Orientation Predicate

computational-geometry geometry

Definition

Orientation Predicate

Let , , and be points in the plane. The orientation predicate is

Its sign tells how lies relative to the directed line :

Intuition

Intuition

The predicate measures the turn from the vector to the vector .

• A positive value means an anticlockwise turn.
• A negative value means a clockwise turn.
• A zero value means no turn, so the three points lie on one line.

The absolute value is twice the area of the triangle :

Why the Formula Works

Move to the origin. This does not change whether is left or right of . The shifted points are

The points and span a parallelogram from the origin. Its signed area is the determinant

The sign records the order of the two shifted points:

• positive if turns anticlockwise to ,
• negative if turns clockwise to ,
• zero if the parallelogram collapses to a line.

Substituting

gives

Coordinate Convention

The left/right interpretation assumes the usual Cartesian plane, where the -axis points upward. In screen coordinates, where the -axis often points downward, the sign interpretation is reversed.

Examples

Left of a horizontal directed line

Let , , and . Then

Hence lies left of .

Right of a horizontal directed line

Let , , and . Then

Hence lies right of .

Collinear points

Let , , and . Then

Hence , , and are collinear.