Curvature-vector pair and its application in displaying CT colon data

2002 ◽  
Author(s):  
Zhan Zhang ◽  
Kenneth R. Hoffmann ◽  
Alan Walczak
Keyword(s):  
Curvature Vector ◽  
Colon Data

Uniqueness of the Geometric Representation in Large Rotation Finite Element Formulations

2010 ◽  
Vol 5 (4) ◽  
Author(s):  
Ahmed A. Shabana
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Computer Aided Design ◽  
Curvature Vector ◽  
Space Curve ◽  
Body Motion ◽  
Large Rotation ◽  
Space Curves ◽  
Curvature Continuity ◽  
Finite Element Formulations

Several finite element formulations used in the analysis of large rotation and large deformation problems employ independent interpolations for the displacement and rotation fields. As explained in this paper, three rotations defined as field variables can be sufficient to define a space curve that represents the element centerline. The frame defined by the rotations can differ from the Frenet frame of the space curve defined by the same rotation field and, therefore, such a rotation-based representation can provide measure of twist shear deformations and captures the rotation of the beam about its axis. However, the space curve defined using the rotation interpolation has a geometry that can significantly differ from the geometry defined by an independent displacement interpolation. Furthermore, the two different space curves defined by the two different interpolations can differ by a rigid body motion. Therefore, in these formulations, the uniqueness of the kinematic representation is an issue unless nonlinear algebraic constraint equations are used to establish relationships between the two independent displacement and rotation interpolations. Nonetheless, significant geometric and kinematic differences between two independent space curves cannot always be reduced by using restoring elastic forces. Because of the nonuniqueness of such a finite element representation, imposing continuity on higher derivatives such as the curvature vector is not straight forward as in the case of the absolute nodal coordinate formulation (ANCF) that defines unique displacement and rotation fields. ANCF finite elements allow for imposing curvature continuity without increasing the order of the interpolation or the number of nodal coordinates, as demonstrated in this paper. Furthermore, the relationship between ANCF finite elements and the B-spline representation used in computational geometry can be established, allowing for a straight forward integration of computer aided design and analysis.

scholarly journals On $L_1$-biharmonic timelike hypersurfaces in pseudo-Euclidean space $E_1^4$

2020 ◽  
Vol 51 (4) ◽  
pp. 313-332
Author(s):  
Firooz Pashaie
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Mean Curvature Vector ◽  
Principal Curvatures ◽  
First Variation ◽  
Chen’S Conjecture ◽  
The First Variation ◽  
Linearized Operator ◽  
Euclidean Submanifolds

A well-known conjecture of Bang Yen-Chen says that the only biharmonic Euclidean submanifolds are minimal ones. In this paper, we consider an extended condition (namely, $L_1$-biharmonicity) on non-degenerate timelike hypersurfaces of the pseudo-Euclidean space $E_1^4$. A Lorentzian hypersurface $x: M_1^3\rightarrow\E_1^4$ is called $L_1$-biharmonic if it satisfies the condition $L_1^2x=0$, where $L_1$ is the linearized operator associated to the first variation of 2-th mean curvature vector field on $M_1^3$. According to the multiplicities of principal curvatures, the $L_1$-extension of Chen's conjecture is affirmed for Lorentzian hypersurfaces with constant ordinary mean curvature in pseudo-Euclidean space $E_1^4$. Additionally, we show that there is no proper $L_1$-biharmonic $L_1$-finite type connected orientable Lorentzian hypersurface in $E_1^4$.

scholarly journals On totally umbilicalCR-submanifolds of a Kaehler manifold

1993 ◽  
Vol 16 (2) ◽  
pp. 405-408
Author(s):  
M. A. Bashir
Keyword(s):  
Sectional Curvature ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Lens Space ◽  
Holomorphic Sectional Curvature ◽  
Mean Curvature Vector ◽  
Kaehler Manifold ◽  
Totally Umbilical ◽  
The Mean

LetMbe a compact3-dimensional totally umbilicalCR-submanifold of a Kaehler manifold of positive holomorphic sectional curvature. We prove that if the length of the mean curvature vector ofMdoes not vanish, thenMis either diffeomorphic toS3orRP3or a lens spaceLp,q3.

scholarly journals Observation of reconnection pulses by Cluster and Double Star

2005 ◽  
Vol 23 (8) ◽  
pp. 2921-2927 ◽  
Author(s):  
X. H. Deng ◽  
R. X. Tang ◽  
R. Nakamura ◽  
W. Baumjohann ◽  
T. L. Zhang ◽  
...  
Keyword(s):  
Magnetic Field ◽  
Energetic Electron ◽  
Curvature Vector ◽  
Double Star ◽  
Neutral Sheet ◽  
Inner Magnetosphere ◽  
Point Analysis ◽  
Earthward Flow ◽  
Tailward Flow ◽  
The Magnetic Field

Abstract. During a reconnection event on 7 August 2004, Cluster and Double Star (TC-1) were near the neutral sheet and simultaneously detected the signatures of the reconnection pulses. AT 22:59 UT tailward flow followed by earthward flow was detected by Cluster at about 15 RE, while earthward plasma flow followed by tailward flow was observed by TC-1 at about 10 RE. During the flow reversal from tailward to earthward, the magnetic field Bz changed sign from mainly negative values to positive, and the X component of the magnetic curvature vector switched sign from the tailward direction to the earthward direction, which indicates that the reconnection site (X-line) moved tailward past the Cluster constellation. By using multi-point analysis and observation of energetic electron and ion flux, we study the movement and structure of the current sheet and discuss the braking effect of the earthward flow bursts in the inner magnetosphere.

scholarly journals Space-like submanifolds with parallel mean curvature in de Sitter spaces

2000 ◽  
Vol 69 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Chongzhen Ouyang ◽  
Zhenqi Li
Keyword(s):  
Fundamental Form ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Second Fundamental Form ◽  
Mean Curvature Vector ◽  
De Sitter ◽  
Parallel Mean Curvature Vector ◽  
Complete Space ◽  
The Second Fundamental Form ◽  
Parallel Mean Curvature

AbstractThis paper investigates complete space-like submainfold with parallel mean curvature vector in the de Sitter space. Some pinching theorems on square of the norm of the second fundamental form are given

Curve Shape Modification and Fairness Evaluation

2007 ◽  
Author(s):  
Tetsuzo Kuragano ◽  
Akira Yamaguchi
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Curvature Vector ◽  
Radius Of Curvature ◽  
Curve Shape ◽  
Linear Quadratic ◽  
Nurbs Curve ◽  
First Derivative ◽  
Curvature Distribution ◽  
The Given

A method to generate a quintic NURBS curve which passes through the given points is described. In this case, there are four more equations than there are positions of the control points. Therefore, four gradients which are the first derivative of a NURBS equation are assigned to the given points. In addition to this method, another method to generate a quintic NURBS curve which passes through the given points and which has the first derivative at these given points is described. In this case, a linear system will be underdetermined, determined or overdetermined depending on the number of given points with gradients. A method to modify NURBS curve shape according to the specified radius of curvature distribution to realize an aesthetically pleasing freeform curve is described. The differences between the NURBS curve radius of curvature and the specified radius of curvature is minimized by introducing the least-squares method. A criterion for a fair curve is proposed. Evaluation whether the designed curve is fair or not is accomplished by a comparison of the designed curve to a curve whose radius of curvature is monotone. The radius of curvature is specified by linear, quadratic, and cubic function using the least-squares method. A curve whose radius of curvature is reshaped by one of these algebraic functions is considered as a fair curve. The curvature vector of the curve is used to evaluate the fairness. The comparison of unit curvature vectors is used to evaluate the directional similarity of the curve. The comparison of the curvature is used to evaluate the similarity of the magnitude of curvature vectors. If the directional similarity of the designed curve is close to the fair curve, and also the similarity of the curvature is close to the fair curve, the designed curve can be judged as a fair curve.

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