• Bulletin of the Korean Mathematical Society
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• v.38 no.4
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• pp.753-761
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• 2001

We introduce the notion of Gauss map of pointwise 1-type on ruled surfaces in the Euclidean 3-space for which vector valued functions is neither trivial nor it extends or coincides with the usual notion of 1-type, in general. We characterize the minimal helicoid in terms of it and give a complete classification of the ruled surfaces with pointwise 1-type Gauss map.

• Journal of the Korean Mathematical Society
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• v.55 no.6
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• pp.1459-1468
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• 2018

We show that an umbilic-free minimal surface in ${\mathbb{R}}^3$ belongs to the associate family of the catenoid if and only if the geodesic curvatures of its lines of curvature have a constant ratio. As a corollary, the helicoid is shown to be the unique umbilic-free minimal surface whose lines of curvature have the same geodesic curvature. A similar characterization of the deformation family of minimal surfaces with planar lines of curvature is also given.

• Journal of the Korean Mathematical Society
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• v.46 no.1
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• pp.215-223
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• 2009

The helicoidal surfaces with pointwise 1-type or harmonic gauss map in Euclidean 3-space are studied. The notion of pointwise 1-type Gauss map is a generalization of usual sense of 1-type Gauss map. In particular, we prove that an ordinary helicoid is the only genuine helicoidal surface of polynomial kind with pointwise 1-type Gauss map of the first kind and a right cone is the only rational helicoidal surface with pointwise 1-type Gauss map of the second kind. Also, we give a characterization of rational helicoidal surface with harmonic or pointwise 1-type Gauss map.

• The Pure and Applied Mathematics
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• v.15 no.3
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• pp.217-220
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• 2008

In this paper, we investigate the harmonic mappings that arise in connection with Scherk's surface and helicoid.

• Communications of the Korean Mathematical Society
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• v.20 no.4
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• pp.861-873
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• 2005

In this paper we approximate a cylindrical helix by bi-conic and bi-quadratic Bezier curves. Each approximation method is $G^1$ end-points interpolation of the helix. We present a sharp upper bound of the Hausdorff distance between the helix and each approximation curve. We also show that the error bound has the approximation order three and monotone increases as the length of the helix increases. As an illustration we give some numerical examples.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.13 no.3
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• pp.217-224
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• 2009

In this paper we present an approximation method of quadric surface using quartic spline. Our method is based on the approximation of quadratic rational B$\acute{e}$zier patch using quartic B$\acute{e}$zier patch. We show that our approximation method yields $G^1$ (tangent plane) continuous quartic spline surface. We illustrate our results by the approximation of helicoid-like surface.

• Journal of the Korean Mathematical Society
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• v.55 no.5
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• pp.1235-1255
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• 2018

We provide definitions for the helicoidal Killing field and the helicoid in arbitrary three-manifolds, and investigate helicoids and ruled minimal surfaces in homogeneous three-manifolds, mainly in $SL_2{\mathbb{R}}$ and Sol(3). In so doing we finish our classification of ruled minimal surfaces in homogeneous three-manifolds with the isometry group of dimension 4.

• Proceedings of the KSME Conference
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• 2001.06c
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• pp.485-491
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• 2001

This paper describes the development of simulation program which is able to design cutter profiles and 3-dimensional geometry for rotors in screw compressor. Based on the symmetric rotor profiles developed previously, cutters are designed and 3-dimensional geometries of rotors are generated used by simulation program. Symmetric rotors are manufactured by a universal milling machine, and surface geometries of them are measured by a 3-dimension scanner. It is shown that simulation program developed is useful to design cutter for rotor manufacturing and to generate the 3-dimensional helicoid geometry of rotor in screw compressor.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.26 no.6
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• pp.1026-1034
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• 2002

This paper describes the development of machining simulation program which is able to design cutter profile and 3-dimensional geometry for rotors in screw compressor. Based on the symmetric rotor profiles developed previously, cutters are designed and 3-dimensional geometries of rotors are generated by using simulation program. Symmetric rotors are manufactured by a universal milling machine, and surface geometries of them are measured by a 3-dimension scanner It is shown that simulation program developed is useful to design cutter for rotor manufacturing and to generate the 3-dimensional helicoid geometry of rotor in screw compressor.

• Bulletin of the Korean Mathematical Society
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• v.53 no.6
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• pp.1887-1892
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• 2016

It is well known that the helicoids are the only ruled minimal surfaces in ${\mathbb{R}}^3$. The similar characterization for ruled minimal surfaces can be given in many other 3-dimensional homogeneous spaces. In this note we consider the product space $M{\times}{\mathbb{R}}$ for a 2-dimensional manifold M and prove that $M{\times}{\mathbb{R}}$ has a nontrivial minimal surface ruled by horizontal geodesics only when M has a Clairaut parametrization. Moreover such minimal surface is the trace of the longitude rotating in M while translating vertically in constant speed in the direction of ${\mathbb{R}}$.