An application of N-Bishop frame to spherical images for direction curves

2017 ◽  
Vol 14 (11) ◽  
pp. 1750162 ◽  
Author(s):  
Ozgur Keskin ◽  
Yusuf Yayli
Keyword(s):  
Unit Sphere ◽  
Integral Curve ◽  
Normal Direction ◽  
Bishop Frame ◽  
Integral Curves ◽  
Spherical Images

In this paper, we first introduce [Formula: see text]-Bishop frame for a normal direction curve which is defined as an integral curve of the principal normal of a curve. We express this new frame and its properties. Afterwards, we obtain new spherical images by translating [Formula: see text]-Bishop frame vectors to the center of unit sphere [Formula: see text] in [Formula: see text]. Then, these new spherical images are called [Formula: see text]-Bishop spherical images. Additionally, we compute the Frénet–Serret equations of these new spherical images. Moreover, we show that integral curves of [Formula: see text]-Bishop spherical images of slant helices are also slant helices. Finally, we present some illustrated examples.

Relation between Darboux and type-2 Bishop frames in Euclidean space

2016 ◽  
Vol 13 (07) ◽  
pp. 1650101 ◽  
Author(s):  
Amine Yilmaz ◽  
Emin Özyilmaz
Keyword(s):  
Euclidean Space ◽  
Transition Matrix ◽  
Geodesic Curvature ◽  
Spherical Image ◽  
Darboux Vector ◽  
Bishop Frame ◽  
Spherical Images

In this work, we investigate relationships between Darboux and type-2 Bishop frames in Euclidean space. Then, we obtain the geodesic curvature of the spherical image curve of the Darboux vector of the type-2 Bishop frame. Also, we give transition matrix between the Darboux and type-2 Bishop frames of the type-2 Bishop frames of the spherical images of the edges [Formula: see text] and [Formula: see text]. Finally, we express some interesting relations and illustrate of the examples by the aid Maple programe.

A parametric study of self-similar blast waves

1972 ◽  
Vol 52 (4) ◽  
pp. 657-682 ◽  
Author(s):  
A. K. Oppenheim ◽  
A. L. Kuhl ◽  
E. A. Lundstrom ◽  
M. M. Kamel
Keyword(s):  
Phase Plane ◽  
Integral Curve ◽  
Blast Waves ◽  
Uniform Density ◽  
Integral Curves ◽  
Comprehensive Examination ◽  
Self Similar ◽  
Two Parameters ◽  
The Waves ◽  
Shock Fronts

The paper presents a comprehensive examination of self-similar blast waves with respect to two parameters, one describing the front velocity and the other the variation of the ambient density immediately ahead of the front. All possible front trajectories are taken into account, including limiting cases of the exponential and logarithmic form. The structure of the waves is analysed by means of a phase plane defined in terms of two reduced co-ordinatesF≡ (t/rμ)uandZ≡ [(t/rμ)a]2, wheretandrare the independent (time and space) variables, μ ≡dlnrn/dIntnthe subscriptndenoting the co-ordinates of the front, anduandaare, respectively, the particle velocity and the speed of sound. Loci of extrema of the integral curves in the phase plane are traced and loci of singularities are determined on the basis of their intersections. Boundary conditions are introduced for the case when the medium into which the waves propagate is at rest. Representative solutions, pertaining to all the possible cases of blast waves bounded by shock fronts propagating into an atmosphere of uniform density, are obtained by evaluating the integral curves and determining the corresponding profiles of the gasdynamic parameters. Particular examples of integral curves for waves bounded by detonations are given and all the degenerate solutions, corresponding to cases where the integral curve is reduced to a point, are delineated.

Quadratic systems having a parabola as an integral curve

1989 ◽  
Vol 112 (1-2) ◽  
pp. 113-134 ◽  
Author(s):  
Colin Christopher
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Integral Curve ◽  
Canonical Forms ◽  
Quadratic Systems ◽  
Integral Curves

SynopsisThe class of quadratic systems having a parabola composed of integral curves is examined. Canonical forms are found for the members of this class, and conditions are obtained, using the Bendixson's Criterion and the Poincaré–Bendixson Theorem, for the existence or non-existence of limit cycles, in the case where there is a limit cycle “inside” the parabola (that is, in the convex component of its compliment).

scholarly journals Cusps of Bishop Spherical Indicatrixes and Their Visualizations

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Haiming Liu ◽  
Donghe Pei
Keyword(s):  
Space Curve ◽  
Singular Points ◽  
Bishop Frame ◽  
Spherical Images

The main result of this paper is using Bishop Frame and “Type-2 Bishop Frame” to study the cusps of Bishop spherical images and type-2 Bishop spherical images which are deeply related to a space curve and to make them visualized by computer. We find that the singular points of the Bishop spherical images and type-2 Bishop spherical images correspond to the point where Bishop curvatures and type-2 Bishop curvatures vanished and their derivatives are not equal to zero. As applications and illustration of the main results, two examples are given.

Vector Fields

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Fixed Point ◽  
Vector Field ◽  
Scalar Field ◽  
Vector Fields ◽  
Integral Curve ◽  
Jacobi Matrix ◽  
Partial Differential Operator ◽  
Material Derivative ◽  
First Order ◽  
Integral Curves

The velocity of a fluid at each point of space-time is a vector field (or flow). It is best to think of it in terms of the effect of fluid flow on some scalar field. A vector field is thus a first order partial differential operator, called the material derivative in fluid mechanics. The path of a speck of dust carried along (advected) by the fluid is the integral curve of the velocity field. Even simple vector fields can have quite complicated integral curves: a manifestation of chaos. Of special interest are incompressible (with zero divergence) and irrotational (with zero curl) flows. A fixed point of a vector field is a point at which it vanishes. The derivative of a vector field at a fixed point is a matrix (the Jacobi matrix) whose spectrum is independent of the choice of coordinates.

The intra-annual streamflow distribution of Ukrainian rivers in different phases of long-term cyclical fluctuations

Energetika ◽  
2015 ◽  
Vol 61 (2) ◽  
Author(s):  
Liudmyla Gorbachova
Keyword(s):  
20Th Century ◽  
Integral Curve ◽  
Annual Streamflow ◽  
Cyclical Fluctuation ◽  
Cyclical Fluctuations ◽  
Integral Curves ◽  
Observation Series ◽  
The Mean

In this article, the analysis of homogeneity and stationarity of the mean monthly discharges of the rivers of Ukraine was carried out by means of hydro-genetic methods with the use of observation series on 305 water gauging stations from the very beginning of the observation till 2010 inclusive. An integral curve was used for the estimates of homogeneity of the observation series. Difference-integral curves were used for estimation of stationarity of the observation series and for the study of their long-term cyclical fluctuations. It turned out that most series of the  mean monthly discharges were homogeneous. The  observation series that have a full cycle of long-term cyclical fluctuations (dry and wet phases) are stationary, whereas other observation series are quasi-stationary. The  intra-annual streamflow redistribution occurs depending on dry and wet phases of long-term cyclical fluctuations. It is shown that the terms, the duration of periods and seasons of intra-annual streamflow distribution should define cyclical fluctuations because the mean monthly discharges change in dry and wet phases. Therefore, it is necessary to carry out development of the schemes of the components of intra-annual streamflow distribution for the rivers of Ukraine because such schemes were defined on the  short observation series in 50–80s of the  20th century without considering a  long-term cyclical fluctuation streamflow.

Energy Distribution in Electron Beams

1972 ◽  
Vol 30 ◽  
pp. 568-569
Author(s):  
Tamotsu Ohno
Keyword(s):  
Electron Beam ◽  
Energy Distribution ◽  
Cross Section ◽  
Integral Curve ◽  
Electron Beams ◽  
Electron Gun ◽  
Angular Divergence ◽  
Apparent Increase ◽  
Integral Width ◽  
The Cross

The energy distribution in an electron; beam from an electron gun provided with a biased Wehnelt cylinder was measured by a retarding potential analyser. All the measurements were carried out with a beam of small angular divergence (<3xl0-4 rad) to eliminate the apparent increase of energy width as pointed out by Ichinokawa.The cross section of the beam from a gun with a tungsten hairpin cathode varies as shown in Fig.1a with the bias voltage Vg. The central part of the beam was analysed. An example of the integral curve as well as the energy spectrum is shown in Fig.2. The integral width of the spectrum ΔEi varies with Vg as shown in Fig.1b The width ΔEi is smaller than the Maxwellian width near the cut-off. As |Vg| is decreased, ΔEi increases beyond the Maxwellian width, reaches a maximum and then decreases. Note that the cross section of the beam enlarges with decreasing |Vg|.

The effects of surface absorbed monolayer microdiffraction patterns

1988 ◽  
Vol 46 ◽  
pp. 680-681
Author(s):  
M. Pan ◽  
J.M. Cowley
Keyword(s):  
Surface Potential ◽  
Electron Probe ◽  
Potential Distribution ◽  
Crystal Surface ◽  
Normal Direction ◽  
Surface Image ◽  
Extended Surface ◽  
Gas Molecules ◽  
Surface Nature ◽  
Electron Microdiffraction

Electron microdiffraction patterns, obtained when a small electron probe with diameter of 10-15 Å is directed to run parallel to and outside a flat crystal surface, are sensitive to the surface nature of the crystals. Dynamical diffraction calculations have shown that most of the experimental observations for a flat (100) face of a MgO crystal, such as the streaking of the central spot in the surface normal direction and (100)-type forbidden reflections etc., could be explained satisfactorily by assuming a modified image potential field outside the crystal surface. However the origin of this extended surface potential remains uncertain. A theoretical analysis by Howie et al suggests that the surface image potential should have a form different from above-mentioned image potential and also be smaller by several orders of magnitude. Nevertheless the surface potential distribution may in practice be modified in various ways, such as by the adsorption of a monolayer of gas molecules.

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