Appendix, containing the Investigation of a Formula for the Rectification of any Arch of an Ellipse

1805 ◽  
Vol 5 (2) ◽  
pp. 271-293
Keyword(s):  
Finite Number ◽  
The Other ◽  
Algebraic Equations ◽  
Conic Sections ◽  
Straight Line ◽  
Curve Line ◽  
Straight Lines

It is now generally understood, that by the rectification of a curve line, is meant, not only the method of finding a straight line exactly equal to it, but also the method of expressing it by certain functions of the other lines, whether straight lines or circles, by which the nature of the curve is defined. It is evidently in the latter sense that we must understand the term rectification, when applied to the arches of conic sections, seeing that it has hitherto been found impossible, either to exhibit straight lines equal to them, or to express their relation to their co-ordinates, by algebraic equations, consisting of a finite number of terms.

scholarly journals Identical appended series of points as invariants in the collocal general-colinear fields

2003 ◽  
Vol 2 (5) ◽  
pp. 387-394
Author(s):  
Sonja Krasic
Keyword(s):  
The Other ◽  
Straight Line ◽  
Similar Series ◽  
Vanishing Line ◽  
Identical Series ◽  
Straight Lines ◽  
General Method

In order to bring the collocal collinear fields from the general into the perspective position, it is required to determine the identical appended series of points. Because of the properties depending on the projectivity that is given by the four appended points (straight lines) the appended identical series of the points and types are ranked among the invariants of general-collinear and perspectively-collinear fields. The procedure of determination of appended identical series of points is comprised of the following: in the set of ?1 of perspectively similar series in one field (whose center of perspective is a point on the vanishing line), find those that are identical to all the series in the set ?1 of perspective identical series of points in the other field (whose center of perspective is the point on the infinitely distant straight line). In the procedure, one begins from the appended similar methods obtained by the general method. The procedure is simplified by the introduction of the specially given similar series of points.

scholarly journals City Shape and the Fractality of Street Patterns

2012 ◽  
Vol 31 (2) ◽  
pp. 29-37 ◽  
Author(s):  
Nahid Mohajeri ◽  
Paul Longley ◽  
Michael Batty
Keyword(s):  
Fractal Dimension ◽  
Power Laws ◽  
Step Length ◽  
The Other ◽  
Main Street ◽  
Straight Line ◽  
Line Slope ◽  
Straight Lines ◽  
The City ◽  
Street Segments

City Shape and the Fractality of Street Patterns This paper discusses, first, the concepts of fractals and power laws in relation to the street patterns of the city of Dundee, East Scotland and, second, the results of the measurement of 6,004 street segments in the city. The trends of the street segments are presented through rose diagrams and show that there are two main street trends in the city: one is parallel with the coast, the other is roughly perpendicular to the coast. It is clear that the coastline largely regulates the street trend, because both the main street trends change along the city so as to be nearly coast-perpendicular and coast-parallel everywhere. The lengths of the street segments follow power laws. When presented on log-log plots, however, the result is not a single straight line but two straight lines. At the break in line slope, the fractal dimension changes from 0.88 to 2.20. The change occurs at the step length of about 100 m, indicating that the short streets belong to a population that is different from that of the longer streets.

scholarly journals Counterexamples to the quadrisecant approximation conjecture

2018 ◽  
Vol 27 (02) ◽  
pp. 1850022
Author(s):  
Sheng Bai ◽  
Chao Wang ◽  
Jiajun Wang
Keyword(s):  
Finite Number ◽  
Line Segment ◽  
Straight Line Segment ◽  
The Other ◽  
Straight Line ◽  
Set Of Points

A quadrisecant line of a knot [Formula: see text] is a straight line which intersects [Formula: see text] in four points, and a quadrisecant is a 4-tuple of points of [Formula: see text] which lie in order along the quadrisecant line. If [Formula: see text] has a finite number of quadrisecants, take [Formula: see text] to be the set of points of [Formula: see text] which are in a quadrisecant. Replace each subarc of [Formula: see text] between two adjacent points of [Formula: see text] along [Formula: see text] with the straight line segment between them. This gives the quadrisecant approximation of [Formula: see text]. It was conjectured that the quadrisecant approximation is always a knot with the same knot type as the original knot. We show that every knot type contains two knots, the quadrisecant approximation of one knot has self-intersections while the quadrisecant approximation of the other knot is a knot with a different knot type.

scholarly journals Note on Legendre's and Bertrand's Proofs of the Parallel-Postulate by Infinite Areas

1911 ◽  
Vol 30 ◽  
pp. 31-36
Author(s):  
D. M. Y. Sommerville
Keyword(s):  
Finite Number ◽  
Plane Surface ◽  
Second Order ◽  
The Other ◽  
Linear Segment ◽  
First Order ◽  
Other Hand ◽  
Straight Lines

One of the most plausible of the host of “proofs” that have ever been offered for Euclid's parallel-postulate is that known as Bertrand's, which is based upon a consideration of infinite areas. The area of the whole plane being regarded as an infinity of the second order, the area of a strip of plane surface bounded by a linear segment AB and the rays AA′, BB perpendicular to AB is an infinity of the first order, since a single infinity of such strips is required to cover the plane. On the other hand, the area contained between two intersecting straight lines is an infinity of the same order as the plane, since the plane can be covered by a finite number of such sectors. Hence if AP is drawn making any angle, however small, with AA′, the area A′AP, an infinity of the second order, cannot be contained within the area A′ABB′, an infinity of the first order, and therefore AP must cut BB′. And this is just Euclid's postulate.

scholarly journals Detour Extra Straight Lines in the Euclidean Plane

2020 ◽  
pp. 237-249
Author(s):  
I. Szalay ◽  
B. Szalay
Keyword(s):  
Euclidean Plane ◽  
Euclidean Geometry ◽  
The Other ◽  
Real Numbers ◽  
Other Hand ◽  
Straight Line ◽  
Axiom Systems ◽  
Straight Lines

Using the theory of exploded numbers by the axiom-systems of real numbers and Euclidean geometry, we explode the Euclidean plane. Exploding the Euclidean straight lines we get super straight lines. The extra straight line is the window phenomenon of super straight line. In general, the extra straight lines are curves in Euclidean sense, but they have more similar properties to Euclidean straight lines. On the other hand, with respect of parallelism we find a surprising property: there are detour straight lines.

scholarly journals III.—On the Extension of Brouncker's Method to the Comparison of Several Magnitudes.

1870 ◽  
Vol 26 (1) ◽  
pp. 59-67
Author(s):  
Edward Sang
Keyword(s):  
The Other ◽  
Exact Science ◽  
Straight Line ◽  
The One ◽  
Definition Of ◽  
Straight Lines

The discovery of those numbers which shall, either truly or approximately, represent the ratio of two magnitudes, necessarily attracted the attention of the earliest cultivators of exact science. The definition of the equality of ratios given in Euclid's compilation clearly exposes the nature of the process used in his time. This process consisted in repeating each of the two magnitudes until some multiple of the one agreed perfectly or nearly with a multiple of the other; the numbers of the repetitions, taken in inverse order, represented the ratio. Thus, if the proposed magnitudes were two straight lines, Euclid would have opened two pairs of compasses, one to each distance, and, beginning at some point in an indefinite straight line, he would step the two distances along, bringing up that which lagged behind, until he obtained an exact or a close coincidence.

CLASSIFYING POLYGONAL CHAINS OF SIX SEGMENTS

2004 ◽  
Vol 13 (04) ◽  
pp. 479-514 ◽  
Author(s):  
THOMAS J. CLARK ◽  
GERARD A. VENEMA
Keyword(s):  
Finite Number ◽  
The Other ◽  
Line Segments ◽  
Polygonal Chain ◽  
Straight Line ◽  
End To End ◽  
Polygonal Chains

A polygonal chain is the union of a finite number of straight line segments in ℝ3 that are connected end-to-end. Two chains are considered to be equivalent if there is an isotopy of ℝ3 that moves one chain to the other while keeping the segments rigid. Each segment must remain straight during the isotopy and the lengths of the segments may not change, but bending and twisting are allowed at the joints between the segments. Chains may be knotted and stuck in this category even though all chains are topologically trivial. Cantarella and Johnston have classified polygonal chains with five or fewer segments. In this paper we classify polygonal chains of six segments.

Chemorheological Treatment of Crosslinked EPDM

1971 ◽  
Vol 44 (5) ◽  
pp. 1334-1340 ◽  
Author(s):  
Kenkichi Murakami ◽  
Saburo Tamura
Keyword(s):  
Stress Relaxation ◽  
Main Chain ◽  
Relaxation Curve ◽  
Oxidative Cleavage ◽  
The Other ◽  
Carbon Bond ◽  
Carbon Carbon Bond ◽  
Straight Line ◽  
Interchange Reaction ◽  
Straight Lines

Abstract Stress relaxation mechanisms were investigated on three types of (EPDM) ethylene-propylene terpolymers in air at 109° C. These polymers differ only by the structure of the crosslinkage in which there is a carbon-carbon bond, a polysulfide linkage −Sx⁁− or a monosulfide linkage (—S—). All the stress relaxation of peroxide-cured EPDM polymer was not due to the oxygen-induced cleavage of the main chain but to a physical flow. In the case of sulfur-cured EPDM polymer, the relaxation curve is divided into three straight lines when the procedure X is used and log f(t)/f(0( is plotted linearly with time. It was concluded that this graph was in agreement with an interchange reaction of the polysulfide linkage by an oxidative cleavage of the monosulfide linkage. On the other hand, a TT-cured EPDM polymer gave a plot with a straight line. This stress relaxation could be explained by an oxidative cleavage of the monosulfide linkage.

scholarly journals Limit Cycles of Planar Piecewise Differential Systems with Linear Hamiltonian Saddles

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1128
Author(s):  
Jaume Llibre ◽  
Claudia Valls
Keyword(s):  
Limit Cycle ◽  
Differential System ◽  
Limit Cycles ◽  
The Other ◽  
Differential Systems ◽  
Other Hand ◽  
Straight Line ◽  
Line Of Discontinuity ◽  
Straight Lines

We provide the maximum number of limit cycles for continuous and discontinuous planar piecewise differential systems formed by linear Hamiltonian saddles and separated either by one or two parallel straight lines. We show that when these piecewise differential systems are either continuous or discontinuous and are separated by one straight line, or are continuous and are separated by two parallel straight lines, they do not have limit cycles. On the other hand, when these systems are discontinuous and separated by two parallel straight lines, we prove that the maximum number of limit cycles that they can have is one and that this maximum is reached by providing an example of such a system with one limit cycle. When the line of discontinuity of the piecewise differential system is formed by one straight line, the symmetry of the problem allows to take this straight line without loss of generality as the line x=0. Similarly, when the line of discontinuity of the piecewise differential system is formed by two parallel straight lines due to the symmetry of the problem, we can assume without loss of generality that these two straight lines are x=±1.

Automatic Tread Gaging Machine

1979 ◽  
Vol 7 (1) ◽  
pp. 31-39
Author(s):  
G. S. Ludwig ◽  
F. C. Brenner
Keyword(s):  
Experimental Tests ◽  
Automatic Machine ◽  
Wear Rates ◽  
Handling Device ◽  
Straight Line ◽  
Component Systems ◽  
Before And After ◽  
Good Repeatability ◽  
The Difference ◽  
Straight Lines

Abstract An automatic tread gaging machine has been developed. It consists of three component systems: (1) a laser gaging head, (2) a tire handling device, and (3) a computer that controls the movement of the tire handling machine, processes the data, and computes the least-squares straight line from which a wear rate may be estimated. Experimental tests show that the machine has good repeatability. In comparisons with measurements obtained by a hand gage, the automatic machine gives smaller average groove depths. The difference before and after a period of wear for both methods of measurement are the same. Wear rates estimated from the slopes of straight lines fitted to both sets of data are not significantly different.

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