A New Approach to Differential Geometry using Clifford's Geometric Algebra

2012 ◽  
Author(s):  
John Snygg
Keyword(s):  
Differential Geometry ◽  
Geometric Algebra ◽  
New Approach ◽  
Clifford’S Geometric Algebra

scholarly journals Digital surface analysis: - A new approach using differential geometry

2013 ◽  
Vol 2013 (1) ◽  
pp. 1-4
Author(s):  
James K. Dirstein ◽  
Pavol Ihring ◽  
Stano Hroncek
Keyword(s):  
Differential Geometry ◽  
Surface Analysis ◽  
New Approach

scholarly journals Applications of the Exponential Function of the Euclidean Motion Group to the Theory of Gearing

2014 ◽  
Vol 6 ◽  
pp. 869580
Author(s):  
Baozhen Lei ◽  
Harald Löwe ◽  
Xunwei Wang
Keyword(s):  
Differential Geometry ◽  
Exponential Function ◽  
Machine Tools ◽  
Basic Equation ◽  
Time Parameter ◽  
Motion Group ◽  
New Approach ◽  
Euclidean Motion Group ◽  
Special Machine ◽  
Euclidean Motion

The present paper provides a first step to a new approach to the theory of gearing, which uses modern differential geometry in order to ensure a strict and coordinate-independent formulation. Here, we are mainly concerned with a basic equation, namely, the equation of meshing, of two rotating surfaces in mesh. Since we are able to solve this equation by the time parameter, we derive parameterizations of the mating pinion from a bevel gear as well as a parameterization for gears produced by special machine tools.

scholarly journals DIFFERENTIAL GEOMETRY AND MECHANICS: APPLICATIONS TO CHAOTIC DYNAMICAL SYSTEMS

2006 ◽  
Vol 16 (04) ◽  
pp. 887-910 ◽  
Author(s):  
JEAN-MARC GINOUX ◽  
BRUNO ROSSETTO
Keyword(s):  
Differential Geometry ◽  
Dynamical Systems ◽  
Slow Manifold ◽  
Van Der Pol ◽  
New Approach ◽  
Metric Properties ◽  
Chaotic Dynamical Systems ◽  
Mechanics Concepts ◽  
The One ◽  
Curvature And Torsion

The aim of this article is to highlight the interest to apply Differential Geometry and Mechanics concepts to chaotic dynamical systems study. Thus, the local metric properties of curvature and torsion will directly provide the analytical expression of the slow manifold equation of slow-fast autonomous dynamical systems starting from kinematics variables (velocity, acceleration and over-acceleration or jerk). The attractivity of the slow manifold will be characterized thanks to a criterion proposed by Henri Poincaré. Moreover, the specific use of acceleration will make it possible on the one hand to define slow and fast domains of the phase space and on the other hand, to provide an analytical equation of the slow manifold towards which all the trajectories converge. The attractive slow manifold constitutes a part of these dynamical systems attractor. So, in order to propose a description of the geometrical structure of attractor, a new manifold called singular manifold will be introduced. Various applications of this new approach to the models of Van der Pol, cubic-Chua, Lorenz, and Volterra–Gause are proposed.

scholarly journals Ideal Spiral Bevel Gears—A New Approach to Surface Geometry

1981 ◽  
Vol 103 (1) ◽  
pp. 127-132 ◽  
Author(s):  
R. L. Huston ◽  
J. J. Coy
Keyword(s):  
Differential Geometry ◽  
Gear Tooth ◽  
Parametric Representation ◽  
Spiral Bevel Gear ◽  
Surface Geometry ◽  
New Approach ◽  
Bevel Gears ◽  
Geometrical Characteristics ◽  
Tooth Surfaces ◽  
Spiral Bevel

This paper discusses the fundamental geometrical characteristics of spiral bevel gear tooth surfaces. The parametric representation of an ideal spiral bevel tooth is developed. The development is based on the elements of involute geometry, differential geometry, and fundamental gearing kinematics. A foundation is provided for the study of nonideal gears and the effects of deviations from ideal geometry on the contact stresses, lubrication, wear, fatigue life, and gearing kinematics.

scholarly journals The double-padlock problem: Is secure classical information transmission possible without key exchange?

2014 ◽  
Vol 33 ◽  
pp. 1460355
Author(s):  
James M. Chappell ◽  
Lachlan J. Gunn ◽  
Derek Abbott
Keyword(s):  
Open Problem ◽  
Geometric Algebra ◽  
Mathematical Description ◽  
The Other ◽  
Mathematical Representation ◽  
Information Channel ◽  
Physical Realization ◽  
Classical Information ◽  
Reduced Complexity ◽  
Clifford’S Geometric Algebra

The idealized Kish-Sethuraman (KS) cipher is theoretically known to offer perfect security through a classical information channel. However, realization of the protocol is hitherto an open problem, as the required mathematical operators have not been identified in the previous literature. A mechanical analogy of this protocol can be seen as sending a message in a box using two padlocks; one locked by the Sender and the other locked by the Receiver, so that theoretically the message remains secure at all times. We seek a mathematical representation of this process, considering that it would be very unusual if there was a physical process with no mathematical description. We select Clifford's geometric algebra for this task as it is a natural formalism to handle rotations in spaces of various dimension. The significance of finding a mathematical description that describes the protocol, is that it is a possible step toward a physical realization having benefits in increased security with reduced complexity.

scholarly journals A new approach to single-phase systems under sinusoidal and non-sinusoidal supply using geometric algebra

2020 ◽  
Vol 189 ◽  
pp. 106605
Author(s):  
Francisco G. Montoya ◽  
Raúl Baños ◽  
Alfredo Alcayde ◽  
Francisco M. Arrabal-Campos
Keyword(s):  
Geometric Algebra ◽  
Single Phase ◽  
New Approach ◽  
Phase Systems
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