collineare

Collineare is an adjective used in geometry to describe a set of points that lie on a common straight line. A set of points is collinear if there exists a single line that contains all of them. Any two distinct points are always collinear, since there is a unique line through them; three or more points may or may not be collinear.

In the plane, three points P1, P2, and P3 are collinear if they lie on the same line. This can be characterized in several equivalent ways: the area of the triangle P1P2P3 is zero, or the determinant formed by their coordinates vanishes, for example x1(y2−y3)+x2(y3−y1)+x3(y1−y2)=0. Equivalently, the vectors P2−P1 and P3−P1 are parallel, or the slopes between pairs of points are equal when defined. A line equation ax+by=c can be satisfied by all three points.

A convenient parametrization of a line through P1 and P2 is P(t) = P1 + t(P2−P1), for t in the real numbers. In higher dimensions, collinearity means that all points lie on a single straight line; for three points P1, P2, P3 in three-space, this is often tested by checking that (P2−P1) and (P3−P1) are linearly dependent, for instance via a zero cross product.

The concept also appears in projective geometry, where collinearity is preserved under projective transformations. In practice, detecting collinearity is a common operation in geometry, computer graphics, computational geometry, and geographic information systems. The term derives from Latin roots meaning “in line” and is translated into many languages, including Italian as collineare.