collineari

Collineari is the Italian term used to describe the geometric property of collinearity, meaning that a set of points lies on a single straight line. In English, the corresponding adjective is collinear; in Italian, collineari is used for phrases such as puntI collineari (collinear points). The concept is fundamental in Euclidean geometry and appears in many mathematical contexts, from basic constructions to analytic geometry.

Three or more points are collineari if there exists one line that contains all of them. With two points, the property is always satisfied, since a unique line passes through any pair of points. In analytic form, several equivalent tests exist: the slopes between pairs of points are equal (when denominators are defined); the area of the triangle formed by the points is zero, or the determinant formed by their coordinates vanishes; equivalently, the cross product of vectors formed from a common point is zero in three dimensions.

Common coordinate criteria can be stated as follows. For points P(x1,y1), Q(x2,y2), and R(x3,y3), the points are collinear if (y2−y1)(x3−x1) = (y3−y1)(x2−x1). In three dimensions, vectors AB and AC are collinear if AB × AC = 0. These formulations generalize to any finite set of points: all points lie on one line if and only if every triple among them is collinear.

Collineari finds use across disciplines, including geometry, computer graphics, geographic information systems, and astronomy, where alignment of objects or bodies is of interest.