Bivector - Projective Geometry

Projective Geometry

Geometric algebra can be applied to projective geometry in a straightforward way. The geometric algebra used is Cℓ_n(ℝ), n ≥ 3, the algebra of the real vector space ℝn. This is used to describe objects in the real projective space ℝℙn - 1. The non-zero vectors in Cℓ_n(ℝ) or ℝn are associated with points in the projective space so vectors that differ only by a scale factor, so their exterior product is zero, map to the same point. Non-zero simple bivectors in Λ2ℝn represent lines in ℝℙn - 1, with bivectors differing only by a (positive or negative) scale factor representing the same line.

A description of the projective geometry can be constructed in the geometric algebra using basic operations. For example given two distinct points in ℝℙn - 1 represented by vectors a and b the line between them is given by a ∧ b (or b ∧ a). Two lines intersect in a point if A ∧ B = 0 for their bivectors A and B. This point is given by the vector

The operation "⋁" is the meet, which can be defined as above in terms of the join, J = A ∧ B for non-zero A ∧ B. Using these operations projective geometry can be formulated in terms of geometric algebra. For example given a third (non-zero) bivector C the point p lies on the line given by C if and only if

So the condition for the lines given by A, B and C to be collinear is

which in Cℓ₃(ℝ) and ℝℙ2 simplifies to

where the angle brackets denote the scalar part of the geometric product. In the same way all projective space operations can be written in terms of geometric algebra, with bivectors representing general lines in projective space, so the whole geometry can be developed using geometric algebra.