geometryblade

Geometryblade is a term used in geometric algebra to denote an algebraic element called a blade, which represents an oriented linear subspace of a vector space. A geometryblade of grade k is the exterior, or wedge, product of k linearly independent vectors: b = u1 wedge u2 wedge ... wedge uk. The subspace spanned by these vectors is encoded by the blade; its orientation and magnitude correspond to the orientation and k-dimensional content of the subspace. The magnitude equals the k-volume of the parallelepiped formed by the vectors, and a unit blade has unit content. In an n-dimensional space, 1-blades are lines, 2-blades are planes, and n-blades are pseudoscalars representing the whole space.

Blades are combined through the geometric product, which factors into a symmetric inner product and an antisymmetric outer (wedge) product. The outer product constructs new blades by wedging vectors; blades can be added and simplified via projection onto basis blades. In practice, geometryblade data structures are used in algorithms for subspace fitting, intersection, and rotation representation via rotors.

Applications include computer graphics for representing planes and volumes, robotics for pose and constraint computation, physics for oriented subspaces in field theories, and computational geometry for collision detection and subspace intersections. The concept is a central element of geometric algebra, developed from Clifford algebra in the 19th century and popularized in modern times by researchers such as David Hestenes. Geometryblade is thus a convenient, if informal, name for blades—the basic units that encode oriented subspaces in geometric algebra.