Comparability Determination of Curved Surface Based on Gaussian Curvature and Normal Vector

2006 ◽  
Author(s):  
Wuying Feng ◽  
Xijian Ping ◽  
Liang Miao
Keyword(s):  
Gaussian Curvature ◽  
Curved Surface ◽  
Normal Vector

Reanalysis of Some CLUSTER Bow-Shock Crossings With an Optimized Timing Method

2020 ◽  
Author(s):  
Gérard M. Chanteur
Keyword(s):  
Condition Number ◽  
A Priori ◽  
Time Lag ◽  
Bow Shock ◽  
Time Interval ◽  
Normal Vector ◽  
Time Lags ◽  
The Matrix ◽  
Pseudo Inverse

<p>When a shock is moving through a cluster of spacecraft, the normal N to the shock and the velocity of the shock along N can be determined from the crossing times of the different spacecraft assuming that the shock is planar and moves without deformation or rotation during the time interval of the encounter. For a cluster of four spacecraft there are six pairs of spacecraft, each one giving raise to a scalar equation relating the vector position R from the first to the second spacecraft, the normal vector N and the time lag Dt : R.N=VDt. This over-determined system of six equations is solved by computing the pseudo inverse of the matrix M acting on the normal vector on the lhs of the equation. Thus the system is modified by attributing a priori a positive weight to each equation (wj, j=1 to 6) the sum being constrained to 1. Then a statistical ensemble of 6-uplets (wj, j=1 to 6) is built ; for each element of this ensemble we compute the condition number of matrix M and we look for the 6-uplet giving the lowest condition number. This procedure warrants the best accuracy of the pseudo-inverse of M and hence the best estimate of the normal vector N. Adding random perturbations to M and to the time lags allows to estimate the uncertainties on N and V through simulations. This optimized timing method is applied to reanalyze some crossings of the terrestrial bow-shock by CLUSTER and the results are compared to the results obtained by the standard method using the reciprocal vectors defined in the ISSI report SR-008 « Multi-Spacecraft Analysis Methods Revisited » published in 2008. A similar method has been applied to the determination of wave vectors of chorus elements observed by MMS in the inner magntosphere.</p>

scholarly journals On the error estimation of multi-spacecraft timing method

2009 ◽  
Vol 27 (10) ◽  
pp. 3949-3955 ◽  
Author(s):  
X.-Z. Zhou ◽  
Z. Y. Pu ◽  
Q.-G. Zong ◽  
P. Song ◽  
S. Y. Fu ◽  
...  
Keyword(s):  
Solar Wind ◽  
Data Analysis ◽  
Error Estimation ◽  
Time Delays ◽  
Normal Vector ◽  
Error Sources ◽  
Data Analysis Technique ◽  
Analysis Technique ◽  
Slow Moving

Abstract. The multi-spacecraft timing method, a data analysis technique based on four-point measurements to obtain the normal vector and velocity of an observed boundary, has been widely applied to various discontinuities in the solar wind and the magnetosphere studies. In this paper, we perform simulations to analyze the errors of the timing method by specifying the error sources to the uncertainties in the determination of the time delays between each spacecraft pair. It is shown that the timing method may have large errors if either the spacecraft tetrahedron is largely elongated and/or flattened, or the discontinuity moves much slower than the constellation itself. The results, therefore, suggest that some of the applications of the timing method require reexamination with special caution, in particular for the studies of the slow-moving discontinuities associated with, for example, the plasmaspheric plumes.

scholarly journals Determination of the geometry of laser-cutting fronts with high spatial and temporal resolution

2021 ◽  
Vol 1135 (1) ◽  
pp. 012013
Author(s):  
Michael Sawannia ◽  
Peter Berger ◽  
Rudolf Weber ◽  
Thomas Graf
Keyword(s):  
Laser Power ◽  
High Speed ◽  
Frame Rate ◽  
Laser Fusion ◽  
Normal Vector ◽  
Local Surface ◽  
Aisi 304 ◽  
High Speed Video ◽  
Laser Fusion Cutting

Abstract The melt flow velocity and the local surface angles of the cutting front during laser fusion cutting of 10 mm AISI 304 were determined for a laser power of 8 kW and a feed rate of 2 m/min. The cut front was recorded with a polarization goniometer, which uses the polarization of the process emission to determine the local surface angles, allowing to calculate the orientation of the normal vector of the surface. The records in this work were carried out with a frame rate of 75 kHz and a spatial resolution of about 30 µm. This allowed to identify big and small structures moving down the cutting front and to determine their velocities. The approximate velocity of the small structures was 9.1 m/s and for the big structures approx. 2.5 m/s. The information of a usual high-speed video was compared with the additionally obtained geometry information.

Four-Point Regulation Algorithm for an End-Effector of a Flexible Drilling Robot

2011 ◽  
Vol 141 ◽  
pp. 298-302 ◽  
Author(s):  
Lai Xi Zhang ◽  
Xing Song Wang
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Curved Surface ◽  
Normal Vector ◽  
Main Body ◽  
End Effector ◽  
Workpiece Surface ◽  
Center Point ◽  
Position Regulation ◽  
The Difference

A new four-point algorithm for orientation and position regulation of an end-effector for a flexible drilling robot is presented in this paper．Four non-coplanar points in three-dimensional space can determine a unique sphere tangent to them, we can find out this sphere’s center point and radii and regard the shape of a workpiece surface in machining area approximately as such a sphere. A vector from the machining point to the center point is regarded as the one normal to the workpiece surface. By this principle, the algorithm firstly measured and calculated a vector normal to drilling machining area on curved surface by four points，and then calculated the difference between this normal vector and the axis of the spindle. If there is difference, the algorithm further figures out the angles of two revolving axes on end-effector and the displacements of three linear axes on robot main body, which should be adjusted．Simulation results of two kinds of curved surface show that the accuracy and efficiency satisfy the practical requirements using this algorithm．

scholarly journals Four-spacecraft determination of magnetopause orientation, motion and thickness: comparison with results from single-spacecraft methods

2004 ◽  
Vol 22 (4) ◽  
pp. 1347-1365 ◽  
Author(s):  
S. E. Haaland ◽  
B. U. Ö. Sonnerup ◽  
M. W. Dunlop ◽  
A. Balogh ◽  
E. Georgescu ◽  
...  
Keyword(s):  
Magnetic Field ◽  
Field Experiment ◽  
Residue Analysis ◽  
Minimum Variance ◽  
Normal Vector ◽  
Current Layer ◽  
Hybrid Techniques ◽  
Cluster Data ◽  
Magnetic Profile

Abstract. In this paper, we use Cluster data from one magnetopause event on 5 July 2001 to compare predictions from various methods for determination of the velocity, orientation, and thickness of the magnetopause current layer. We employ established as well as new multi-spacecraft techniques, in which time differences between the crossings by the four spacecraft, along with the duration of each crossing, are used to calculate magnetopause speed, normal vector, and width. The timing is based on data from either the Cluster Magnetic Field Experiment (FGM) or the Electric Field Experiment (EFW) instruments. The multi-spacecraft results are compared with those derived from various single-spacecraft techniques, including minimum-variance analysis of the magnetic field and deHoffmann-Teller, as well as Minimum-Faraday-Residue analysis of plasma velocities and magnetic fields measured during the crossings. In order to improve the overall consistency between multi- and single-spacecraft results, we have also explored the use of hybrid techniques, in which timing information from the four spacecraft is combined with certain limited results from single-spacecraft methods, the remaining results being left for consistency checks. The results show good agreement between magnetopause orientations derived from appropriately chosen single-spacecraft techniques and those obtained from multi-spacecraft timing. The agreement between magnetopause speeds derived from single- and multi-spacecraft methods is quantitatively somewhat less good but it is evident that the speed can change substantially from one crossing to the next within an event. The magnetopause thickness varied substantially from one crossing to the next, within an event. It ranged from 5 to 10 ion gyroradii. The density profile was sharper than the magnetic profile: most of the density change occured in the earthward half of the magnetopause. Key words. Magnetospheric physics (magnetopause, cusp and boundary layers; instruments and techniques) – Space plasma physics (discontinuities)

scholarly journals A note on the harmonic quasiconformal diffeomorphisms of the unit disc

Filomat ◽  
2015 ◽  
Vol 29 (2) ◽  
pp. 335-341
Author(s):  
Miljan Knezevic
Keyword(s):  
Gaussian Curvature ◽  
Unit Disc ◽  
Quasiconformal Mappings ◽  
Conformal Metrics

We analyze the properties of harmonic quasiconformal mappings and by comparing some suitably chosen conformal metrics defined in the unit disc we obtain some geometrically motivated inequalities for those mappings (see for instance [15, 17, 20]). In particular, we obtain the answers to many questions concerning these classes of functions which are related to the determination of different properties that are of essential importance for validity of the results such as those that generalize famous inequalities of the Schwarz-Pick type. The approach used is geometrical in nature, via analyzing the properties of the Gaussian curvature of the conformal metrics we are dealing with. As a consequence of this approach we give a note to the co-Lipschicity of harmonic quasiconformal self mappings of the unit disc at the origin.

scholarly journals Conceptual contributions to the determination of the curvature in Engineering courses

2019 ◽  
pp. 9-19
Author(s):  
Benjamín Vázquez-González ◽  
Homero Jiménez-Rabiela ◽  
Adrian Gustavo Bravo-Acosta ◽  
María Berenice Guadalupe Quintana-Diaz
Keyword(s):  
Differential Geometry ◽  
Radius Of Curvature ◽  
Normal Vector ◽  
Planar Curves

Differential geometry began with the study of the characteristics of planar curves, then the behavior of the curves in space was analyzed, which led to the postulates of Frenet, and hence differential geometry evolved due to the contributions of Gauss. At the highly specialized undergraduate courses, most of the literature presents this topic based on definitions, which can be understood with some difficulty by both students and even some teachers. This work presents a detailed description of the terms defined in the concept of curvature. It is of great importance that students from engineering courses understand this concept with certainty and confidence, because it will allow perceiving abstract terms, such as radius of curvature, osculating circle, normal vector; thus, they will have complete handling in the basic description of the movement of bodies. Some examples are presented.

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