An augmented coordinate representation for points in projective space that adds an extra scaling coordinate so that projective equivalence classes are represented linearly; a point in projective n-space is represented by (n+1)-tuples up to nonzero scalar multiplication.

Principle

Embed affine n-space into projective n-space by adjoining a homogeneous coordinate (often denoted w) and identify points (x_1,...,x_n) with tuples (w,x_1,...,x_n) modulo multiplication by any nonzero scalar, enabling linear handling of projective transformations and points at infinity.

Demonstration

In the projective plane P^2, a finite point (x,y) corresponds to homogeneous coordinates (1,x,y); a line not through the origin is given by a linear homogeneous equation aX+bY+cZ=0, and parallel affine lines meet at a point with Z=0 (a point at infinity).

Misapplication

Using homogeneous coordinates without modding out scalar multiples, treating (1,2,3) and (2,4,6) as distinct points and thereby miscounting intersections or failing to represent points at infinity correctly.

Consequence

Using homogeneous coordinates converts projective maps into linear maps on homogeneous tuples, unifies treatment of finite and infinite points, and simplifies algebraic geometry computations like intersection multiplicities.

Reversal

Instead of projective homogeneous coordinates, work in an affine chart with explicit inhomogeneous coordinates (x_i/w); this avoids the extra component but loses global linearity and explicit representation of points at infinity.

Boundary

Applicable to projective geometry and algebraic geometry contexts where scalar equivalence is allowed; inappropriate for contexts requiring absolute scale (e.g., Euclidean distances unchanged by scaling) unless scale is fixed by normalization.

Semantic Tension

Confused with barycentric coordinates: both use extra coordinates and projective-like invariance, but barycentric coordinates are weights relative to a simplex summing to one, whereas homogeneous coordinates are defined up to any nonzero scalar and represent projective points.

Synthesis

Homogeneous coordinates = (n+1)-tuples defined up to nonzero scalar that linearly represent points of projective n-space, enabling unified algebraic treatment of finite and infinite configurations.