Principal curvatures and parallel surfaces of wave fronts

2019 ◽  
Vol 19 (4) ◽  
pp. 541-554 ◽  
Author(s):  
Keisuke Teramoto
Keyword(s):  
Principal Curvature ◽  
Singular Points ◽  
Principal Curvatures ◽  
Wave Fronts ◽  
Geometric Invariants ◽  
Parallel Surfaces

Abstract We give criteria for which a principal curvature becomes a bounded C∞-function at non-degenerate singular points of wave fronts by using geometric invariants. As applications, we study singularities of parallel surfaces and extended distance squared functions of wave fronts. Moreover, we relate these singularities to some geometric invariants of fronts.

DESIGN AND ANALYSIS OF GENEVA MECHANISM WITH CURVED SLOTS

2014 ◽  
Vol 38 (4) ◽  
pp. 557-567 ◽  
Author(s):  
Jung-Fa Hsieh
Keyword(s):  
Analytical Description ◽  
Singular Points ◽  
Principal Curvatures ◽  
Surface Theory ◽  
Double Points ◽  
Pressure Angle ◽  
Comprehensive Method ◽  
Analytical Formulae ◽  
Indexing Mechanism

A simple yet comprehensive method is proposed for the design of a Geneva indexing mechanism with curved slots. In the proposed approach, conjugate surface theory is used to derive an analytical description of the profile of the curved slots with and without an offset feature. Analytical formulae are then presented for the pressure angle of the Geneva mechanism and the principal curvatures of the curved slots. The effectiveness of an appropriate offset angle in eliminating the singular points and double-points on the curved slot profile is then demonstrated. Finally, a Geneva mechanism is fabricated in order to demonstrate the feasibility of the proposed approach.

scholarly journals Palm Vein Recognition Algorithm using Multilobe Differential Filters

2019 ◽  
Author(s):  
Екатерина Сафронова ◽  
Ekaterina Safronova ◽  
Елена Павельева ◽  
Elena Pavelyeva
Keyword(s):  
Principal Curvature ◽  
Spatial Configuration ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Gaussian Kernel ◽  
Branch Points ◽  
Principal Curvatures ◽  
Feature Maps ◽  
Vein Recognition ◽  
Palm Vein

In this article the new algorithm for palm vein recognition using multilobe differential filters is proposed. After palm vein image preprocessing vein structure is detected based on principal curvatures. The image is considered as a surface in a three-dimensional space. Some vein points are selected using the maximum principal curvature values, and the other vein points are found from starting points by moving along the direction of minimum principal curvature. Multilobe differential filters are used to extract feature maps for vein images. These filters are flexible in terms of basic lobe choice and spatial configuration of lobes. The multilobe differential filters used in the article simulate vein branch points, and Gaussian kernel is used as the basic lobe. The normalized root-mean-square error is applied for image matching. Experimental results using CASIA multi-spectral palmprint image database demonstrate the effectiveness of the proposed method. The value of EER=0.01693 is obtained.

scholarly journals Dupin's indicatrix: a tool for quantifying periclinal folds on maps

2003 ◽  
Vol 140 (6) ◽  
pp. 721-726 ◽  
Author(s):  
RICHARD J. LISLE
Keyword(s):  
Remote Sensing ◽  
Aspect Ratio ◽  
Principal Curvature ◽  
Quantitative Assessment ◽  
Tangent Plane ◽  
New Method ◽  
Principal Curvatures ◽  
Remote Sensing Images ◽  
Geometrical Figure ◽  
Anticlastic Curvature

The elliptical and hyperbolic outcrop patterns characteristic of periclinal folds can be used to classify structures according to different curvature attributes. Elliptical patterns indicate domal-basinal structures with synclastic curvature, that is, principal curvatures of the same sign. Hyperbolic patterns are diagnostic of anticlastic curvature (saddle-like structures). Such outcrop geometries are geological examples of Dupin's indicatrix, the geometrical figure obtained by sectioning a curved surface on a plane parallel and almost coincident with the tangent plane. The aspect ratio of Dupin's indicatrix is theoretically related to the ratio of the principal curvature values for the part of the structure being considered. This new method allows quantitative assessment of structures on maps and on remote sensing images. Illustrations are given from Wyoming, USA, and Yorkshire, England.

Study on Induced Principal Curvature and Induced Principal Direction Based on Normal Vector of Instantaneous Contact Line

2015 ◽  
Author(s):  
Yaping Zhao ◽  
Chenru Xi ◽  
Yimin Zhang
Keyword(s):  
Contact Line ◽  
Principal Curvature ◽  
Principal Direction ◽  
Normal Curvature ◽  
Normal Vector ◽  
Principal Curvatures ◽  
Proof Techniques ◽  
Principal Directions ◽  
Transient Contact ◽  
Worm Drive

The computing formulae, in different forms, for the normal vector of the instantaneous contact line are summarized systematically. For some of them, the distinct and sententious proof techniques are put forward. Based on the normal vector of the transient contact line, the computing formulae for the induced normal curvature and the induced geodesic torsion are deduced laconically and strictly. Owing to making use of the normal vector of the transient contact line, the style of the obtained formulae is more elegant. Particularly, a novel developing approach for the computing formula of the induced geodesic torsion is proposed. On the basis of the induced geodesic torsion, the computing formulae for the induced principal directions are derived. From this, the calculating formulae for the induced principal curvatures are obtained rigorously and conveniently. All these work reveal the pivotal position of the normal vector of the momentary contact line in the meshing theory for the line conjugate gearing. By right of the meshing theory established, the meshing analysis for the modified TA worm drive is performed. A number of basic and important formulae are attained and the numerical outcome of the induced principal curvature is given out.

scholarly journals FOCAL SURFACES OF WAVE FRONTS IN THE EUCLIDEAN 3-SPACE

2018 ◽  
Vol 61 (2) ◽  
pp. 425-440 ◽  
Author(s):  
KEISUKE TERAMOTO
Keyword(s):  
Wave Fronts ◽  
Geometric Properties ◽  
Geometric Invariants ◽  
Initial Wave

AbstractWe characterise singularities of focal surfaces of wave fronts in terms of differential geometric properties of the initial wave fronts. Moreover, we study relationships between geometric properties of focal surfaces and geometric invariants of the initial wave fronts.

scholarly journals Lines of principal curvature on canal surfaces in R³

2006 ◽  
Vol 78 (3) ◽  
pp. 405-415 ◽  
Author(s):  
Ronaldo Garcia ◽  
Jaume Llibre ◽  
Jorge Sotomayor
Keyword(s):  
Differential Equations ◽  
Principal Curvature ◽  
Riccati Equations ◽  
Principal Curvatures ◽  
Regular Curve ◽  
Variable Radius ◽  
Canal Surfaces ◽  
Curvature Lines ◽  
Principal Lines

In this paper are determined the principal curvatures and principal curvature lines on canal surfaces which are the envelopes of families of spheres with variable radius and centers moving along a closed regular curve in R³. By means of a connection of the differential equations for these curvature lines and real Riccati equations, it is established that canal surfaces have at most two isolated periodic principal lines. Examples of canal surfaces with two simple and one double periodic principal lines are given.

ON THE RIEMANN CURVATURE OPERATORS IN RANDERS SPACES

2013 ◽  
Vol 10 (09) ◽  
pp. 1350044
Author(s):  
M. RAFIE-RAD
Keyword(s):  
Principal Curvature ◽  
Finsler Geometry ◽  
Linear Operators ◽  
Curvature Operator ◽  
Riemann Curvature ◽  
Principal Curvatures ◽  
Underlying Space ◽  
Algebraic Properties ◽  
Tangent Manifold ◽  
Randers Metrics

The Riemann curvature in Riemann–Finsler geometry can be regarded as a collection of linear operators on the tangent spaces. The algebraic properties of these operators may be linked to the geometry and the topology of the underlying space. The principal curvatures of a Finsler space (M, F) at a point x are the eigenvalues of the Riemann curvature operator at x. They are real functions κ on the slit tangent manifold TM0. A principal curvature κ(x, y) is said to be isotropic (respectively, quadratic) if κ(x, y)/F(x, y) is a function of x only (respectively, κ(x, y) is quadratic with respect to y). On the other hand, the Randers metrics are the most popular and prominent metrics in pure and applied disciplines. Here, it is proved that if a Randers metric admits an isotropic principal curvature, then F is of isotropic S-curvature. The same result is also established for F to admit a quadratic principal curvature. These results extend Shen's verbal results about Randers metrics of scalar flag curvature K = K(x) as well as those Randers metrics with quadratic Riemann curvature operator. The Riemann curvature [Formula: see text] may be broken into two operators [Formula: see text] and [Formula: see text]. The isotropic and quadratic principal curvature are characterized in terms of the eigenvalues of [Formula: see text] and [Formula: see text].

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