##  [Homogeneous Coordinates](/homogeneous-coordinates-0) 

 Definition

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

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

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

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

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

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

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

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

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.