• Journal of the Korean Mathematical Society
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• v.56 no.3
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• pp.805-823
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• 2019

We introduce polynomial PH-MPH transitive mappings which transform planar PH curves to MPH curves in ${\mathbb{R}}^{2,1}$, and prove that parameterizations of Enneper surfaces of the 1st and the 2nd kind and conjugates of Enneper surfaces of the 2nd kind are PH-MPH transitive. We show how to solve $C^1$ Hermite interpolation problems in ${\mathbb{R}}^{2,1}$, for an admissible $C^1$ Hermite data-set, by using the parametrization of Enneper surfaces of the 1st kind. We also show that we can obtain interpolants for at least some inadmissible data-sets by using MPH biarcs on Enneper surfaces of the 1st kind.

• Communications of the Korean Mathematical Society
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• v.37 no.1
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• pp.303-326
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• 2022

Our aim is to construct Hermite-type exponentially fitted interpolation formulas that use not only the pointwise values of an 𝜔-dependent function f but also the values of its first derivative at three unequally spaced nodes. The function f is of the form, f(x) = g₁(x) cos(𝜔x) + g₂(x) sin(𝜔x), x ∈ [a, b], where g₁ and g₂ are smooth enough to be well approximated by polynomials. To achieve such an aim, we first present Hermite-type exponentially fitted interpolation formulas I_N built on the foundation using N unequally spaced nodes. Then the coefficients of I_N are determined by solving a linear system, and some of the properties of these coefficients are obtained. When N is 2 or 3, some results are obtained with respect to the determinant of the coefficient matrix of the linear system which is associated with I_N. For N = 3, the errors for I_N are approached theoretically and they are compared numerically with the errors for other interpolation formulas.

• Communications of the Korean Mathematical Society
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• v.23 no.2
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• pp.295-305
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• 2008

This paper demonstrates how composite-exponential-fitting interpolation rules can be constructed to fit an oscillatory function using not only pointwise values of that function but also of that functions's derivative on a closed and bounded interval of interest. This is done in the framework of exponential-fitting techniques. These rules extend the classical composite cubic Hermite interpolating polynomials in the sense that they become the classical composite polynomials as a parameter tends to zero. Some examples are provided to compare the newly constructed rules with the classical composite cubic Hermite interpolating polynomials (or recently developed interpolation rules).

• Journal of applied mathematics & informatics
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• v.20 no.1_2
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• pp.575-584
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• 2006

In this paper we introduce a discrete tangent vector of a polygon defined on each vertex by a linear combination of forward difference and backward difference, and show that if the polygon is originated from a smooth curve then direction of the discrete tangent vector is a second order approximation of the direction of the tangent vector of the original curve. Using this discrete tangent vector, we also introduced the geometric cubic Hermite interpolation of a polygon with controlled initial and terminal speed of the curve segments proportional to the edge length. In this case the whole interpolation is $C^1$. Experiments suggest that about $90\%$ of the edge length is the best fit for the initial and terminal speeds.

• Journal of the Korean Mathematical Society
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• v.57 no.4
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• pp.957-971
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• 2020

In this paper, we introduce a new method to construct a parametric surface in terms of curves and points lying on Euclidean 3-space, called a C⁰-Hermite surface interpolation. We also prove the existence of a C⁰-Hermite interpolation of isoparametric surfaces with the so-called marching scale functions, and give some examples. Finally, we construct ruled surfaces and surfaces foliated by a circle as an isoparametric surface.

• Communications of the Korean Mathematical Society
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• v.21 no.2
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• pp.397-407
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• 2006

We construct polynomial-fitting interpolation rules to agree with a function f and its first derivative f' at equally spaced nodes on the interval of interest by introducing a linear functional with which we produce systems of linear equations. We also introduce a matrix whose determinant is not zero. Such a property makes it possible to solve the linear systems and then leads to a conclusion that the rules are uniquely determined for the nodes. An example is investigated to compare the rules with Hermite interpolating polynomials.

• Transactions of the Korean Society of Mechanical Engineers B
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• v.27 no.7
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• pp.859-866
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• 2003

The Lagrangian dispserion of fluid particles in inhomogeneous turbulence is investigated by a direct numerical simulation of turbulent channel flow. Fluid particle velocity and acceleration along a particle trajectory are computed by employing several interpolation schemes such as linear interpolation, high-order Lagrange polynomial interpolation and the Hermite interpolation schemes. The performances of the schemes are evaluated through comparison of errors in computed particle positions, velocities and accelerations against spectral interpolation. Adopting the four-point Hermite interpolation in the homogeneous directions and Chebyshev polynomials in the wall-normal direction appears to produce most reliable Lagrangian statistics including acceleration correlations with a reasonable amount of computational overhead.

• Journal of computational fluids engineering
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• v.12 no.1
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• pp.35-42
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• 2007

This paper describes a recent development on the divergence free basis function based on a hermite stream function and verifies its validity by comparing results with those from a modified residual method known as one of stabilized finite element methods. It can be shown that a proper choice of degrees of freedom at a node with a proper arrangement of the hermite interpolation functions can yield solenoidal or divergent free interpolation functions for the velocities. The well-known cavity problem has been chosen for validity of the present algorithm. The comparisons from numerical results between the present and the modified residual showed the present method yields better results in both the velocity and the pressure within modest Reynolds numbers(Re = 1,000).

• Transactions of the Korean Society of Machine Tool Engineers
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• v.12 no.5
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• pp.67-72
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• 2003

For the analysis of Kirchhoff plate bending problems, a new meshless method is implemented. For the satisfaction of the $C^1$ continuity condition in which the first derivative is treated an another primary variable, Hermite interpolation is enforced on standard reproducing kernel particle method. In order to impose essential boundary conditions on solving $C^1$ continuity problems, shape function modifications are adopted. Through numerical tests, the characteristics and accuracy of the HRKPM are investigated and compared with the finite element analysis. By this implementatioa it is shown that high accuracy is achieved by using HRKPM for solving Kirchhoff plate bending problems.

• Journal of the Korean Data and Information Science Society
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• v.17 no.4
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• pp.1289-1297
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• 2006

Before other statistical data analysis the preprocessing steps should be performed adequately to have meaningful results. These steps include processes such as baseline correction, normalization, denoising, and multiple alignment. In this paper an algorithm for baseline correction is proposed with using the piecewise cubic Hermite interpolation with block-selected points and local minima after denoising for SELDI or MALDI mass spectrometry data.