Frenet Frame-Based Generalized Space Curve Representation for Pose-Invariant Classification and Recognition of 3-D Face

2016 ◽  
Vol 46 (4) ◽  
pp. 522-533 ◽  
Author(s):  
Manar D. Samad ◽  
Khan M. Iftekharuddin
Keyword(s):  
Space Curve ◽  
Frenet Frame ◽  
Curve Representation

scholarly journals A Curve Satisfying K/T = s with constant K>0

2015 ◽  
Vol 12 (2) ◽  
Author(s):  
Yun Myung Oh ◽  
Ye Lim Seo
Keyword(s):  
Linear Function ◽  
Explicit Formula ◽  
Series Solution ◽  
Space Curve ◽  
Frenet Frame ◽  
Rectifying Curve

In the present paper, we investigate a space curve in which the curvature is constant and the torsion is a linear function. The aim of this paper is to find an explicit formula for this space curve when the ratioof the torsion to the curvature is a linear function when the curvature is constant. KEYWORDS: Space Curve, Curvature, Torsion, General Helix, Frenet Frame, Series Solution, Rectifying Curve

scholarly journals New Topological Configurations in the Continuous Heisenberg Spin Chain: Lower Bound for the Energy

2015 ◽  
Vol 2015 ◽  
pp. 1-3 ◽  
Author(s):  
Rossen Dandoloff
Keyword(s):  
Lower Bound ◽  
Heisenberg Model ◽  
Unit Vector ◽  
Space Curve ◽  
The Other ◽  
Frenet Frame ◽  
Heisenberg Spin ◽  
One Dimensional ◽  
Heisenberg Spin Chain ◽  
New Class

In order to study the spin configurations of the classical one-dimensional Heisenberg model, we map the normalized unit vector, representing the spin, on a space curve. We show that the total chirality of the configuration is a conserved quantity. If, for example, one end of the space curve is rotated by an angle of 2πrelative to the other, the Frenet frame traces out a noncontractible loop inSO(3)and this defines a new class of topological spin configurations for the Heisenberg model.

Geometry Design and Contact Ratio Analysis for Circular Arc Parallel-Axis Helical Gears

2016 ◽  
Author(s):  
Z. Chen ◽  
B. Lei ◽  
Q. Zhao
Keyword(s):  
Rolling Process ◽  
Helical Gear ◽  
Space Curve ◽  
Impact Factors ◽  
Contact Ratio ◽  
Circular Arc ◽  
Practical Applications ◽  
Geometry Design ◽  
Gear Mechanism ◽  
The Impact

Based on space curve meshing theory, in this paper, we present a novel geometric design of a circular arc helical gear mechanism for parallel transmission with convex-concave circular arc profiles. The parameter equations describing the contact curves for both the driving gear and the driven gear were deduced from the space curve meshing equations, and parameter equations for calculating the convex-concave circular arc profiles were established both for internal meshing and external meshing. Furthermore, a formula for the contact ratio was deduced, and the impact factors influencing the contact ratio are discussed. Using the deduced equations, several numerical examples were considered to validate the contact ratio equation. The circular arc helical gear mechanism investigated in this study showed a high gear transmission performance when considering practical applications, such as a pure rolling process, a high contact ratio, and a large comprehensive strength.

scholarly journals A Note on Parametric Surfaces in Minkowski 3-Space

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Esra Betul Koc Ozturk
Keyword(s):  
Linear Combination ◽  
Sufficient Conditions ◽  
Ruled Surface ◽  
Necessary And Sufficient Conditions ◽  
Frenet Frame ◽  
Parametric Surface ◽  
Parametric Surfaces ◽  
Null Curve ◽  
Necessary And Sufficient

With the help of the Frenet frame of a given pseudo null curve, a family of parametric surfaces is expressed as a linear combination of this frame. The necessary and sufficient conditions are examined for that curve to be an isoparametric and asymptotic on the parametric surface. It is shown that there is not any cylindrical and developable ruled surface as a parametric surface. Also, some interesting examples are illustrated about these surfaces.

Designing and Mapping Trimming Curves on Surfaces Using Orthogonal Projection

1990 ◽  
Author(s):  
Joseph Pegna ◽  
Franz-Erich Wolter
Keyword(s):  
Differential Equation ◽  
Orthogonal Projection ◽  
Broad Class ◽  
Design Tool ◽  
Space Curve ◽  
Computationally Efficient ◽  
Curves On Surfaces ◽  
Surface Representations ◽  
Differential Properties ◽  
Surface Curve

Abstract In the design and manufacturing of shell structures it is frequently necessary to construct trimming curves on surfaces. The novel method introduced in this paper was formulated to be coordinate independent and computationally efficient for a very general class of surfaces. Generality of the formulation is attained by solving a tensorial differential equation that is formulated in terms of local differential properties of the surface. In the method proposed here, a space curve is mapped onto the surface by tracing a surface curve whose points are connected to the space curve via surface normals. This surface curve is called to be an orthogonal projection of the space curve onto the surface. Tracing of the orthogonal projection is achieved by solving the aforementionned tensorial differential equation. For an implicitely represented surface, the differential equation is solved in three-space. For a parametric surface the tensorial differential equation is solved in the parametric space associated with the surface representation. This method has been tested on a broad class of examples including polynomials, splines, transcendental parametric and implicit surface representations. Orthogonal projection of a curve onto a surface was also developed in the context of surface blending. The orthogonal projection of a curve onto two surfaces to be blended provides not only a trimming curve design tool, but it was also used to construct smooth natural maps between trimming curves on different surfaces. This provides a coordinate and representation independent tool for constructing blend surfaces.

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.

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