Editing Homogeneous coordinates (section)

==Line coordinates and duality==
{{Main|Duality (projective geometry)}}
The equation of a line in the projective plane may be given as {{nowrap|<math>sx + ty + uz = 0</math>}} where <math>s</math>,
<math>t</math> and <math>u</math> are constants. Each triple {{nowrap|<math>(s, t, u)</math>}} determines a line, the line determined is
unchanged if it is multiplied by a non-zero scalar, and at least one of <math>s</math>, <math>t</math> and <math>u</math> must be non-zero. So the
triple {{nowrap|<math>(s, t, u)</math>}} may be taken to be homogeneous coordinates of a line in the projective plane,
that is [[line coordinates]] as opposed to point coordinates. If in <math>
sx+ty+uz=0</math> the letters <math>s</math>, <math>t</math> and <math>u</math> are taken as variables and
<math>x</math>, <math>y</math> and <math>z</math> are taken as constants then the equation becomes an equation of a set of lines in the space of
all lines in the plane. Geometrically it represents the set of lines that pass through the point {{nowrap|<math>(x, y,
z)</math>}} and may be interpreted as the equation of the point in line-coordinates. In the same way, planes in 3-space may
be given sets of four homogeneous coordinates, and so on for higher dimensions.<ref>{{harvnb|Bôcher|1907|pp=
    107&ndash;108}} (adapted to the plane according to the footnote on p. 108)</ref>

The same relation, {{nowrap|<math>sx + ty + uz = 0</math>}}, may be regarded as either the equation of a line or the
equation of a point. In general, there is no difference either algebraically or logically between homogeneous
coordinates of points and lines. So plane geometry with points as the fundamental elements and plane geometry with lines
as the fundamental elements are equivalent except for interpretation. This leads to the concept of duality in
projective geometry, the principle that the roles of points and lines can be interchanged in a theorem in projective
geometry and the result will also be a theorem. Analogously, the theory of points in projective 3-space is dual to the
theory of planes in projective 3-space, and so on for higher dimensions.<ref>{{harvnb|Woods|1922|pp= 2, 40}}</ref>