collinear

Collinear describes a set of points that lie on a single straight line. By definition, any two points are collinear, and a set of three or more points is collinear if and only if there exists a straight line containing all of them.

In the plane, given points A(x1,y1), B(x2,y2), and C(x3,y3), they are collinear if the area of triangle ABC is zero, which is equivalent to a zero determinant. A practical condition is (x2 - x1)(y3 - y1) - (y2 - y1)(x3 - x1) = 0. Another common test is that the slopes of AB and AC are equal (with a caveat for vertical lines, where x1 = x2 = x3). In vector form, AB and AC are linearly dependent, meaning there exists a scalar t such that AB = t·AC.

In three-dimensional space, three points A, B, and C are collinear if AB and AC are parallel, which can be checked by a zero cross product AB × AC. More generally, points lie on the same line if they can be written in a parametric form r = p0 + t v, where p0 is a point on the line and v is a direction vector.

Collinearity has several applications in geometry and computational geometry. It is used to detect degenerate configurations, simplify problems, and influence algorithms for convex hulls, polygon validity, and coordinate geometry. Testing collinearity often involves determinants or cross products to avoid division and maintain precision, especially in exact arithmetic or when dealing with floating-point coordinates. See also alignment and linear dependence.