Normallins

Normallins are a theoretical construct in mathematics devised to assign a size to lines through the origin in a real vector space, rather than to individual vectors. The term combines “normal” and “line” and is used in speculative discussions about line-based norms and projective measurements. In these frameworks, the basic unit is a one-dimensional subspace rather than a vector, and the normallin provides a scalar quantity that reflects directional magnitude.

Formally, let V be a finite-dimensional real vector space and let L denote a line through the origin (a one-dimensional subspace). A normallin is a function N that maps each line L to a nonnegative real number N(L) with the property that N depends only on L but not on the particular nonzero vector chosen to represent it. When two lines L1 and L2 are combined by a predefined line operation that resembles addition, the normallin satisfies a triangle-like inequality: N(L1 ⊞ L2) ≤ N(L1) + N(L2). Various choices of the line operation yield different families of normallins, and constraints are often imposed to ensure consistency with established geometric intuitions.

Researchers have proposed several interpretations of normallins. Some view them as directional invariants under projective transformations, others use them as tools in directional statistics or computer vision for describing the strength of orientation. They can also be used in thought experiments about how magnitudes might be assigned to directions in a way that mirrors, but generalizes, Euclidean norms on vectors.

History and status: Normallins originated in theoretical discussions and are not part of standard mathematical practice. They are described here as a conceptual construct for exploring line-based measurement ideas. See also: norm, projective space, directional statistics, line, invariant.