Striction Curve Research Articles

Instantaneous Screws of Weight-Bearing Knee: What Can the Screws Tell Us About the Knee Motion

There are several ways to represent a given object's motion in a 3D space having 6DOF i.e., three translations and three rotations. Some of the methods that are used are mathematical and do not provide any geometrical insight into the nature of the motion. Screw theory is a mathematical, while at the same time, geometrical method in which the 6DOF motion of an object can be represented. We describe the 6DOF motion of a weight-bearing knee by its screw parameters, that are extracted from 3D Optical Reflective motion capture data. The screw parameters which describe the transformation of the shank with respect to the thigh in each two successive frames, is represented as the instantaneous screw axis of the motion given in its Plücker line coordinate, along with its corresponding pitch and intensity values. Moreover, the Striction curve associated with the motion provides geometrical insight into the nature of the motion and its repeatability. We describe the theoretical background and demonstrate what the screw can tell us about the motion of healthy subjects' knee.

On Time-like Ruled Surfaces in Minkowski 3-space

In this paper, a study of time-like ruled surfaces in Minkowski 3-space is investigated by strictly connected time-like straight line moving with Darboux's frame along a differentiable space-like curve. By using the striction curve and the distribution parameter of time-like ruled surfaces, some theorems related to the geodesic curvature and the second fundamental form tensor are obtained.

Some Characteristic Properties of Parallelz-Equidistant Ruled Surfaces

Some characteristic properties of two ruled surfaces whose principal normal vectors are parallel along their striction curves inE3are examined by assuming that the distance between two central planes at suitable points is constant,E3. In case of which two ruled surfaces are close, the relationship between the integral invariants of this ruled surfaces is computed.

On Certain Surfaces and Curves Associated with the Bennett Linkage

Although the subject of many more investigations than any other skew linkage, the Bennett loop does not yield up its properties easily. Recent work has revealed the axodes, which putatively determine its higher-order kinematics, and some of the derivable geometric features, but the study remains incomplete. The present paper uncovers previously hidden characteristics of significance, namely, the operative regulus in the fixed axode's central tangent surface and the consequent parametric equation of its curve of striction. The approach is extended to an examination of the hyperboloid which contains the loop's joint axes.

Formulas for computing axode normal and curvatures with derivation based on fundamental concepts of engineering differential geometry

The formulas for computing surface normal and surface curvatures/GAVs of axodes are derived. The surface normal of axodes involves second-order time derivatives of the relevant conjugate motion, instead of only first order. The surface curvatures of axodes involve third-order time derivatives, instead of only first and second. The existence of curve of striction on axode is explained. An exemplar of computation for verification of the formulas is cited.

The Curve of Striction of the Bennett Loop’s Fixed Axode

In previous work, the algebraic representation of a fixed axode of the Bennett linkage has been revealed as extraordinarily cumbersome. In this sequel we use properties of the ruled surface to determine the central point of a typical generator of the axode and hence its curve of striction as the intersection of two comparatively simple surfaces. Because of its special significance in this case, we also obtain the equation to the central tangent surface. A feature of the investigation is the direct employment of screw vectors in dual format rather than unit line vectors.

Simulation of response of sodium layer to the propagation of gravity wave

A unified approach to kinematic synthesis of mechanism is presented in this paper. Firstly a new approach to adaptive curve fitting is presented, which leads the normal fitting error to be minimum for a series of given discrete points, including a plane curve fitting, a spherical curve fitting and a ruled surface fitting in terms of invariants of ruled surface. Approximate characteristic points and lines are defined, such as an approximate circle point, an approximate slide point, an approximate spherical cone point and an approximate constant axis with an approximate spherical image cone point and an approximate striction curve. Then, the ruled surface fitting will be converted into a space curve fitting and a spherical curve fitting by differential geometry. Based on these definitions and the adaptive curve fitting approaches, the unified mathematical model is set up for the kinematic synthesis of mechanism from planar, spherical to spatial motion. Finally, a planar mechanism or a spatial mechanism can be synthesized by means of searching for two approximate characteristic points, or a characteristic point and a characteristic line, even two characteristic lines. This puper lays a theoretic base for the existence of the best solution and the convergence of the optimum algorithm.

Magnetostrictive Solids: Macroscopic Response and Microstructure Evolution under Applied Magnetic Fields and Loads

The macroscopic behavior of magnetostrictive materials results from domain evolu tions occurring at a microscopic scale. We show how to compute virgin magnetization and magneto striction curves by appraising the behavior of the underlying microstructures. Our method hinges on an averaging device (Young measures), which allows us to pass from the microscopic to the macro scopic scale. We take into account the kinematical constraints on the accommodation of elastic effects, and we highlight the role of material symmetry in the selection of energetically optimal microstructures. Finally, we analyze the influence of applied surface tractions on the magnetoelastic response of a magnetostrictive specimen.

�ber Zindler-Kurven in euklidischen R�umen

Zindler-curves in Euclidean plane ℝ2 are closed curves with a one-parametric set of congruent “main chords” which bisect both the length and the enclosed area of the curve. Related curves z ⊂ ℝn have been studied by J.Hoschek [2], [3] (in the case n=3) and B.Wegner [8] (for arbitrary n⩾2). In this note we generalize the results on Zindler-curves in ℝn. These curves can simply be generated since the midpoints of the main chords are situated on the striction curve of the main chord-surface.

On the dual spherical motions—I

In this paper it is shown that if the axes of curvature of the ruled surfaces U 1(t) in the moving and fixed spaces intersect each other perpendicularly then the dralls (or distribution parameter) of the ruled surfaces, which are generated by the geodesic trihedron U 1, U 2, U 3s of a dual curve U 1(t) , satify the A. Mannheim and R. Hamilton's formulae. Furthermore, the “explicit characterization” of the spherical curve of striction is given in terms of U 3-surface and the skewness of distribution of the generator of U 1-surface.

über die Striktionslinien von einparametrigen Scharen linearer Räume

A generalisation of a ruled surface in n-dimensional euclidean space may be generated by euclidean motion of a s-plane As. For this one-parametric family {As} the curve of striction is defined and the following theorems are proved: (I) The generators As are parallel along the curve of striction, i.e. the multivectors representing As form a parallel vector field along the curve of striction. (II) If the curve of striction is geodesic on {As}, it is also an isogonal trajectory of the family of generators {As}.

Die algebraischen Konoide mit ebener Striktionslinie

Among the algebraic ruled surfaces a theorem ofH. Brauner characterizes certain helicoids of projective screw motions by the property to possess continuous sets of plane shadow curves. Making use of this theorem all algebraic conoids with a plane striction curve are determined in euclidean 3-space and some properties of these curves are discussed.

A Note upon some Properties of the Curve of Striction

Certain properties of Ruled Surfaces relating to their Curves of Striction are suggested and immediately proved by the use of a particular kind of coordinates, known as Dual Coordinates, for the Generating Lines. These coordinates, were introduced by Prof. E. Study; the theory of them is fully explained and many beautiful applications are made in his treatise, Geometrie der Dynamen, Teubner, 1903, and in his article, Complexe Grössen in the Encyclopädie der math. Wissenschaften, Bd. I., pp. 147–183, and specially p. 166. In applying the method here I have ventured to regard the fundamental quantity c in a light which, if less rigorous, has the advantage of being familiar to most of us in other mathematical work.