parallelem

Parallelem is a term used in some mathematical discussions to denote a class of geometric mappings that preserve the parallel relation among lines. It is not a standard object in Euclidean geometry, and its exact definition varies across sources. In one common interpretation, a parallelem P is a map that assigns to each line l a line P(l) such that whenever l1 and l2 are parallel, P(l1) and P(l2) are also parallel. If P is defined on all lines of the plane and is bijective, this condition characterizes affine transformations, which preserve straightness, incidence, and parallelism, though they may alter distances and angles.

Properties: Parallelems preserve the parallelism structure of a figure but generally do not preserve metric properties. They can rearrange shapes, alter size or orientation, and include familiar transformations such as translations, rotations, reflections, and shears. However, more exotic P that are not affine can also satisfy the parallelism-preserving condition on a given subset of lines.

Examples: A translation by a fixed vector is a parallelem; a rotation about a point is a parallelem; a shear is a parallelem as it preserves parallelism. Projective transformations typically do not qualify, as they can map parallel lines to intersecting lines.

Applications: The concept is used mainly in theoretical discussions about parallelism and in graphics or tiling where preserving directionality is important, without requiring preservation of distances or angles.

See also: affine transformation, parallelism, geometry.