Planar and Spatial Rigid Motion as Special Cases of Spherical and 3-Spherical Motion

1983 ◽  
Vol 105 (3) ◽  
pp. 569-575 ◽  
Author(s):  
J. M. McCarthy
Keyword(s):  
Rigid Motion ◽  
Arc Length ◽  
Spatial Motion ◽  
Special Cases ◽  
Curvature Theory ◽  
Rigid Motions ◽  
Spherical Motion ◽  
Smooth Transformation

This paper examines spherical and 3-spherical rigid motions with instantaneous invariants approaching zero. It is shown that these motions may be identified with planar and spatial motion, respectively. The instantaneous invariants are ratios of arc-length along the surface of the sphere to its radius, thus the process of shrinking their value may be viewed as expanding the sphere while bounding the instantaneous displacements on the sphere. This allows a smooth transformation of the results of the curvature theory of spherical and 3-spherical motion into their planar and spatial counterparts.

The Curvature Theory of Line Trajectories in Spatial Kinematics

1981 ◽  
Vol 103 (4) ◽  
pp. 718-724 ◽  
Author(s):  
J. M. McCarthy ◽  
B. Roth
Keyword(s):  
Planar Motion ◽  
Spatial Motion ◽  
Third Order ◽  
Local Shape ◽  
The Third ◽  
Moving Body ◽  
Physical Representation ◽  
Curvature Theory ◽  
Spatial Kinematics ◽  
Differential Properties

This paper develops the differential properties of ruled surfaces in a form which is applicable to spatial kinematics. Derivations are presented for the three curvature parameters which define the local shape of a ruled surface. Related parameters are also developed which allow a physical representation of this shape as generated by a cylindric-cylindric crank. These curvature parameters are then used to define all the lines in the moving body which instantaneously generate speciality shaped trajectories. Such lines may be used in the synthesis of spatial motions in the same way that the points on the inflection circle and cubic of stationary curvature are used to synthesize planar motion. As an example of this application several special sets of lines are defined: the locus of all lines which for a general spatial motion instantaneously generate helicoids to the second order and the locus of lines generating right hyperboloids to the third order.

Planar Tooth Profile Synthesis for Relative Curvature

2017 ◽  
Author(s):  
Zhiyuan Yu ◽  
Kwun-Lon Ting
Keyword(s):  
Contact Stress ◽  
Tooth Profile ◽  
Hertzian Contact ◽  
Rolling Motion ◽  
Special Cases ◽  
Systematic Methodology ◽  
Hertzian Contact Stress ◽  
Curvature Theory ◽  
The Given ◽  
Tooth Pair

This paper is the first that uses the new conjugation curvature theory [1] to directly synthesize conjugate tooth profiles with the given relative curvature that determines the Hertzian contact stress. Conjugation curvature theory offers a systematic methodology to synthesize the relative curvature for a tooth pair. For any given relative curvature between the contact tooth profiles, a generating point can be located on an auxiliary body. Under the rolling motion among the pinion pitch, the gear pitch and the pitch on the auxiliary body, the generating point will trace fully conjugate profiles on the pinion and gear bodies with the given relative curvature at the instant of the contact. Full conjugation throughout the contact of the profiles is guaranteed with the three instant centers remaining coincident [1]. The methodology is demonstrated with a planar tooth profile synthesis with given relative curvature. One may find that the Wildhaber-Novikov tooth profile, which is known to have low relative curvature and Hertzian contact stress, and its variations become special cases under such methodology.

scholarly journals MAXIMIZING THE AREA OF OVERLAP OF TWO UNIONS OF DISKS UNDER RIGID MOTION

2009 ◽  
Vol 19 (06) ◽  
pp. 533-556 ◽  
Author(s):  
SERGIO CABELLO ◽  
MARK DE BERG ◽  
PANOS GIANNOPOULOS ◽  
CHRISTIAN KNAUER ◽  
RENÉ VAN OOSTRUM ◽  
...  
Keyword(s):  
Approximation Algorithms ◽  
Approximation Algorithm ◽  
High Probability ◽  
Rigid Motion ◽  
Maximum Area ◽  
Empty Intersection ◽  
Rigid Motions ◽  
Unit Disks

Let A and B be two sets of n resp. m disjoint unit disks in the plane, with m ≥ n. We consider the problem of finding a translation or rigid motion of A that maximizes the total area of overlap with B. The function describing the area of overlap is quite complex, even for combinatorially equivalent translations and, hence, we turn our attention to approximation algorithms. We give deterministic (1 - ∊)-approximation algorithms for translations and for rigid motions, which run in O((nm/∊2) log (m/∊)) and O((n2m2/∊3) log m)) time, respectively. For rigid motions, we can also compute a (1 - ∊)-approximation in O((m2n4/3Δ1/3/∊3) log n log m) time, where Δ is the diameter of set A. Under the condition that the maximum area of overlap is at least a constant fraction of the area of A, we give a probabilistic (1 - ∊)-approximation algorithm for rigid motions that runs in O((m2/∊4) log 2(m/∊) log m) time and succeeds with high probability. Our results generalize to the case where A and B consist of possibly intersecting disks of different radii, provided that (i) the ratio of the radii of any two disks in A ∪ B is bounded, and (ii) within each set, the maximum number of disks with a non-empty intersection is bounded.

(Non-)rigid motion interpretation : a regularized approach

1988 ◽  
Vol 233 (1271) ◽  
pp. 217-234 ◽  
Keyword(s):  
Multiple Scales ◽  
Rigid Motion ◽  
Image Data ◽  
Small Region ◽  
General Assumption ◽  
Naturally Occurring ◽  
Special Cases ◽  
Ill Posed ◽  
Motion Discontinuities ◽  
Variational Condition

Determining 3D motion from a time-varying 2D image is an ill-posed problem; unless we impose additional constraints, an infinite number of solutions is possible. The usual constraint is rigidity, but many naturally occurring motions are not rigid and not even piecewise rigid. A more general assumption is that the parameters (or some of the parameters) characterizing the motion are approximately (but not exactly) constant in any sufficiently small region of the image. If we know the shape of a surface we can uniquely recover the smoothest motion consistent with image data and the known structure of the object, through regularization. This paper develops a general paradigm for the analysis of nonrigid motion. The variational condition we obtain includes many previously studied constraints as ‘special cases’. Among them are isometry, rigidity and planarity. If the variational condition is applied at multiple scales of resolution, it can be applied to turbulent motion. Finally, it is worth noting that our theory does not require the computation of correspondence (optic flow or discrete displacements), and it is effective in the presence of motion discontinuities.

scholarly journals Efficient algorithms for the reverse shortest path problem on trees under the hamming distance

2017 ◽  
Vol 27 (1) ◽  
pp. 46-60 ◽  
Author(s):  
Javad Tayyebi ◽  
Massoud Aman
Keyword(s):  
Shortest Path ◽  
Hamming Distance ◽  
Minimum Cost ◽  
Shortest Path Problem ◽  
Arc Length ◽  
Bound Constraints ◽  
Shortest Distance ◽  
Special Cases ◽  
Minimum Cost Flow Problem ◽  
Minimum Cut Problem

Given a network G(V,A,c) and a collection of origin-destination pairs with prescribed values, the reverse shortest path problem is to modify the arc length vector c as little as possible under some bound constraints such that the shortest distance between each origin-destination pair is upper bounded by the corresponding prescribed value. It is known that the reverse shortest path problem is NP-hard even on trees when the arc length modifications are measured by the weighted sum-type Hamming distance. In this paper, we consider two special cases of this problem which are polynomially solvable. The first is the case with uniform lengths. It is shown that this case transforms to a minimum cost flow problem on an auxiliary network. An efficient algorithm is also proposed for solving this case under the unit sum-type Hamming distance. The second case considered is the problem without bound constraints. It is shown that this case is reduced to a minimum cut problem on a tree-like network. Therefore, both cases studied can be solved in strongly polynomial time.

scholarly journals Faster backtracking algorithms for the generation of symmetry-invariant permutations

2002 ◽  
Vol 2 (6) ◽  
pp. 277-287 ◽  
Author(s):  
Oscar Moreno ◽  
John Ramírez ◽  
Dorothy Bollman ◽  
Edusmildo Orozco
Keyword(s):  
Parallel Implementations ◽  
Special Cases ◽  
Rigid Motions

A new backtracking algorithm is developed for generating classes of permutations, that are invariant under the groupG4of rigid motions of the square generated by reflections about the horizontal and vertical axes. Special cases give a new algorithm for generating solutions of the classicaln-queens problem, as well as a new algorithm for generating Costas sequences, which are used in encoding radar and sonar signals. Parallel implementations of this latter algorithm have yielded new Costas sequences for lengthn,19≤n≤24.

The Instantaneous Kinematics of Line Trajectories in Terms of a Kinematic Mapping of Spatial Rigid Motion

1987 ◽  
Vol 109 (1) ◽  
pp. 95-100 ◽  
Author(s):  
J. M. McCarthy
Keyword(s):  
Rigid Motion ◽  
Distribution Parameter ◽  
Spatial Motion ◽  
Euler Parameters ◽  
Kinematic Theory ◽  
Unit Hypersphere ◽  
Kinematic Mapping ◽  
Curvature And Torsion ◽  
Curve Form

The dual Euler parameters of a rigid spatial motion are used to define a curve on a dual unit hypersphere. The dual velocity, curvature, and torsion of this curve form a set of instantaneous parameters related to the instantaneous invariants of the motion. In this paper these new parameters are used to reformulate the kinematic theory of line trajectories. The distribution parameter, Disteli formulas, and the inflection congruence are examined in detail.

Curvature Theory of Envelope Curve in Two-Dimensional Motion and Envelope Surface in Three-Dimensional Motion

2015 ◽  
Vol 7 (3) ◽  
Author(s):  
Wei Wang ◽  
Delun Wang
Keyword(s):  
Three Dimensional ◽  
Two Dimensional ◽  
Spatial Motion ◽  
Envelope Curve ◽  
Contact Conditions ◽  
Numerical Examples ◽  
Envelope Surface ◽  
Straight Line ◽  
Curvature Theory ◽  
Fixed Line

The curvature theories for envelope curve of a straight line in planar motion and envelope ruled surface of a plane in spatial motion are systematically presented in differential geometry language. Based on adjoint curve and adjoint surface methods as well as quasi-fixed line and quasi-fixed plane conditions, the centrode and axode are taken as two logical starting-points to study kinematic and geometric properties of the envelope curve of a line in two-dimensional motion and the envelope surface of a plane in three-dimensional motion. The analogical Euler–Savary equation of the line and the analogous infinitesimal Burmester theories of the plane are thoroughly revealed. The contact conditions of the plane-envelope and some common surfaces, such as circular and noncircular cylindrical surface, circular conical surface, and involute helicoid are also examined, and then the positions and dimensions of different osculating ruled surfaces are given. Two numerical examples are presented to demonstrate the curvature theories.

scholarly journals Some Subclasses of Uniformly Univalent Functions with Respect to Symmetric Points

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 287 ◽  
Author(s):  
Shahid Mahmood ◽  
Hari Srivastava ◽  
Sarfraz Malik
Keyword(s):  
Growth Rate ◽  
Univalent Functions ◽  
Series Representation ◽  
Coefficient Estimate ◽  
Arc Length ◽  
Conic Domain ◽  
Special Cases ◽  
Symmetric Points ◽  
Inclusion Results ◽  
Taylor’S Series

This article presents the study of certain analytic functions defined by bounded radius rotations associated with conic domain. Many geometric properties like coefficient estimate, radii problems, arc length, integral representation, inclusion results and growth rate of coefficients of Taylor’s series representation are investigated. By varying the parameters in results, several well-known results in literature are obtained as special cases.

scholarly journals The Frenet-Serret description of Born rigidity and its application to the Dirac equation

2020 ◽  
Vol 66 (2 Mar-Apr) ◽  
pp. 180
Author(s):  
J.B. Formiga
Keyword(s):  
Dirac Equation ◽  
Special Relativity ◽  
Physical Theory ◽  
Rigid Motion ◽  
Noether Theorem ◽  
Max Born ◽  
Inertial Frames ◽  
Rigid Motions ◽  
To Come ◽  
Definition Of

The role played by non-inertial frames in physics is one of the most interesting subjects that we can study when dealing with a physical theory. It does not matter whether we are studying classical theories such as special relativity or quantum theory, the idea of an accelerated frame is always one of the first ideas to come to our minds. In the case of special relativity, a problem with the concept of rigidity emerged as soon as Max Born gave a reasonable definition of rigid motion: the Herglotz-Noether theorem imposes a strong restriction on the possible rigid motions. In this paper, the equivalence of this theorem with another one that is formulated with the help of Frenet-Serret formalism is proved, showing the connection between the rigid motion and the curvatures of the observer's trajectory in spacetime. In addition, the Dirac equation in the Frenet-Serret frame for an arbitrary observer is obtained and applied to the rotating observers. The solution in the rotating frame is given in terms of that of an inertial one.

Sign in / Sign up

Export Citation Format

Share Document