The Real Projective Plane
In projective geometry, the real projective plane is defined as the collection of lines through the origin in . The distance function on is most readily understood from this point of view. Namely, the distance between two lines through the origin is by definition the angle between them (measured in radians), or more precisely the lesser of the two angles. This distance function corresponds to the metric of constant Gaussian curvature +1.
Alternatively, can be defined as the surface obtained by identifying each pair of antipodal points on the 2-sphere.
Other metrics on can be obtained by quotienting metrics on imbedded in 3-space in a centrally symmetric way.
Topologically, can be obtained from the Möbius strip by attaching a disk along the boundary.
Among closed surfaces, the real projective plane is the simplest non-orientable such surface.
Read more about this topic: Introduction To Systolic Geometry
Famous quotes containing the words real and/or plane:
“The words themselves are clean, so are the things to which they apply. But the mind drags in a filthy association, calls up some repulsive emotion. Well, then, cleanse the mind, that is the real job. It is the mind which is the Augean stables, not language.”
—D.H. (David Herbert)
“It was the most ungrateful and unjust act ever perpetrated by a republic upon a class of citizens who had worked and sacrificed and suffered as did the women of this nation in the struggle of the Civil War only to be rewarded at its close by such unspeakable degradation as to be reduced to the plane of subjects to enfranchised slaves.”
—Anna Howard Shaw (1847–1919)