Plücker Plane Coordinates
Encode a plane with six minors and test whether a perturbed bivector still represents a plane.
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Plücker Plane Coordinates
Plücker coordinates encode a subspace of a vector space using alternating forms. Specifically, for a 2-dimensional plane in a 4-dimensional space, we can represent the plane using six minors.
Worked Example
Consider two vectors in a 4-dimensional space: u = (1, 2, 0, 1) and v = (0, 1, 2, 1).
The six 2x2 minors are formed by choosing pairs of columns:
- p12 = u1v2 - u2v1 = 1(1) - 2(0) = 1
- p13 = u1v3 - u3v1 = 1(2) - 0(0) = 2
- p14 = u1v4 - u4v1 = 1(1) - 1(0) = 1
- p23 = u2v3 - u3v2 = 2(2) - 0(1) = 4
- p24 = u2v4 - u4v2 = 2(1) - 1(1) = 1
- p34 = u3v4 - u4v3 = 0(1) - 1(2) = -2
A bivector represents a decomposable plane if and only if it satisfies the Plücker relation:
p12*p34 - p13*p24 + p14*p23 = (1)(-2) - (2)(1) + (1)(4) = -2 - 2 + 4 = 0
The norm of this bivector is the square root of the sum of the squares of the minors, which represents the area of the parallelogram formed by u and v:
Area = √(12 + 22 + 12 + 42 + 12 + (-2)2) = √(1 + 4 + 1 + 16 + 1 + 4) = √27 ≈ 5.196152
If we apply a scale factor of 1, the relation holds at 0, indicating decomposability.
Experiment
You can test the Plücker relation and bivector norms using the interactive tool at https://riano.app/module/plucker-plane-coordinates.
- Set the Rows control to
Genericand Row scale to1. Observe that the Plücker relation is 0, and the status is Decomposable. - Change Minor perturbation to
1. This modifies p34, breaking the relation. The status becomes Nondecomposable, meaning the new six values no longer represent a simple 2-plane. - Change Rows to
Dependent. Observe that the relation is 0, but the norm is also 0, yielding aZero bivectorstatus.
Limitations
The implementation only considers planes in a 4-dimensional vector space (Grassmannian Gr(2, 4)), limiting the visualization to six coordinates. Numerical inaccuracies might result in algebraically zero floating residuals being slightly off, so a tolerance of 1e-11 is used for determining decomposability. If all coordinates are exactly zero, it represents no 2-plane (a zero bivector). Additionally, the module specifically isolates a deliberate perturbation on p34 rather than an arbitrary bivector transformation.
References
For more information on Plücker coordinates and the Grassmannian relation, refer to Wolfram MathWorld: Plücker Coordinates.
Experience it interactively
Adjust parameters, observe in real time, and build deep intuition with Riano’s interactive Plücker Plane Coordinates module.
Try Plücker Plane Coordinates on Riano →