geometry Representing Lines Normal form / homogeneous equation
Example
A line in the plane can be described as
a*x + b*y + c = 0
This uses a three-element parameter vector [a, b, c] to describe the line. Sometimes the constant term c is moved to the right hand side of the equation instead. The representation is not unique, since the length of that vector is arbitrary (as long as it is not zero). Such a vector is called homogeneous.
The vector [a, b] is perpendicular to the line, hence the name “normal form”. If the length of that vector is one, this is called the Hesse normal form which can be used to easily compute the distance of arbitrary points to the line in question. Even that doesn't make the representation fully unique, since one may negate all three parameters and obtain the same set of points satisfying the equation.
The concept does generalize to higher dimensions, but then it no longer describes a line, but instead a plane in 3d and a hyperplane in general.
