Plücker Coordinates and Lines in Space

Abstract

Lines, especially in ℝ³, 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 non-linear. To illustrate this, we briefly study the problem of determining the set of lines that intersect four given lines ℓ ₁,…,ℓ ₄⊆ℝ³. (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 non-linear manner, with points in a higher dimensional space. This then leads to linear intersection conditions. The so-called 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 three-dimensional case at the end of this chapter.