• Korean Journal of Mathematics
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• v.26 no.4
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• pp.545-559
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• 2018

In this paper, we study the position vector of a developable q-slant ruled surface in the Euclidean 3-space $E^3$ in means of the Frenet frame of a q-slant ruled surface. First, we determinate the natural representations for the striction curve and ruling of a q-slant ruled surface. Then we obtain general parameterization of a developable q-slant ruled surface with respect to the conical curvature of the surface. Finally, we introduce some examples for the obtained result.

• Proceedings of the Korean Society of Precision Engineering Conference
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• 1997.04a
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• pp.862-867
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• 1997

A 5-axis NC milling technology is presented on ruled surface. Problems in 5-axis NC machining are such as tool interference,tool collision and change of tool attitude,etc. The change of tool attitude causes rotation of cutter and variation of feedrate to overcut part surface. This poor control of tool attitude is the primary problem in multi-axis NC milling. This paper observes ruled surface for control of tool attitude. Ruled surface is composed of directrix and ruling, line of constant magnitude. Directrix corresponds to points on part surface and Ruling cutting tool. Trajectory of tool movement corresponds to ruled surface.

• Korean Journal of Mathematics
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• v.24 no.4
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• pp.613-626
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• 2016

In this article, we study skew ruled surfaces by using the geodesic Frenet trihedron of its generator. We obtained some conditions on this surface to ensure that this ruled surface is flat, II-flat, minimal, II-minimal and Weingarten surface. Moreover, the parametric equations of asymptotic and geodesic lines on this ruled surface are determined and illustrated through example using the program of mathematica.

• Journal of the Korean Society of Clothing and Textiles
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• v.28 no.11
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• pp.1536-1543
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• 2004

Garment is made by a 2D pattern and should be fitted to a human body which has 3D characteristics. Therefore, to design a pattern more effectively, the use of 3D information of a human body and the investigation of relationship between the 3D garment and 2D pattern are necessary. In this work, ruled surface method was used to reflect the 3D information of a human body for a pattern design. The images of the brassiere line on the woman's dress form were captured by phase-shifting projection moire system and the 3D information on the design line was obtained. 2D patterns on the various parts of the brassiere were developed directly from the 3D data by the ruled surface method. In addition, design line, the area and the amount of dart were quantified. And then we verify the appropriateness of the ruled surface method to the 2D pattern development by measuring the distribution of the space between women's figure and segmented clothing item. It was found that the ruled surface method is useful to transform the 3D design line to the 2D pattern, if we followed the steps suggested in this paper.

• Bulletin of the Korean Mathematical Society
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• v.49 no.3
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• pp.635-645
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• 2012

In this study, we investigate the curvature functions of ruled surface with lightlike ruling for a null curve with Cartan frame in Minkowski 3-space. Also, we give relations between the curvature functions of this ruled surface and curvature functions of central normal surface. Finally, we use the curvature theory of the ruled surface for determine differential properties of a robot end-effector motion.

• Journal of the Korean Mathematical Society
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• v.43 no.6
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• pp.1339-1355
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• 2006

This paper develops in detail the differential geometry of ruled surfaces from two perspectives, and presents the underlying relations which unite them. Both scalar and dual curvature functions which define the shape of a ruled surface are derived. Explicit formulas are presented for the computation of these functions in both formulations of the differential geometry of ruled surfaces. Also presented is a detailed analysis of the ruled surface which characterizes the shape of a general ruled surface in the same way that osculating circle characterizes locally the shape of a non-null Lorentzian curve.

• Journal of the Korean Mathematical Society
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• v.55 no.4
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• pp.987-1004
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• 2018

In this paper, we get several centroaffine invariant properties for a ruled surface in ${\mathbb{R}}^4$ with centroaffine theories of codimension two. Then by solving certain partial differential equations and studying a centroaffine surface with some centroaffine invariant properties in ${\mathbb{R}}^4$, we obtain such a surface is centroaffinely equivalent to a ruled surface or one of the flat centered cyclic surfaces. Furthermore, some centroaffine invariant properties for centered cyclic surfaces are considered.

• Proceedings of the Korean Society of Precision Engineering Conference
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• 2002.10a
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• pp.482-487
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• 2002

This paper presents a robot trajectory generation method based on the dual curvature theory of ruled surfaces. Robot trajectory can be represented as a ruled surface generated by the TCP(Tool Center Point) and my unit vector among the tool frame. Dual curvature theory of ruled surfaces provides the robot control algorithm with the motion property parameters. With the differential properties of the ruled surface, the linear and angular motion properties of the robot end effector can be utilized in the robot trajectory planning.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.24 no.1 s.173
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• pp.182-189
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• 2000

This paper presents the real time control method of 5-axis NC machine for high precision and productivity based on the curvature theory, of a ruled surface. The trajectory, of NC machine is described by, way of a ruled surface generated by the points on part surface and tool axis direction vector. The curvature theory, of a ruled surface is then applied to deter-mine the motion parameters of the 5-axis machine for control. The controller computes position, orientation, and differential motion parameters of the tool in each sampling period. The real-time approach produces smoother surfaces and requires substantially less machining time compared to conventional off-line approaches. The propose real-time control method based of the curvature theory of a ruled surface may give new methodology of precision 5-axis machine control.

• Journal of the Korea Computer Graphics Society
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• v.4 no.2
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• pp.69-75
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• 1998

Ruled surfaces are useful concept for surface design because they are defined by the one-parameter family of lines. In this paper, we prove that the offsets of a developable surface (a special class of ruled surfaces) are developable surfaces. Moreover, we prove that the offsets of a non-developable ruled surface cannot be ruled surfaces.