Editing Homogeneous coordinates (section)

==Homogeneity==
Homogeneous coordinates are not uniquely determined by a point, so a function defined on the coordinates, say
{{nowrap|<math>f(x, y, z)</math>}}, does not determine a function defined on points as with Cartesian coordinates.
But a condition {{nowrap|<math>f(x, y, z) = 0</math>}} defined on the coordinates, as might be used to describe a
curve, determines a condition on points if the function is [[Homogeneous function|homogeneous]]. Specifically, suppose
there is a <math>k</math> such that

<math display="block">f(\lambda x, \lambda y, \lambda z) = \lambda^k f(x,y,z).</math>

If a set of coordinates represents the same point as {{nowrap|<math>(x, y, z)</math>}} then it can be written
{{nowrap|<math>(\lambda x, \lambda y, \lambda z)</math>}} for some non-zero value of <math>\lambda</math> . Then

<math display="block"> f(x,y,z)=0 \iff f(\lambda x, \lambda y, \lambda z) = \lambda^k f(x,y,z)=0.</math>

A [[polynomial]] {{nowrap|<math>g(x, y)</math>}} of degree <math>k</math> can be turned into a [[homogeneous polynomial]] by
replacing <math>x</math> with <math>x/z</math>, <math>y</math> with <math>y/z</math> and multiplying by <math>z^k</math>, in other words by
defining

<math display="block">f(x, y, z)=z^k g(x/z, y/z).</math>

The resulting function <math>f</math> is a polynomial, so it makes sense to extend its domain to triples where {{nowrap|<math>z =
0</math>}}. The process can be reversed by setting {{nowrap|<math>z = 1</math>}}, or

<math display="block">g(x, y)=f(x, y, 1).</math>

The equation {{nowrap|<math>f(x, y, z) = 0</math>}} can then be thought of as the homogeneous form of
{{nowrap|<math>g(x, y) = 0</math>}} and it defines the same curve when restricted to the Euclidean plane. For example,
the homogeneous form of the equation of the line {{nowrap|<math>ax + by + c = 0</math>}} is {{nowrap|<math>ax + by
+ cz = 0.</math>}}<ref>For the section: {{harvnb|Miranda|1995|p= 14}} and {{harvnb|Jones|1912|p= 120}}</ref>