Search results
Results from the WOW.Com Content Network
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. Points are shown as dots; lines are shown as lines or circles. The Fano plane is the projective plane arising from the field of two elements. It is the smallest projective plane, with only seven points and seven lines.
Projective geometry is not necessarily concerned with curvature and the real projective plane may be twisted up and placed in the Euclidean plane or 3-space in many different ways. [1] Some of the more important examples are described below. The projective plane cannot be embedded (that is without intersection) in three-dimensional Euclidean space.
Projective geometry was instrumental in the validation of speculations of Lobachevski and Bolyai concerning hyperbolic geometry by providing models for the hyperbolic plane: [12] for example, the Poincaré disc model where generalised circles perpendicular to the unit circle correspond to "hyperbolic lines" , and the "translations" of this ...
C ∗ is also a projective plane, called the dual plane of C. If C and C ∗ are isomorphic, then C is called self-dual. The projective planes PG(2, K) for any field (or, more generally, for every division ring (skewfield) isomorphic to its dual) K are self-dual. In particular, Desarguesian planes of finite order are always self-dual.
Hanfried Lenz gave a classification scheme for projective planes in 1954, [6] which was refined by Adriano Barlotti in 1957. [7] This classification scheme is based on the types of point–line transitivity permitted by the collineation group of the plane and is known as the Lenz–Barlotti classification of projective planes.
If n is one or two, a projective space of dimension n is called a projective line or a projective plane, respectively. The complex projective line is also called the Riemann sphere. All these definitions extend naturally to the case where K is a division ring; see, for example, Quaternionic projective space.
The use of real numbers gives homogeneous coordinates of points in the classical case of the real projective spaces, however any field may be used, in particular, the complex numbers may be used for complex projective space. For example, the complex projective line uses two homogeneous complex coordinates and is known as the Riemann sphere.
In mathematics, real projective space, denoted or (), is the topological space of lines passing through the origin 0 in the real space +. It is a compact , smooth manifold of dimension n , and is a special case G r ( 1 , R n + 1 ) {\displaystyle \mathbf {Gr} (1,\mathbb {R} ^{n+1})} of a Grassmannian space.