Pisteparit

Pisteparit, literally “point pairs” in Finnish, refer to pairs formed from a finite set of points in a Euclidean space or, more generally, in a metric space. They can be considered as ordered pairs (p, q) or as unordered two-element subsets {p, q}, depending on the mathematical context.

In formal terms, if S is a set of points, the set of unordered pisteparit consists of all two-element subsets {p, q} with p and q in S and p ≠ q. The set of ordered pisteparit consists of all pairs (p, q) with p and q in S and p ≠ q. For a set of n points, there are n(n−1)/2 unordered pairs and n(n−1) ordered pairs.

Geometrically, each pistepari defines a segment connecting the two points, and it determines related quantities such as the distance d(p, q), the midpoint ((p+q)/2 in coordinate form), and the difference vector q − p. The collection of all pisteparit on a given point set forms the edge set of the complete geometric graph on that set.

Applications of pisteparit appear across mathematics and computer science. They are used to build distance matrices, study pairwise relationships among points, formulate clustering and nearest-neighbor problems, and construct graphs for pattern recognition, computer vision, and geographical information systems. They also underpin various algorithms in computational geometry, such as those for triangulations and matching.

Example: given three points A, B, and C, the unordered pisteparit are {A, B}, {A, C}, and {B, C}; the ordered pisteparit are (A, B), (B, A), (A, C), (C, A), (B, C), and (C, B). See also graph theory and computational geometry.