Parallelshifted

Parallelshifted is an adjective used to describe a relationship between two geometric objects, signals, or data sets that differ only by a constant translation. The term emphasizes that the two entities are parallel in orientation and offset by a fixed vector, with no rotation, scaling, or distortion involved. The concept is common in geometry, computer vision, and signal processing as a basic form of equivalence under translation.

Formal definition: Let A be a subset of Euclidean space R^n. A set B is parallelshifted from A if there exists a displacement vector v in R^n such that B = A + v, where A + v = { x + v | x in A }. In this case B is a translated copy of A; corresponding points differ by the same vector v.

Properties: translations preserve length and angle, so parallelshifted objects are congruent and share the same shape. No rotation or scaling occurs. If A has symmetries, B inherits them under the same translation. When A is finite, B has the same cardinality; for infinite A, both are unbounded similarly.

Applications: in image processing, shifting an image by a fixed pixel amount yields a parallelshifted image used in alignment and motion compensation. In computer graphics, parallelshifted copies create translating patterns and tessellations. In data analysis, translating a time series corresponds to a parallelshifted sequence when interpreted in a phase space.

See also: translation, congruence, isometry, affine transformation, pattern registration.