Collinear
GeometryCollinear points are three or more points that all lie on the same straight line.
Definition
Collinear means "on the same line." If three or more points are collinear, you can draw one straight line that passes through all of them.
Example
Points A, B, and C are collinear if they all sit on the same straight line. If you place three dots on a ruler and they all touch the ruler's edge, those dots are collinear.
Key Insight
The word "collinear" comes from Latin: "co-" means together, and "linear" means line. So collinear simply means "together on a line." Non-collinear points are needed to define a triangle or a plane.
Definition
Three or more points are collinear if and only if they all lie on a single straight line. Any two points are always collinear (since two points determine a line). The interesting case is three or more points: they may or may not be collinear.
Example
Points $(0, 0)$, $(1, 2)$, and $(2, 4)$ are collinear because they all satisfy $y = 2x$. Points $(0, 0)$, $(1, 1)$, and $(2, 3)$ are NOT collinear - no single line passes through all three.
Key Insight
To test collinearity of three points $A, B, C$: check if the area of triangle $ABC$ equals zero. Area $= (1/2)|x_A(y_B - y_C) + x_B(y_C - y_A) + x_C(y_A - y_B)|$. If this equals $0$, the points are collinear.
Definition
Points $P_1, P_2, \ldots, P_n$ in $\mathbb{R}^n$ are collinear if they all lie on a common affine line, i.e., if the vectors $(P_2 - P_1), (P_3 - P_1), \ldots, (P_n - P_1)$ are all scalar multiples of a single vector. Equivalently, the rank of the matrix formed by $(P_i - P_1)$ is at most $1$.
Example
Three points $A, B, C$ in $\mathbb{R}^2$ are collinear iff $\det([B-A, C-A]) = 0$, i.e., the $2 \times 2$ determinant $(B_x-A_x)(C_y-A_y) - (B_y-A_y)(C_x-A_x) = 0$. This determinant equals twice the signed area of triangle $ABC$.
Key Insight
In projective geometry, collinearity is a projective invariant: projections and perspectivities map collinear points to collinear points. This is the foundation of cross-ratio theory and projective transformations used in computer vision.