pistevälien

Pistevälien is a Finnish term referring to the distances or gaps between points (pisteet) in various mathematical, data, and spatial contexts. The concept is used to describe how far apart two points are, or how evenly points are spaced along a domain.

In geometry and analytic geometry, pisteväli describes the length of the segment joining two points. The standard measure is the Euclidean distance. For points A(x1, y1) and B(x2, y2) in a plane, the distance is d = sqrt((x2 − x1)^2 + (y2 − y1)^2). In n-dimensional space, d = sqrt(sum_i (xi − yi)^2). More generally, distances can be defined by p‑norms: d_p(A,B) = (sum_i |xi − yi|^p)^(1/p). Distances with these definitions satisfy the properties of a metric: non-negativity, identity of indiscernibles, symmetry, and the triangle inequality.

Beyond pure geometry, pistevälit are important in data sampling and time series, where they describe the spacing between consecutive data points or measurements. A uniform grid on a domain has a typical pisteväli equal to the grid step h. In geographic information systems and spatial analysis, pistevälit refer to the gaps between observation points, sampling locations, or nodes in a network.

Applications of pistevälien include clustering, nearest-neighbor searches, interpolation, mesh generation, and path planning, where the size and distribution of gaps influence accuracy and efficiency. When comparing distances, the choice of metric and units matters, as different metrics yield different notions of proximity.

See also: distance, metric, sampling, interpolation, spatial analysis.