scholarly journals On powers of Plücker coordinates and representability of arithmetic matroids

2020 ◽  
Vol 112 ◽  
pp. 101911 ◽  
Author(s):  
Matthias Lenz
Keyword(s):  
Plücker Coordinates

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.

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.

scholarly journals INTERSECTION COMPUTATION IN PROJECTIVE SPACE USING HOMOGENEOUS COORDINATES

2008 ◽  
Vol 08 (04) ◽  
pp. 615-628 ◽  
Author(s):  
VACLAV SKALA
Keyword(s):  
Projective Space ◽  
Space Representation ◽  
Linear System Of Equations ◽  
Homogeneous Coordinates ◽  
Plücker Coordinates ◽  
Parametric Line ◽  
Cad Cam ◽  
Principle Of Duality ◽  
New Formulation ◽  
Cam Systems

There are many algorithms based on computation of intersection of lines, planes etc. Those algorithms are based on representation in the Euclidean space. Sometimes, very complex mathematical notations are used to express simple mathematical solutions. This paper presents solutions of some selected problems that can be easily solved by the projective space representation. Sometimes, if the principle of duality is used, quite surprising solutions can be found and new useful theorems can be generated as well. It will be shown that it is not necessary to solve linear system of equations to find the intersection of two lines in the case of E2 or the intersection of three planes in the case of E3. Plücker coordinates and principle of duality are used to derive an equation of a parametric line in E3 as an intersection of two planes. This new formulation avoids division operations and increases the robustness of computation. The presented approach for intersection computation is well suited especially for applications where robustness is required, e.g. large GIS/CAD/CAM systems etc.

A Geometrical treatment of the Correspondence between Lines in Threefold Space and Points of a Quadric in Fivefold space

1925 ◽  
Vol 22 (5) ◽  
pp. 694-699 ◽  
Author(s):  
H. W. Turnbull
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Quadratic Relation ◽  
Four Dimensions ◽  
Homogeneous Coordinates ◽  
Plücker Coordinates ◽  
Straight Line ◽  
Set Up ◽  
Image Position ◽  
Three Dimensional Space

§ 1. The six Plücker coordinates of a straight line in three dimensional space satisfy an identical quadratic relationwhich immediately shows that a one-one correspondence may be set up between lines in three dimensional space, λ, and points on a quadric manifold of four dimensions in five dimensional space, S5. For these six numbers pij may be considered to be six homogeneous coordinates of such a point.

scholarly journals Using Plücker Coordinates for Pumping Speed Evaluation of Molecular Pump in the DSMC Method

2001 ◽  
Vol 7 (1) ◽  
pp. 11-20 ◽  
Author(s):  
Fong-Zhi Chen ◽  
Ming-June Tsai ◽  
Yu-Wen Chang ◽  
Rong-Yuan Jou ◽  
Hong-Ping Cheng
Keyword(s):  
Three Dimensional ◽  
Direct Simulation Monte Carlo ◽  
Flow Simulation ◽  
Vacuum Pump ◽  
Ruled Surface ◽  
Molecular Flow ◽  
Dsmc Method ◽  
Plücker Coordinates ◽  
Straight Line ◽  
Speed Performance

In this study, the Plücker coordinates representation is used to formulate the ruled surface and the molecular path for pumping speed performance evaluation of a molecular vacuum pump. The ruled surface represented by the Pliicker coordinates is used to develop a criterion for when gas molecules hit the pump surface wall. The criterion is applied to analyze the flow rate of a new developed vacuum pump in transition regimes by using the DSMC (Direct Simulation Monte Carlo) method. When a molecule flies in a neutral electrical field its path is a straight line. If the molecular path and the generators of a ruled surface are both represented by the Pliicker coordinates, the position of the molecular hit on the wall can be verified by the reciprocal condition of the lines. The Plücker coordinates representation is quite convenient in the DSMC method for this three-dimensional molecular flow simulation.

Sign in / Sign up

Export Citation Format

Share Document