Middle-school students’ concept images of geometric translations

This study explored sixth grade students’ concept images of geometric translations and the possible sources of their conceptions in a non-technological environment. The data were gathered through a written instrument, student and teacher interviews and document analyses. Analyses of student responses revealed two major concept images of geometric translations: (a) translation as translational motion, and (b) translation as both translational and rotational motion. Students who held these conceptions showed various levels of understanding, such as conceiving translations as undefined motion, partially-defined motion, and defined-motion of a single geometric figure on the plane. The findings of the study suggested, in general, consistencies between students’ concept images and their concept definitions. However, most of the students’ concept definitions were inconsistent with the formal concept definition of geometric translations.Data analyses also revealed five interpretations of a translation vector: (a) vector as a reference line, (b) vector as a symmetry line, (c) vector as a direction indicator, (d) vector as a parameter, and (e) vector as an abstract tool. Furthermore, classroom instruction, mathematics and science textbooks, real-life examples and everyday language were the major sources of students’ concept images of geometric translations.     

Controllability and stabilization of programmed motions of an automobile-type transport robot

A non-linear model of the motion of an automobile-type transport robot (TR) with absolutely rigid wheels, a steering device and actuators based on DC motors, is considered. Such a model for TR motion is a non-holonomic electromechanical system and, if the dynamics of the actuators and the steering device (forces of elasticity and attenuation in its elements) is ignored, corresponds to the model of automobile motion devised by Lineikin [1]. Non-linear canonical transformations of the state and control space coordinates are constructed which reduce the initial equations of motion of the TR to a simpler canonical form, convenient for the analysis and synthesis of control systems for the TR. These transformations are used to find the conditions for the controllability of the TR as a controlled object. Algorithms are given for constructing programmed controls and programmed motions of the TR. Stabilizing control laws are synthesized that make the programmed motions of the TR asymptotically stable and guarantee that the transients will have preassigned properties     

A distance on curves modulo rigid transformations

We propose a geometric method for quantifying the difference between parametrized curves in Euclidean space by introducing a distance function on the space of parametrized curves up to rigid transformations (rotations and translations). Given two curves, the distance between them is defined as the infimum of an energy functional which, roughly speaking, measures the extent to which the jet field of the first curve needs to be rotated to match up with the jet field of the second curve. We show that this energy functional attains a global minimum on the appropriate function space, and we derive a set of first-order ODEs for the minimizer.     

The Moving-Frame Method for the Iterated-Integrals Signature: Orthogonal Invariants

Geometric, robust-to-noise features of curves in Euclidean space are of great interest for various applications such as machine learning and image analysis. We apply Fels–Olver’s moving-frame method (for geometric features) paired with the log-signature transform (for robust features) to construct a set of integral invariants under rigid motions for curves in \({\mathbb {R}}^d\) from the iterated-integrals signature. In particular, we show that one can algorithmically construct a set of invariants that characterize the equivalence class of the truncated iterated-integrals signature under orthogonal transformations, which yields a characterization of a curve in \({\mathbb {R}}^d\) under rigid motions (and tree-like extensions) and an explicit method to compare curves up to these transformations.

     

Probabilistic matching of planar regions

We analyze a probabilistic algorithm for matching shapes modeled by planar regions under translations and rigid motions (rotation and translation). Given shapes A and B, the algorithm computes a transformation t such that with high probability the area of overlap of t(A) and B is close to maximal. In the case of polygons, we give a time bound that does not depend significantly on the number of vertices, but on perimeter and area of the shapes and, in the case of rigid motions, also on the diameter.     

Frameworks with forced symmetry II: orientation-preserving crystallographic groups

We give a combinatorial characterization of minimally rigid planar frameworks with orientation-preserving crystallographic symmetry, under the constraint of forced symmetry. The main theorems are proved by extending the methods of the first paper in this sequence from groups generated by a single rotation to groups generated by translations and rotations. The proof makes use of new families of matroids and submodular functions defined on crystallographic groups.     

High school students’ understandings of geometric transformations in the context of a technological environment

This study investigated the nature of students’ understandings of geometric transformations, which included translations, reflections, rotations, and dilations, in the context of the technological tool, The Geometer’s Sketchpad. The researcher implemented a seven-week instructional unit on geometric transformations within an Honors Geometry class. Students’ conceptions of transformations as functions were analyzed using the APOS theory and were informed by an analysis of students’ interpretations and uses of representations of geometrical objects using the constructs of drawing and figure. The analysis suggests students’ understandings of key concepts including domain, variables and parameters, and relationships and properties of transformations were critical for supporting the development of deeper understandings of transformations as functions.     

第二梯度流体的蠕变流和热传导相似解

给出了在笛卡儿坐标系中,忽略惯性的缓慢流动的二维运动方程和二阶梯度流体的传热方程．当Re1时,若从运动方程中简单地省略惯性项,则结果方程的解仍然近似有效．事实上,从无量纲的动量和能量方程也可导出这一结论．利用李群分析,知道求得的方程是对称的．李代数包括4个有限参数和一个无限参数组成的李群变换,其中一个是比例对称变换,另一个是平移变换．利用对称性求得两种不同形式的解．利用x和y坐标的平移,给出了指数形式的精确解．对于比例对称变换,更多地涉及到常微分方程,只能给出级数形式的近似解,最后讨论了某些边值问题．     

Exploring the interplay between memory effects and vesicle dynamics: A five-dimensional analysis using rigid sphere models and mapping techniques

The memory effect in vesicle dynamics refers to the persistence of shape changes in lipid vesicles, a type of lipid bilayer membrane that mimics some features of real cells, particularly red blood cells (RBCs). To study this effect, a fractional rigid sphere model in five dimensions has been investigated, which maps the dynamics of a vesicle using the Caputo operator. This model provides new insights into the mechanisms behind the memory effect as well as the emergence of two other motions, namely, tank-treading with underdamped and tank-treading with overdamped vesicle oscillations, in addition to the tank-treading mode where the vesicle rotates about its axis while maintaining a constant shape. Overall, this work represents a significant advance in our understanding of vesicle dynamics and has important implications for understanding the behavior of RBCs.     

Symmetry and integrable canonical flows

Following the method of a group theoretic formulation of rigid body dynamics, we construct an elementary proof that f commuting generators of symmetries of an f degree-of-freedom Hamiltonian system yield integrability of the dynamics in the form of f independent translations in phase space. The integrability of the dynamical system follows directly from the trivial integrability of a particular set of group parameter velocities that are nonintegrable in the absence of symmetry, and does not rely at all upon any assumption of separability of the Hamiltonian-Jacobi partial differential equation. Our method relies upon Hamel's explanation of when one can and cannot choose group parameters as generalized coordinates, and uses the Poisson bracket formulation of mechanics that is familiar to physicists.

We formally extend Euler's theorem on rigid body motions to other transformation groups for Hamiltonian flows in phase space, and also note the analogy between nonholonomic coordinates in classical mechanics and uncertainty principles in quantum mechanics.     

On conditions of existence of permanent rotations of connected rigid bodies system about vertical vector

In contrast to the classical problem of motion of a heavy rigid body about a fixed point where the permanent rotations are well known and completely investigated [7, 3] as the most simple and good visually demonstrated type of motions, in multibody mechanics under an increasing of quantity of the system bodies, mechanical parameters and the order of differential motion equations the study of such motions is more complicated problem. The problem on permanent rotations of two connected rigid bodies under influence of gravity force was investigated in [2, 4]. In this paper a system consisting of arbitrary constant quantity, n, of heavy rigid bodies which are sequentially jointed in a chain is considered. The conditions of existence of motions when each body permanently rotates about the vertical vector are determined. These conditions are analyzed in a general case when the bodies angular velocities are different.     

Geodesics on the Group of Semi-Affine Transformations of the Euclidean Plane

We obtain parameterized representations of geodesics of a sub-Riemannian metric on the three-dimensional Lie group of semi-affine transformations of the Euclidean plane, i.e., those that act as orientation preserving affine mappings on one axis, and as translations on the other one.     

Motion planning for planar bipeds using flatness: the single support phase

Trajectory planning and control of planar motions of biped robots is considered. The robot is modeled as a hierarchical structure of rigid links with rotational joints, which may be seen as a pendulum tree. Motors are available at all rotational joints. However, by the absence of control torques at the contact points with the ground, the system is underactuated. It is shown how differential flatness and time scaling can be helpful for the design of walking motions. Emphasis is put on the single support phase, when the robot touches the floor at a single point. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Partial surface matching by using directed footprints

In this paper we present a new technique for partial surface and volume matching of images in three dimensions. In this problem, we are given two objects in 3-space, each represented as a set of points, scattered uniformly along its boundary or inside its volume. The goal is to find a rigid motion of one object which makes a sufficiently large portion of its boundary lying sufficiently close to a corresponding portion of the boundary of the second object. This is an important problem in pattern recognition and in computer vision, with many industrial, medical, and chemical applications. Our algorithm is based on assigning a directed footprint to every point of the two sets, and locating all the pairs of points (one of each set) whose undirected components of the footprints are sufficiently similar. The algorithm then computes for each such pair of points all the rigid transformations that map the first point to the second, while making the respective direction components of their footprints coincide. A voting scheme is employed for computing transformations which map significantly large number of points of the first set to points of the second set. Experimental results on various examples are presented and show the accurate and robust performance of our algorithm.     

Single rub-impacting periodic motions of a rigid constrained rotor system

This paper discusses the existence of single rub-impacting period-n motions for a kind of rotor systems with rigid constraints. The ranges of parameters for period-2 motions are also given. An example of this method is given.     

Controllability and stabilization of the programmed motions of a transport robot

A non-linear mathematical model for the motion of a transport robot (TR) with a caterpillar chassis and with drives based on DC motors, which is a non-holonomic electromechanical system, is considered. Non-linear canonical transformations of the coordinates of the state and control space are constructed, which reduce the initial equations of motion of the TR to a simpler canonical form, which is convenient for analysing and synthesizing control systems for the TR. The conditions for the TR to be controllable as a controlled object are found. Algorithms are given for constructing programmed motions (PMs) of the TR. Stabilizing control laws are synthesized under which the PMs of the TR are asymptotically stable and transients of a specified nature are ensured.     

Oscillations of a rigid body containing an elastic element with distributed parameters

The motions of a hybrid (discrete-continual) system, consisting of a carrier rigid body and an elastic element with distributed parameters fastened to it are investigated. Two types of fastening are considered: (1) both ends are clamped, and (2) one of the ends is clamped while the other is free. A closed system of integro-differential equations is obtained which describes the state of the system under arbitrary initial conditions and forces applied to the rigid body. The perturbed motion of the rigid body in the case of a quasi-linear restoring force is investigated using asymptotic methods. The motions are studied both when there is internal resonance between the oscillations of the rigid body and the natural oscillations of the element, and when there are no such resonances. Qualitative effects are found.     

Codimension two bifurcations of fixed points in a class of vibratory systems with symmetrical rigid stops

A vibratory system having symmetrically placed rigid stops and subjected to periodic excitation is considered. Local codimension two bifurcations of the vibratory system with symmetrical rigid stops, associated with double Hopf bifurcation and interaction of Hopf and pitchfork bifurcation, are analyzed by using the center manifold theorem technique and normal form method of maps. Dynamic behavior of the system, near the points of codimension two bifurcations, is investigated by using qualitative analysis and numerical simulation. Hopf-flip bifurcation of fixed points in the vibratory system with a single stop are briefly analyzed by comparison with unfoldings analyses of Hopf-pitchfork bifurcation of the vibratory system with symmetrical rigid stops. Near the value of double Hopf bifurcation there exist period-one double-impact symmetrical motion and quasi-periodic impact motions. The quasi-periodic impact motions are represented by the closed circle and “tire-like” attractor in projected Poincaré sections. With change of system parameters, the quasi-periodic impact motions usually lead to chaos via “tire-like” torus doubling.     

The mass centre,the gravity centre and their duals on the elliptic plane

In this article the mass centres belonging to the force-free motions of a rigid particle system of the elliptic plane are defined. We examine the elementary geometrical connection between the mass centres and the gravity centre of any triangle on the elliptic plane. We search for those lines of an elliptic triangle that can be considered as the dual notions of the gravity centre, the centre of the incircle and the mass centre.