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The only projective geometry of dimension 0 is a single point. A projective geometry of dimension 1 consists of a single line containing at least 3 points. The geometric construction of arithmetic operations cannot be performed in either of these cases. For dimension 2, there is a rich structure in virtue of the absence of Desargues' Theorem.
Self-dual. In finite geometry, the Fano plane (named after Gino Fano) is a finite projective plane with the smallest possible number of points and lines: 7 points and 7 lines, with 3 points on every line and 3 lines through every point. These points and lines cannot exist with this pattern of incidences in Euclidean geometry, but they can be ...
The field planes are usually denoted by PG(2, q) where PG stands for projective geometry, the "2" is the dimension and q is called the order of the plane (it is one less than the number of points on any line). The Fano plane, discussed below, is denoted by PG(2, 2). The third example above is the projective plane PG(2, 3). The Fano plane.
In projective geometry, Desargues's theorem, named after Girard Desargues, states: Two triangles are in perspective axially if and only if they are in perspective centrally. Denote the three vertices of one triangle by a, b and c, and those of the other by A, B and C. Axial perspectivity means that lines ab and AB meet in a point, lines ac and ...
In projective geometry, duality or plane duality is a formalization of the striking symmetry of the roles played by points and lines in the definitions and theorems of projective planes. There are two approaches to the subject of duality, one through language (§ Principle of duality) and the other a more functional approach through special ...
Projective space. In graphical perspective, parallel (horizontal) lines in the plane intersect at a vanishing point (on the horizon). In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet at infinity. A projective space may thus be viewed as the extension of a ...