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Homogeneous Coordinates

Homogeneous coordinates (x_1,x_2,x_3) of a finite point (x,y) in the plane are any three numbers for which

(x_1)/(x_3)=x
(1)
(x_2)/(x_3)=y.
(2)

Coordinates (x_1,x_2,0) for which

(x_2)/(x_1)=lambda
(3)

describe the point at infinity in the direction of slope lambda.

In homogeneous coordinates, the equation of a line

a_1x+a_2y+a_3=0
(4)

is given by

a_1x_1+a_2x_2+a_3x_3=0.
(5)

Two points expressed using homogeneous coordinates (a_1,a_2,a_3) and (b_1,b_2,b_3) are identical iff

|a_2 a_3; b_2 b_3|=|a_3 a_1; b_3 b_1|=|a_1 a_2; b_1 b_2|=0.
(6)

Two lines expressed using homogeneous coordinates

a_1x_1+a_2x_2+a_3x_3=0
(7)
b_1x_1+b_2x_2+b_3x_3=0
(8)

are identical iff

|a_2 a_3; b_2 b_3|=|a_3 a_1; b_3 b_1|=|a_1 a_2; b_1 b_2|=0.
(9)

The intersection of the two lines above is given by

x_1=|a_2 a_3; b_2 b_3|
(10)
x_2=|a_3 a_1; b_3 b_1|
(11)
x_3=|a_1 a_2; b_1 b_2|.
(12)

See also

Barycentric Coordinates, Trilinear Coordinates

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References

Graustein, W. C. "Homogeneous Cartesian Coordinates. Linear Dependence of Points and Lines." Ch. 3 in Introduction to Higher Geometry. New York: Macmillan, pp. 29-49, 1930.

Referenced on Wolfram|Alpha

Homogeneous Coordinates

Cite this as:

Weisstein, Eric W. "Homogeneous Coordinates." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HomogeneousCoordinates.html

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