scholarly journals The Number of Lines that may lie upon a Surface of given Order

1930 ◽  
Vol 26 ◽  
pp. xi-xii
Author(s):  
H. W. Richmond
Keyword(s):  
Point Contact ◽  
Ruled Surface ◽  
Correct Number ◽  
Upper Limit ◽  
Straight Line ◽  
Straight Lines

The greatest number of straight lines that can lie upon a surface of order n (not being a ruled surface) is unknown, except if n is three. Salmon and Clebsch have shown that the points of contact of lines which have a four-point contact with the surface lie upon a locus of order n (11n – 24), the intersection of the surface of order n with another of order 11n – 24. Since a straight line lying wholly on the former surface must form a part of this locus, the number n (11n – 24) is an upper limit to the number of lines; if n is three, this gives 27, the correct number. But for values of n > 3, it is improbable that this limit1 can be reached.

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.

scholarly journals An Improved Path-Generating Regulator for Two-Wheeled Robots to Track the Circle/Arc Passage

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Jun Dai ◽  
Naohiko Hanajima ◽  
Toshiharu Kazama ◽  
Akihiko Takashima
Keyword(s):  
Control Method ◽  
Convergence Property ◽  
Tangential Direction ◽  
Navigation Problem ◽  
Wheeled Robots ◽  
Robot Trajectory ◽  
Look Ahead ◽  
Straight Line ◽  
The Family ◽  
Straight Lines

The improved path-generating regulator (PGR) is proposed to path track the circle/arc passage for two-wheeled robots. The PGR, which is a control method for robots so as to orient its heading toward the tangential direction of one of the curves belonging to the family of path functions, is applied to navigation problem originally. Driving environments for robots are usually roads, streets, paths, passages, and ridges. These tracks can be seen as they consist of straight lines and arcs. In the case of small interval, arc can be regarded as straight line approximately; therefore we extended the PGR to drive the robot move along circle/arc passage based on the theory that PGR to track the straight passage. In addition, the adjustable look-ahead method is proposed to improve the robot trajectory convergence property to the target circle/arc. The effectiveness is proved through MATLAB simulations on both the comparisons with the PGR and the improved PGR with adjustable look-ahead method. The results of numerical simulations show that the adjustable look-ahead method has better convergence property and stronger capacity of resisting disturbance.

A new geometry definition and generation method for a face gear meshed with a spur pinion

2021 ◽  
pp. 095440542110609
Author(s):  
Xian-Long Peng
Keyword(s):  
Ruled Surface ◽  
Tooth Surface ◽  
Face Gear ◽  
Absolute Deviation ◽  
Numerical Examples ◽  
Gear Teeth ◽  
The Face ◽  
Geometry Feature ◽  
Tooth Flank ◽  
Straight Lines

The conventional tooth surface of a face gear is difficult to manufacture, and the cutter for the face gear cutting is not uniform even though the parameters of the pinion mating with the face gear slightly change. Based on the analysis of the geometry features of the tooth surface, a new developable ruled surface is defined as the tooth flank of the face gear, for which the most important geometry feature is that the flank could be represented by a family of straight lines, hence it could be generated by a straight-edged cutter. The mathematical models of the new ruled tooth surface, the cutter and the generation method are presented, the deviation between the ruled surface and the conventional surface, the correction of the ruled surface to reduce the deviation are investigated through numerical examples. The manufacturing process is simulated by VERICUT software, and the results demonstrate that even when the principle deviation is added to the machined deviation, the absolute deviation is on the micro-scale. The meshing and contact simulation shows that the new surface could obtain good meshing performance when the number of face gear teeth is greater than three times the number of pinion teeth. This research provides a new method for manufacturing face gears.

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.

A theorem of M. L. Urquhart's and some consequences

2007 ◽  
Vol 91 (520) ◽  
pp. 39-50
Author(s):  
R. T. Leslie
Keyword(s):  
Euclidean Geometry ◽  
Straight Line ◽  
Elementary Theorem ◽  
Straight Lines

In an obituary of M. L. Urquhart in [1], David Elliott quotes him as claiming that Urquhart's theorem (below) is the most elementary theorem of Euclidean Geometry ‘since it involves only the concepts of straight line and distance’.Urquhart's theoremLet AC and AE be two straight lines.Let B be a point on AC, D a point on AE, and suppose that BE and CD intersect at F.If AB + BF = AD + DF then AC + CF = AE + EF. (1)

scholarly journals II. Second memoir on plane stigmatics

1867 ◽  
Vol 15 ◽  
pp. 192-203
Keyword(s):  
General Expression ◽  
Fundamental Lemma ◽  
Double Points ◽  
Straight Line ◽  
Straight Lines ◽  
Stigmatic Point ◽  
Coordinate Geometry

Let there be two groups of points upon a plane, termed, for distinction, indices and stigmata respectively, bearing such relations to each other that any one index determines the position of n stigmata, and any one stigma determines the position of m indices. The theory of these relations between indices and stigmata constitutes plane stigmatics . Each related pair of index X and stigma Y constitutes a stigmatic point , henceforth written “the s. point ( xy )." The straight lines joining any index with each of its corresponding stigmata are termed ordinates . If, when the index moves upon a straight line, the ordinate remains parallel to some other straight line, the relation between index and stigma is that expressed by the relation between abscissa and ordinate in the coordinate geometry of Descartes. When only one index corresponds to one stigma and conversely, and both indices and stigmata lie always on one and the same straight line, or the indices upon one and the stigmata upon another, the relations between indices and stigmata are those between homologous points in the homographic geometry of Chasles. The general expression of the stigmatic relation is obtained by a generalization of Chasles’s fundamental lemma in his theory of characteristics ( Comptes Rendus , June 27, 1864, vol. lviii. p. 1175), clinants being substituted for scalars. It results that in certain forms of the law of coordination , which “ coordinates ” the stigmata with the indices, there may be solitary indices which have no corresponding stigmata, and solitary stigmata which have no corresponding indices, and also double points in which the index coincides with its stigma (76). The particular case in which one index corresponds to one stigma and conversely, and no solitary index or stigma occurs, is termed a stigmatic line (henceforth written “s. line”), because the Cartesian case is that of a Cartesian straight line in ordinary coordinate geometry, but in the general s. line the figures described by index and stigma may be any directly similar plane figures (77). The investigation of this particular case occupies almost the whole of the Introductory Memoir . When one index corresponds to one stigma and conversely, but there is one solitary index and one solitary stigma, we have s. homography , provided the solitary index is distinct from the solitary stigma (79), and s. involution when the solitary index coincides with the solitary stigma (78), so called because they generalize the relations treated of under these names by Chasles.

scholarly journals A general vector analysis, with applications to electrodynamical theory

1925 ◽  
Vol 108 (747) ◽  
pp. 418-455 ◽  
Keyword(s):  
Three Dimensions ◽  
Vector Analysis ◽  
Cartesian Coordinates ◽  
Straight Line ◽  
Straight Lines ◽  
General Vector

The vector analyses in use up to the present, as a rule, are concerned with quantities which are represented by straight lines, and the space to which they are applicable is Euclidean in its properties. The straight line, AB, in space of three dimensions, is represented by a vector a, and if B has Cartesian coordinates ( x, y, z ) with respect to A, we write: a = i x + j y + k z , where i, j, k, are fundamental vectors. An account will be given of a vector analysis in which a vector is represented by δa' = Σ n i n δx n . The vector is of infinitesimal length and represents a component measured in any system of co-ordinates.

Mathematical Model for Face-Hobbed Straight Bevel Gears

2012 ◽  
Vol 134 (9) ◽  
Author(s):  
Yi-Pei Shih
Keyword(s):  
Mathematical Model ◽  
Bevel Gear ◽  
Tooth Contact Analysis ◽  
Rolling Circle ◽  
High Productivity ◽  
Bevel Gears ◽  
Straight Line ◽  
Base Circle ◽  
Straight Line Mechanism ◽  
Straight Lines

Face hobbing, a continuous indexing and double-flank cutting process, has become the leading method for manufacturing spiral bevel gears and hypoid gears because of its ability to support high productivity and precision. The method is unsuitable for cutting straight bevel gears, however, because it generates extended epicycloidal flanks. Instead, this paper proposes a method for fabricating straight bevel gears using a virtual hypocycloidal straight-line mechanism in which setting the radius of the rolling circle to equal half the radius of the base circle yields straight lines. This property can then be exploited to cut straight flanks on bevel gears. The mathematical model of a straight bevel gear is developed based on a universal face-hobbing bevel gear generator comprising three parts: a cutter head, an imaginary generating gear, and the motion of the imaginary generating gear relative to the work gear. The proposed model is validated numerically using the generation of face-hobbed straight bevel gears without cutter tilt. The contact conditions of the designed gear pairs are confirmed using the ease-off topographic method and tooth contact analysis (TCA), whose results can then be used as a foundation for further flank modification.

Investigating the Gap Between Actual and Perceived Distance from a Nuclear Power Plant: A Case Study in Japan

2015 ◽  
Vol 10 (4) ◽  
pp. 627-634 ◽  
Author(s):  
Takaaki Kato ◽  
◽  
Shogo Takahara ◽  
Toshimitsu Homma ◽  
Keyword(s):  
Power Plant ◽  
Nuclear Power Plant ◽  
Nuclear Power ◽  
Tohoku Earthquake ◽  
Personal Attributes ◽  
Fukushima Accident ◽  
Straight Line ◽  
Straight Line Distance ◽  
Accident Response ◽  
Straight Lines

This study investigates factors in gaps between perceived and actual straight-line distance to Japan’s Kashiwazaki-Kariwa nuclear power plant (KKNPP). The distance to areas in the official accident response plan is defined using straight lines from the NPP, making it important to determine whether area residents understand these distances correctly. Adults living in the two municipalities cohosting the NPP were surveyed randomly in 2005, 2010 and 2011. In this study, we consider three groups of factors — geographical features, personal attributes, and experience in events highlighting nuclear safety. The Niigata-ken Chuetsu-oki earthquake hit the NPP between the first and second of these three surveys, and the Tohoku earthquake and the March 2011 Fukushima nuclear accident occurred between the second and the third surveys. Before the Fukushima accident, overestimations of straight-line distance were common among respondents, and geographical features such as lack of NPP visibility aggravated bias between actual and perceived distance. After the Fukushima accident, underestimation of the distance became common and personal attributes became more influential as the factor of the perceived-actual distance gap.

Three-dimensional extension of Bresenham's algorithm and its application in straight-line interpolation

2002 ◽  
Vol 216 (3) ◽  
pp. 459-463 ◽  
Author(s):  
X-W Liu ◽  
K Cheng
Keyword(s):  
Line Segment ◽  
Three Dimensional ◽  
Straight Line Segment ◽  
Numerical Control ◽  
Interpolation Algorithm ◽  
Straight Line ◽  
Actual Error ◽  
Potential Applications ◽  
Sample Position ◽  
Straight Lines

Conventional straight-line generating algorithms, such as the digital differential analyser (DDA), Bresenham's algorithm and the mid-point algorithm, are suitable only for planer straight lines on the coordinate planes, of which Bresenham's algorithm is the most efficient. In this paper, the authors have extended Bresenham's algorithm to spatial straight lines. Given a spatial straight-line segment with two end-points, the authors have applied Bresenham's algorithm to the projections of the line segment on two of the three coordinate planes, which is determined by the largest of the coordinate lengths of the line segment, thereby obtaining a three-dimensional extension of the algorithm. In a case study, the authors calculated the distance between each sample position and the given line segment. The result reveals that the actual error at each sample position is smaller than the maximum theoretical error, and the performance of the three-dimensional extension of Bresenham's algorithm is as good as that of Bresenham's original planer algorithm. One of its potential applications is the three-dimensional step straight-line interpolation used in computer numerical control (CNC) systems of machine tools and rapid prototyping machines. Application of the algorithm is contrasted with that of the traditional DDA step straight-line interpolation algorithm. The result confirms that the three-dimensional extension of Bresenham's algorithm is much better than the DDA straight-line interpolation algorithm.

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