Plücker Coordinates and Lines in Space
 Michael Joswig,
 Thorsten Theobald
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Abstract
Lines, especially in ℝ^{3}, play a significant role in the modeling of geometric problems in computer graphics and machine vision. For example, a point b is visible from a point a if the line segment from a to b does not intersect another object of the scene.
Although a line is an affine subspace of the original space, the conditions for the intersection of lines are intrinsically nonlinear. To illustrate this, we briefly study the problem of determining the set of lines that intersect four given lines ℓ _{1},…,ℓ _{4}⊆ℝ^{3}. (These intersection lines are called transversals.) If this problem were a linear or an affine linear problem, the number of solutions would always be 0, 1, or infinite. Actually, we will see below that for lines in general position, there exist exactly two (in general complex) lines with this property.
For many problems which involve the configurations of lines, it is useful to identify the lines, in a nonlinear manner, with points in a higher dimensional space. This then leads to linear intersection conditions. The socalled Plücker coordinates achieve this.
Before we begin to study the line configurations, we will first define Plücker coordinates for arbitrary subspaces of a projective space. We study the linear intersection conditions mentioned above in this general setting before we return to the threedimensional case at the end of this chapter.
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 Title
 Plücker Coordinates and Lines in Space
 Book Title
 Polyhedral and Algebraic Methods in Computational Geometry
 Pages
 pp 193207
 Copyright
 2013
 DOI
 10.1007/9781447148173_12
 Print ISBN
 9781447148166
 Online ISBN
 9781447148173
 Series Title
 Universitext
 Series ISSN
 01725939
 Publisher
 Springer London
 Copyright Holder
 SpringerVerlag London
 Additional Links
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 Authors

 Michael Joswig ^{(1)}
 Thorsten Theobald ^{(2)}
 Author Affiliations

 1. Fachbereich Mathematik, Technische Universität Darmstadt, Darmstadt, Germany
 2. Institut für Mathematik, FB 12, Johann Wolfgang GoetheUniversität, Frankfurt am Main, Germany
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