain vector field normal to the curve, while the latter contains spherical and plane curves. Finally, we establish a formal mapping between rectifying curves in an (m + 2)-dimensional space and spherical curves in an (m + 1)-dimensional space.