Center-point Curves Through Six Arbitrary Points

1997 ◽  
Vol 119 (1) ◽  
pp. 36-39 ◽  
Author(s):  
A. P. Murray ◽  
J. Michael McCarthy
Keyword(s):  
Rigid Body ◽  
Kinematic Synthesis ◽  
Cubic Curve ◽  
Center Point ◽  
Point Curve ◽  
Non Linear ◽  
Pass Through ◽  
The Given

A circular cubic curve called a center-point curve is central to kinematic synthesis of a planar 4R linkage that moves a rigid body through four specified planar positions. In this paper, we show the set of circle-point curves is a non-linear subset of the set of circular cubics. In general, seven arbitrary points define a circular cubic curve; in contrast, we find that a center-point curve is defined by six arbitrary points. Furthermore, as many as three different center-point curves may pass through these six points. Having defined the curve without identifying any positions, we then show how to determine sets of four positions that generate the given center-point curve.

Assessing Position Order in Rigid Body Guidance: An Intuitive Approach to Fixed Pivot Selection

2008 ◽  
Vol 131 (1) ◽  
Author(s):  
David H. Myszka ◽  
Andrew P. Murray ◽  
James P. Schmiedeler
Keyword(s):  
Rigid Body ◽  
Single Point ◽  
New Method ◽  
Entire Plane ◽  
Fixed Frame ◽  
Center Point ◽  
Point Curve ◽  
Pivot Selection ◽  
Line Passing ◽  
Pass Through

Several established methods determine if an RR dyad will pass through a set of finitely separated positions in order. The new method presented herein utilizes only the displacement poles in the fixed frame to assess whether a selected fixed pivot location will yield an ordered dyad solution. A line passing through the selected fixed pivot is rotated one-half revolution about the fixed pivot, in a manner similar to a propeller with infinitely long blades, to sweep the entire plane. Order is established by tracking the sequence of displacement poles intersected. With four or five positions, fixed pivot locations corresponding to dyads having any specified order are readily found. Five-position problems can be directly evaluated to determine if any ordered solutions exist. Additionally, degenerate four-position cases for which the set of fixed pivots corresponding to ordered dyads that collapse to a single point on the center point curve can be identified.

A Parameterization of the Central Axis Congruence Associated with Four Positions of a Rigid Body in Space

1993 ◽  
Vol 115 (3) ◽  
pp. 547-551 ◽  
Author(s):  
J. M. McCarthy
Keyword(s):  
Rigid Body ◽  
The Body ◽  
Central Axis ◽  
Spatial Mechanisms ◽  
Center Point ◽  
Point Curve ◽  
Screw Surface ◽  
Computer Based

Given four positions of a rigid body in space, there is a congruence of lines that can be used as the central axes of cylindric cranks to guide the body through the four positions. This “central axis congruence” is a generalization of the center point curve of planar kinematics. It is known that this congruence is identical to the screw congruence which arises in the study of complementary screw quadrilateral. It is less well-known that the screw congruence is the “screw surface” of the 4C linkage formed by the complementary screw quadrilateral, and it is this relationship that we use to obtain a parameterization for the screw congruence and in turn, the central axis congruence. This parameterization should facilitate the use of this congruence in computer based design of spatial mechanisms.

An extension of Wallace's, Miguel's and Clifford's theorems on circles

1925 ◽  
Vol 22 (5) ◽  
pp. 684-687 ◽  
Author(s):  
F. P. White
Keyword(s):  
The Other ◽  
Three Dimensions ◽  
The Third ◽  
Cubic Curve ◽  
Pass Through ◽  
The Given

The theorem of Wallace that if four arbitrary lines he taken in a plane the circumcircles of the four triangles formed by them in threes meet in a point—in which for circles we may equally well take conies drawn through two arbitrary points I, J of the plane —may be proved very easily from three dimensions. Through the four given lines draw four arbitrary planes, forming a tetrahedron X, Y, Z, T. A unique cubic curve can be drawn through I, J and the four vertices. Projecting this curve from the point T upon the original plane, we get a conic which passes through I, J and the vertices of the triangle formed by the lines in which the plane meets the planes TYZ, TZX, TXY, that is, by three.of the given lines. This conic, which is one of the conics of the theorem, also passes through the third point K, other than I, J, in which the original plane meets the cubic curve. Projecting in turn from the other three vertices of the tetrahedron, we get the other three conics of the theorem, which also pass through K.

A Unified Analysis and Implementation of Four Multiply Separated Position Synthesis of Four-Bar Linkages

1992 ◽  
Author(s):  
P. Srikrishna ◽  
Kenneth J. Waldron
Keyword(s):  
Rigid Body ◽  
Design Procedure ◽  
Unified Approach ◽  
Center Point ◽  
Point Curve ◽  
Position Synthesis

Abstract The objective of this paper is to derive analytically the circle-point and center-point curve equations for the synthesis of four-bar linkages for rigid body guidance through four multiply separated design positions. A unified approach is evolved to deal with the different combinations of four finitely and infinitesimally separated design position, namely the PP-P-P, PP-PP and PPP-P cases. The design procedure incorporates the rectification procedures developed by Waldron (1977) to eliminate the branch and order problems and is implemented in the interactive synthesis package RECSYN.

Assessing Position Order in Rigid Body Guidance: An Intuitive Approach to Fixed Pivot Selection

2007 ◽  
Author(s):  
David H. Myszka ◽  
Andrew P. Murray ◽  
James P. Schmiedeler
Keyword(s):  
Rigid Body ◽  
Single Point ◽  
Entire Plane ◽  
Fixed Frame ◽  
Point Curve ◽  
Solution Methods ◽  
Pivot Selection ◽  
Line Passing ◽  
Pass Through ◽  
Degenerate Cases

This paper presents a new method for determining whether an RR dyad will pass through a set of finitely separated positions in order. Several established solution methods have been previously documented for this problem. This method utilizes only the displacement poles in the fixed frame to assess in an intuitive fashion whether a selected fixed pivot location will result in an ordered dyad solution. A line passing through the selected fixed pivot is rotated one-half revolution about the fixed pivot, in a manner similar to a propeller with infinitely long blades, to sweep the entire plane. Order is established by tracking the sequence of the displacement poles intersected by the rotating line. With four or five positions, fixed pivot locations corresponding to dyads having any specified order are readily found. Five-position problems can be directly evaluated to determine if any ordered solutions exist, and degenerate cases of four positions for which the set of fixed pivots corresponding to ordered dyads collapses to a single point on the center point curve can be identified.

CLASS OF BOOLEAN FUNCTIONS CONSTRUCTED USING SIGNIFICANT BITS OF LINEAR RECURRENCES OVER THE RING ℤ2n

2019 ◽  
Vol 6 (2) ◽  
pp. 90-94
Author(s):  
Hernandez Piloto Daniel Humberto
Keyword(s):  
Linear Function ◽  
Boolean Functions ◽  
Hamming Distance ◽  
Linear Recurrences ◽  
Maximum Period ◽  
Non Linear ◽  
The Given ◽  
Class Of Functions

In this work a class of functions is studied, which are built with the help of significant bits sequences on the ring ℤ2n. This class is built with use of a function ψ: ℤ2n → ℤ2. In public literature there are works in which ψ is a linear function. Here we will use a non-linear ψ function for this set. It is known that the period of a polynomial F in the ring ℤ2n is equal to T(mod 2)2α, where α∈ , n01- . The polynomials for which it is true that T(F) = T(F mod 2), in other words α = 0, are called marked polynomials. For our class we are going to use a polynomial with a maximum period as the characteristic polyomial. In the present work we show the bounds of the given class: non-linearity, the weight of the functions, the Hamming distance between functions. The Hamming distance between these functions and functions of other known classes is also given.

Geometric Characteristics of the Center-Point Curve Based on the Kinematics of the Compatibility Linkage

1992 ◽  
Author(s):  
J. A. Schaaf ◽  
J. A. Lammers
Keyword(s):  
Complex Number ◽  
Change Point ◽  
Double Point ◽  
Classification Scheme ◽  
Center Point ◽  
Degenerate Form ◽  
Point Curve ◽  
Position Synthesis ◽  
Point Form

Abstract In this paper we develop a method of characterizing the center-point curves for planar four-position synthesis. We predict the five characteristic shapes of the center-point curve using the kinematic classification of the compatibility linkage obtained from a complex number formulation for planar four-position synthesis. This classification scheme is more extensive than the conventional Grashof and non-Grashof classifications in that the separate classes of change point compatibility linkages are also included. A non-Grashof compatibility linkage generates a unicursal form of the center-point curve; a Grashof compatibility linkage generates a bicursal form; a single change point compatibility linkage generates a double point form; and a double or triple change point compatibility linkage generates a circular-degenerate or a hyperbolic-degenerate form.

scholarly journals Kinematic synthesis of planar, shape-changing, rigid body mechanisms for design profiles with significant differences in arc length

2013 ◽  
Vol 70 ◽  
pp. 425-440 ◽  
Author(s):  
Shamsul A. Shamsudin ◽  
Andrew P. Murray ◽  
David H. Myszka ◽  
James P. Schmiedeler
Keyword(s):  
Rigid Body ◽  
Arc Length ◽  
Kinematic Synthesis ◽  
Planar Shape
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