scholarly journals Characterization of Rectifying and Sphere Curves in ℝ3

2017 ◽  
Vol 14 (2) ◽  
Author(s):  
Julie Logan ◽  
Yun Myung Oh
Keyword(s):  
Tangent Vector ◽  
Normal Vector ◽  
Extra Condition ◽  
Space Curves ◽  
3D Space ◽  
Nonzero Curvature ◽  
Unit Speed ◽  
Curvature And Torsion ◽  
Rectifying Curve

Studies of curves in 3D-space have been developed by many geometers and it is known that any regular curve in 3D space is completely determined by its curvature and torsion, up to position. Many results have been found to characterize various types of space curves in terms of conditions on the ratio of torsion to curvature. Under an extra condition on the constant curvature, Y. L. Seo and Y. M. Oh found the series solution when the ratio of torsion to curvature is a linear function. Furthermore, this solution is known to be a rectifying curve by B. Y. Chen’s work. This project, uses a different approach to characterize these rectifying curves. This paper investigates two problems. The first problem relates to figuring out what we can say about a unit speed curve with nonzero curvature if every rectifying plane of the curve passes through a fixed point in ℝ3. Secondly, some formulas of curvature and torsion for sphere curves are identified. KEYWORDS: Space Curve; Rectifying Curve; Curvature; Torsion; Rectifying Plane; Tangent Vector; Normal Vector; Binormal Vector

scholarly journals Parametric Equations for Space Curves Whose Spherical Images Are Slant Helices

2019 ◽  
Vol 11 (5) ◽  
pp. 82
Author(s):  
Abderrazzak EL HAIMI ◽  
Malika IZID ◽  
Amina OUAZZANI CHAHDI
Keyword(s):  
Vector Fields ◽  
Parametric Representation ◽  
Normal Vector ◽  
Third Order ◽  
Regular Curve ◽  
Space Curves ◽  
Slant Helix ◽  
Spherical Images ◽  
Vector Differential Equation

The curve whose tangent and binormal indicatrices are slant helices is called a slant-slant helix. In this paper, we give a new characterization of a slant-slant helix and determine a vector differential equation of the third order satisfied by the derivative of principal normal vector fields of a regular curve. In terms of solution, we determine the parametric representation of the slant-slant helix from the intrinsic equations. Finally, we present some examples of slant-slant helices by means of intrinsic equations.

scholarly journals A special interpretation of the concept constant breadth for a space curve

2019 ◽  
Vol 23 (Suppl. 1) ◽  
pp. 371-382
Author(s):  
Tuba Agirman-Aydin
Keyword(s):  
Variable Coefficients ◽  
Space Curve ◽  
Normal Vector ◽  
Exit Point ◽  
Constant Breadth ◽  
Normal Vectors ◽  
Linear Differential ◽  
Curve Of Constant Breadth ◽  
Definition Of ◽  
Curvature And Torsion

The definition of curve of constant breadth in the literature is made by using tangent vectors, which are parallel and opposite directions, at opposite points of the curve. In this study, normal vectors of the curve, which are parallel and opposite directions are placed at the exit point of the concept of curve of constant breadth. In this study, on the concept of curve of constant breadth according to normal vector is worked. At the conclusion of the study, is obtained a system of linear differential equations with variable coefficients characterizing space curves of constant breadth according to normal vector. The coefficients of this system of equations are functions depend on the curvature and torsion of the curve. Then is obtained an approximate solution of this system by using the Taylor matrix collocation method. In summary, in this study, a different interpretation is made for the concept of space curve of constant breadth, the first time. Then this interpretation is used to obtain a characterization. As a result, this characterization we?ve obtained is solved.

scholarly journals On characterization of horizontal biharmonic curves in H^2 × R

2020 ◽  
Vol 15 (2) ◽  
pp. 35-41
Author(s):  
Talat Körpinar ◽  
Essin Turhan
Keyword(s):  
Curvature And Torsion ◽  
Biharmonic Curves

In this paper, we study biharmonic curves in H2 × R. We show that all of them are helices. By using the curvature and torsion of the curves, we give some characterizations of horizontal biharmonic curves in H2 × R.

Rational space curves are not “unit speed”

2007 ◽  
Vol 24 (4) ◽  
pp. 238-240 ◽  
Author(s):  
Rida T. Farouki ◽  
Takis Sakkalis
Keyword(s):  
Rational Space ◽  
Space Curves ◽  
Unit Speed

scholarly journals OSCAR: a framework to identify and quantify cells in densely packed three-dimensional biological samples

2021 ◽  
Author(s):  
Mario ledesma-terron ◽  
Diego perez-dones ◽  
david G Miguez
Keyword(s):  
Biological Samples ◽  
Object Segmentation ◽  
Three Dimensional ◽  
Cell Types ◽  
Biological Tissues ◽  
Nuclear Marker ◽  
Quantitative Characterization ◽  
3D Space ◽  
Immunofluorescence Labeling

We have developed an Object Segmentation, Counter and Analysis Resource (OSCAR) that is designed specifically to quantify densely packed biological samples with reduced signal-to-background ratio. OSCAR uses as input three dimensional images reconstructed from confocal 2D sections stained with dies such as nuclear marker and immunofluorescence labeling against specific antibodies to distinguish the cell types of interest. Taking advantage of a combination of arithmetic, geometric and statistical algorithms, OSCAR is able to reconstruct the objects in the 3D space bypassing segmentation errors due to the typical reduced signal to noise ration of biological tissues imaged in toto. When applied to the zebrafish developing retina, OSCAR is able to locate and identify the fate of each nuclei as a cycling progenitor or a terminally differentiated cell, providing a quantitative characterization of the dynamics of the developing vertebrate retina in space and time with unprecedented accuracy.

scholarly journals ESTIMATING TORSION OF DIGITAL CURVES USING 3D IMAGE ANALYSIS

2016 ◽  
Vol 35 (2) ◽  
pp. 81 ◽  
Author(s):  
Christoph Blankenburg ◽  
Christian Daul ◽  
Joachim Ohser
Keyword(s):  
A Priori ◽  
Three Dimensional ◽  
Synthetic Data ◽  
3D Image ◽  
Continuous Curve ◽  
Space Curves ◽  
Discrete Curves ◽  
Curvature Estimation ◽  
Tomographic Data ◽  
Curvature And Torsion

Curvature and torsion of three-dimensional curves are important quantities in fields like material science or biomedical engineering. Torsion has an exact definition in the continuous domain. However, in the discrete case most of the existing torsion evaluation methods lead to inaccurate values, especially for low resolution data. In this contribution we use the discrete points of space curves to determine the Fourier series coefficients which allow for representing the underlying continuous curve with Cesàro’s mean. This representation of the curve suits for the estimation of curvature and torsion values with their classical continuous definition. In comparison with the literature, one major advantage of this approach is that no a priori knowledge about the shape of the cyclic curve parts approximating the discrete curves is required. Synthetic data, i.e. curves with known curvature and torsion, are used to quantify the inherent algorithm accuracy for torsion and curvature estimation. The algorithm is also tested on tomographic data of fiber structures and open foams, where discrete curves are extracted from the pore spaces.

scholarly journals CHARACTERIZATION OF SPACELIKE BIHARMONIC CURVES WITH TIMELIKE BINORMAL ACCORDING TO FLAT METRIC IN LORENTZIAN HEISENBERG GROUP Heis³ - doi: 10.5269/bspm.v30i2.14706

2011 ◽  
Vol 30 (2) ◽  
pp. 101-107
Author(s):  
Talat Körpınar ◽  
Essin Turhan
Keyword(s):  
Heisenberg Group ◽  
Parametric Representation ◽  
Flat Metric ◽  
Curvature And Torsion ◽  
Biharmonic Curves

In this paper, we study spacelike biharmonic curves with timelike binormal according to flat metric in the Lorentzian Heisenberg group Heis³. We characterize spacelike biharmonic curves with timelike binormal in terms of their curvature and torsion. Additionally, we determine the parametric representation of the spacelike biharmonic curves with timelike binormal according to flat metric from this characterization.

scholarly journals PARABOLIC GEODESICS AS PARALLEL CURVES IN PARABOLIC GEOMETRIES

2013 ◽  
Vol 24 (09) ◽  
pp. 1350067
Author(s):  
MARC HERZLICH
Keyword(s):  
Tangent Vector ◽  
Natural Connection ◽  
Simple Characterization ◽  
Parabolic Geometries ◽  
Definition Of

We give a simple characterization of the parabolic geodesics introduced by Čap, Slovák and Žádník for all parabolic geometries. This goes through the definition of a natural connection on the space of Weyl structures. We then show that parabolic geodesics can be characterized as the following data: a curve on the manifold and a Weyl structure along the curve, so that the curve is a geodesic for its companion Weyl structure and the Weyl structure is parallel along the curve and in the direction of the tangent vector of the curve.

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