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Graßmann-Plücker Relations / Grassmann-Plücker Relations.

I found the Graßmann-Plücker Relations / Grassmann-Plücker Relations / Grassmann-Pluecker Relations surprising when I read about them in Jürgen Richter-Gebert’s book (see below for the reference).

The relations are: for arbitrary five points (A-E) in the projective plane, using homogeneous coordinates, the following relationship in terms of determinants of combinations of the points is:

\[\det(ABC)\det(ADE) - \det(ABD)\det(ACE) + \det(ABE)\det(ACD) = 0\label{eq:relations}\]

Related: three points in the projective plane, using homogeneous coordinates, are collinear if their determinant is zero. In other words, if for points A-C,

\[\det(ABC) = 0\label{eq:collinear}\]

then A, B, and C are collinear.

Where I learned this:

Jürgen Richter-Gebert’s Perspectives on Projective Geometry: A Guided Tour Through Real and Complex Geometry 2011, ISBN 9783642172854, page 15 in the softcover version.

Other reference:

Wikipedia: in German, which does not include this case but does discuss the relations in \(\mathbb{R}^2\) and \(\mathbb{R}^3\).

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