• Honam Mathematical Journal
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• v.29 no.3
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• pp.475-480
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• 2007

In this paper we find the point at which the rational B$\'{e}$zier curve has the given tangent direction. We also analyze the geometric properties of the point of quadratic rational B$\'{e}$zier curve.

• Korean Journal of Computational Design and Engineering
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• v.5 no.2
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• pp.168-174
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• 2000

A biarc is a curve connecting two circular arcs with the constraints of tangent continuity so that it can represent the free form currie approximately connecting several biarcs with the tangent continuity. Since a biarc consists of circular arcs, the offset curve of the curve represented by biarcs can be easily obtained. Besides. if the tool path is represented by biarcs, the efficiency of machining is improved and the amount of data is decreased. When approximating a curve with biarcs, the location of the point where two circular arcs meet each other plays an important part in determining the shape of a biarc. In this thesis, the optimum point where two circular arcs meet is calculated using the tangent information of the curve to approximate so that it takes less calculation time to approximate due to the decrease of the number of iterations.

• 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.

• Korean Journal of Mathematics
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• v.32 no.2
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• pp.259-284
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• 2024

In present study, we introduce ruled surfaces whose base curve is the Salkowski curve in Euclidean 3-space and whose generating lines consist of the Frenet vectors of this curve (tangent, principal normal and binormal vectors). Then, we produce regular surfaces from a vector with real coefficients, which is a linear combination of these vectors, and we examine some special cases for these surfaces. Moreover, we present some geometric properties and graphics of all these surfaces.

• International Journal of CAD/CAM
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• v.9 no.1
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• pp.55-60
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• 2010

Quadratic B$\acute{e}$zier curves are important geometric entities in many applications. However, it was often ignored by the literature the fact that a single segment of a quadratic B$\acute{e}$zier curve may fail to fit arbitrary endpoint unit tangent vectors. The purpose of this paper is to provide a solution to this problem, i.e., constructing $G^1$ quadratic B$\acute{e}$zier curves satisfying given endpoint (positions and arbitrary unit tangent vectors) conditions. Examples are given to illustrate the new solution and to perform comparison between the $G^1$ quadratic B$\acute{e}$zier cures and other curve schemes such as the composite geometric Hermite curves and the biarcs.

• Journal of the Chungcheong Mathematical Society
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• v.34 no.1
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• pp.37-53
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• 2021

In this paper we define scaled visual curvature and visual Frenet frame that can be visually accepted for discrete space curves. Scaled visual curvature is relatively simple compared to multi-scale visual curvature and easy to control the influence of noise. We adopt scaled minimizing directions of height functions on each neighborhood. Minimizing direction at a point of a curve is a direction that makes the point a local minimum. Minimizing direction can be given by a small noise around the point. To reduce this kind of influence of noise we exmine the direction whether it makes the point minimum in a neighborhood of some size. If this happens we call the direction scaled minimizing direction of C at p ∈ C in a neighborhood Br(p). Normal vector of a space curve is a second derivative of the curve but we characterize the normal vector of a curve by an integration of minimizing directions. Since integration is more robust to noise, we can find more robust definition of discrete normal vector, visual normal vector. On the other hand, the set of minimizing directions span the normal plane in the case of smooth curve. So we can find the tangent vector from minimizing directions. This lead to the definition of visual tangent vector which is orthogonal to the visual normal vector. By the cross product of visual tangent vector and visual normal vector, we can define visual binormal vector and form a Frenet frame. We examine these concepts to some discrete curve with noise and can see that the scaled visual curvature and visual Frenet frame approximate the original geometric invariants.

• Journal of Educational Research in Mathematics
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• v.14 no.2
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• pp.171-185
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• 2004

Students are exposed to a concept of tangent from a specific context of the relation between a circle and straight lines at the 7th grade. This initial experience might cause epistemological obstacles regarding learning concepts of tangent to additional curves. The paper provides a method of how to introduce a series of concepts of tangent in order to lead students to revise and improve the concept of tangent which they have. As students have chance to reflect and revise a series of concepts of tangent step by step, they realize the facts that the properties such as 'meeting the curve at one point' and 'touching but not cutting the curve' may be regarded as the proper definition of tangent in some limited contexts but are not essential in more general contexts. And finally students can grasp and appreciate that concept of tangent as the limit of secants and the relation between tangent and derivative.

• Proceedings of the KSR Conference
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• 1999.11a
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• pp.542-549
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• 1999

The purpose of this study is to investigate stability of a high speed train and to propose optimal design using sensitivity analysis of suspension design parameters. A form of equations of motion in tangent track and curve track is obtained based on each creep force. Tangent track and curve track equations include lateral, rolling and yawing motions of wheel sets, bogies, and carbodies. Three track cases have been chosen to stability assesment of a high speed train analysis. Sensitivity equations are set up by directly differentiating the equations of motion. This study def'.led Stability performance index of a high speed train in tangent track and curve track. The relative magnitude of the effect of suspension parameters on the critical speed is computed, and by adjusting these parameters, the increase of the critical speed is achieved.

• Communications of Mathematical Education
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• v.23 no.3
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• pp.625-642
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• 2009

In the school mathematics the concept of tangent is introduced in several steps in suitable contexts. Students are required to reflect and revise their concepts of tangent in order to apply the improved concept to wider range of contexts. In this paper we consider the tangent as the optimal linear approximation to a curve at a given point and make three discussions on pedagogical aspects of it. First, it provides a method of finding roots of real numbers which can be used as an application of tangent. This may help students improve their affective variables such as interest, attitude, motivation about the learning of tangent. Second, this concept reflects the modern point of view of tangent, the linear approximation of nonlinear problems. Third, it gives precise meaning of two tangent lines appearing two sides of a cusp point of a curve.

• Honam Mathematical Journal
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• v.37 no.4
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• pp.487-504
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• 2015

For a smooth plane curve, the curvature can be characterized by the rate of change of the angle between the tangent vector and a fixed vector. In this article we prove that the curvature of a space curve can also be given by the rate of change of the locally defined angle between the tangent vector at a point and the nearby point. By using height functions, we introduce turning angle of a space curve and characterize the curvature by the rate of change of the turning angle. The main advantage of the turning angle is that it can be used to characterize the curvature of discrete curves. For this purpose, we introduce a discrete turning angle and a discrete curvature called visual curvature for space curves. We can show that the visual curvature is an approximation of curvature for smooth curves.