Homogeneity
Homogeneous coordinates are not uniquely determined by a point, so a function defined on the coordinates, say f(x, y, z), does not determine a function defined on points as with Cartesian coordinates. But a condition f(x, y, z) = 0 defined on the coordinates, as might be used to describe a curve, determines a condition on points if the function is homogeneous. Specifically, suppose there is a k such that
If a set of coordinates represent the same point as (x, y, z) then it can be written (λx, λy, λz) for some non-zero value of λ. Then
A polynomial g(x, y) of degree k can be turned into a homogeneous polynomial by replacing x with x/z, y with y/z and multiplying by zk, in other words by defining
The resulting function f is a polynomial so it makes sense to extend its domain to triples where z = 0. The process can be reversed by setting z = 1, or
The equation f(x, y, z) = 0 can then be thought of as the homogeneous form of g(x, y) = 0 and it defines the same curve when restricted to the Euclidean plane. For example, the homogeneous form of the equation of the line ax + by + c = 0 is ax + by + cz = 0.
Read more about this topic: Homogeneous Coordinates
Famous quotes containing the word homogeneity:
“Dissonance between family and school, therefore, is not only inevitable in a changing society; it also helps to make children more malleable and responsive to a changing world. By the same token, one could say that absolute homogeneity between family and school would reflect a static, authoritarian society and discourage creative, adaptive development in children.”
—Sara Lawrence Lightfoot (20th century)
“Seems fairly clear that you fix a breed by LIMITING the amount of alien infiltration. You make a race by homogeneity and by avoiding INbreeding.... No argument has ever been sprouted against it. You like it in dogs and horses.”
—Ezra Pound (1885–1972)