# Homogeneous Coordinates

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## homogeneous coordinates

[‚hä·mə′jē·nē·əs kō′ȯrd·ən·əts] (mathematics)

To a point in the plane with cartesian coordinates (

*x,y*) there corresponds the homogeneous coordinates (*x*_{1},*x*_{2},*x*_{3}), where*x*_{1}/*x*_{3}=*x, x*_{2}/*x*_{3}=*y;*any polynomial equation in cartesian coordinates becomes homogeneous if a change into these coordinates is made.## Homogeneous Coordinates

of a point, line, and so on, coordinates that have the property that the object determined by them does not change when all coordinates are multiplied by a nonzero number. For example, the homogeneous coordinates of a point *M* in the plane are three numbers *X, Y*, and *Z*, related by the equation *X:Y:Z = x:y:1*, where *x* and *y* are its Cartesian coordinates. The introduction of homogeneous coordinates makes it possible to extend the class of points of the Euclidean plane by the addition of points whose third homogeneous coordinate is zero (ideal points, or points at infinity). This is important in projective geometry.