scholarly journals Geometrical revelation of correlated characteristics of the ray and axis order of the Plücker coordinates in line geometry

2020 ◽  
Vol 153 ◽  
pp. 103983
Author(s):  
Jian S Dai ◽  
Jie Sun
Keyword(s):  
Line Geometry ◽  
Plücker Coordinates

XVIII.—The Invariant Theory of Forms in Six Variables relating to the Line Complex

1927 ◽  
Vol 46 ◽  
pp. 210-222 ◽  
Author(s):  
H. W. Turnbull
Keyword(s):  
Invariant Theory ◽  
Line Geometry ◽  
Five Dimensions ◽  
Ordinary Space ◽  
Homogeneous Coordinates ◽  
Plücker Coordinates ◽  
Algebraic Theories ◽  
Straight Line ◽  
Image Position ◽  
Theory Of Forms

It is well known that the Plücker coordinates of a straight line in ordinary space satisfy a quadratic identitywhich may also be considered as the equation of a point-quadric in five dimensions, if the six coordinates Pij are treated as six homogeneous coordinates of a point. Projective properties of line geometry may therefore be treated as projective properties of point geometry in five dimensions. This suggests that certain algebraic theories of quaternary forms (corresponding to the geometry of ordinary space) can best be treated as algebraic theories of senary forms: that is, forms in six homogeneous variables.

Geometric tolerances: A new application for line geometry and screws

2002 ◽  
Vol 216 (1) ◽  
pp. 95-103 ◽  
Author(s):  
J K Davidson ◽  
J J Shah
Keyword(s):  
Mathematical Model ◽  
Cylindrical Surface ◽  
Line Geometry ◽  
Tolerance Zone ◽  
New Model ◽  
Plücker Coordinates ◽  
Geometric Tolerances ◽  
Size Tolerance

A new mathematical model is introduced for the tolerances of cylindrical surfaces. The model is compatible with the ISO/ANSI/ASME standard for geometric tolerances. Central to the new model is a Tolerance-Map®†, a hypothetical volume of points that corresponds to all possible locations and variations of a segment of a line (the axis) that can arise from tolerances on size, location and orientation of the cylindrical surface. Each axis in a tolerance zone will be represented with the six Plücker coordinates. Cylindrical surfaces in a tolerance zone for the same hole can then be treated by attaching a size tolerance to each of the lines, thereby forming a screw. Relationships for the content of line solids for a tolerance zone are developed to correspond to the variations of locations. These are then used to obtain a measure for the increment in cost when a more refined tolerance is specified. This model is one part of a bilevel model that is under development for geometric tolerances.

scholarly journals Algebras of quasi-Plücker coordinates are Koszul

2018 ◽  
Vol 222 (9) ◽  
pp. 2810-2822
Author(s):  
Robert Laugwitz ◽  
Vladimir Retakh
Keyword(s):  
Plücker Coordinates

scholarly journals Robust Visibility Surface Determination in Object Space via Plücker Coordinates

2021 ◽  
Vol 7 (6) ◽  
pp. 96
Author(s):  
Alessandro Rossi ◽  
Marco Barbiero ◽  
Paolo Scremin ◽  
Ruggero Carli
Keyword(s):  
Finite Number ◽  
State Of The Art ◽  
Lossless Compression ◽  
3D Models ◽  
Optimal Result ◽  
Object Space ◽  
Plücker Coordinates ◽  
Determination Methods

Industrial 3D models are usually characterized by a large number of hidden faces and it is very important to simplify them. Visible-surface determination methods provide one of the most common solutions to the visibility problem. This study presents a robust technique to address the global visibility problem in object space that guarantees theoretical convergence to the optimal result. More specifically, we propose a strategy that, in a finite number of steps, determines if each face of the mesh is globally visible or not. The proposed method is based on the use of Plücker coordinates that allows it to provide an efficient way to determine the intersection between a ray and a triangle. This algorithm does not require pre-calculations such as estimating the normal at each face: this implies the resilience to normals orientation. We compared the performance of the proposed algorithm against a state-of-the-art technique. Results showed that our approach is more robust in terms of convergence to the maximum lossless compression.

The Effect of Line Geometry on Void Growth in Thin, Narrow Aluminum Lines

1991 ◽  
Vol 226 ◽  
Author(s):  
P. Borgesen ◽  
J. K. Lee ◽  
M. A. Korhonen ◽  
C.-Y. Li
Keyword(s):  
Stress Relaxation ◽  
Void Growth ◽  
Room Temperature ◽  
Line Geometry

AbstractThe stress induced growth of individual voids in passivated Al-lines at room temperature was monitored in-situ without removing the passivation. The kinetics was strongly influenced by variations in line gec.etry, even over distances of many Am, indicating variations in the stress relaxation as well.

scholarly journals Characterizations of the $G_2(4)$ and $L_3(4)$ Near Octagons

10.37236/8476 ◽  
2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Bart De Bruyn
Keyword(s):  
Generalized Quadrangle ◽  
Line Geometry ◽  
Near Polygon ◽  
Generalized Polygon ◽  
Generalized Hexagon ◽  
Split Cayley Hexagon ◽  
Point Line ◽  
Line Spread

A triple $(\mathcal{S},S,\mathcal{Q})$ consisting of a near polygon $\mathcal{S}$, a line spread $S$ of $\mathcal{S}$ and a set $\mathcal{Q}$ of quads of $\mathcal{S}$ is called a polygonal triple if certain nice properties are satisfied, among which there is the requirement that the point-line geometry $\mathcal{S}'$ formed by the lines of $S$ and the quads of $\mathcal{Q}$ is itself also a near polygon. This paper addresses the problem of classifying all near polygons $\mathcal{S}$ that admit a polygonal triple $(\mathcal{S},S,\mathcal{Q})$ for which a given generalized polygon $\mathcal{S}'$ is the associated near polygon. We obtain several nonexistence results and show that the $G_2(4)$ and $L_3(4)$ near octagons are the unique near octagons that admit polygonal triples whose quads are isomorphic to the generalized quadrangle $W(2)$ and whose associated near polygons are respectively isomorphic to the dual split Cayley hexagon $H^D(4)$ and the unique generalized hexagon of order $(4,1)$.

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