Homogeneous Coordinates - Duality

Duality

The same relation, sx + ty + uz = 0, may be regarded either the equation of a line or the equation of a point. In general, there is no difference either algebraically or logically between the 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.