Space-Antispace Transform Correspondence in Projective Geometric Algebra

By Eric Lengyel
May 20, 2022

The concept of duality can be understood geometrically in an n-dimensional projective setting by considering both the subspace that an object occupies and the complementary subspace that the object concurrently does not occupy. The dimensionalities of these two components always sum to n, and they represent the space and antispace associated with the object. (Antispace is also known as negative space or counterspace.) The example shown in Figure 1 demonstrates the duality between homogeneous points and lines in a three-dimensional projective space. The triplet of coordinates $$(p_x, p_y, p_z)$$ can be interpreted as a vector pointing from the origin toward a specific location on the projection plane $$z = 1$$. This vector corresponds to the one-dimensional space of the point that it represents. The dual of a point materializes when we consider all of the directions of space that are orthogonal to the single direction $$(p_x, p_y, p_z)$$. As illustrated by the figure, these directions span an $$(n - 1)$$-dimensional subspace that intersects the projection plane at a line when $$n = 3$$. In this way, the coordinates $$(p_x, p_y, p_z)$$ can be interpreted as both a point and a line, and they are duals of each other.