Topological and Differential Geometric Notions
The one-point compactification of the plane is homeomorphic to a sphere (see stereographic projection); the open disk is homeomorphic to a sphere with the "north pole" missing; adding that point completes the (compact) sphere. The result of this compactification is a manifold referred to as the Riemann sphere or the complex projective line. The projection from the Euclidean plane to a sphere without a point is a diffeomorphism and even a conformal map.
The plane itself is homeomorphic (and diffeomorphic) to an open disk. For the Lobachevsky plane such diffeomorphism is conformal, but for the Euclidean plane it is not.
Read more about this topic: Plane (geometry)
Famous quotes containing the words differential, geometric and/or notions:
“But how is one to make a scientist understand that there is something unalterably deranged about differential calculus, quantum theory, or the obscene and so inanely liturgical ordeals of the precession of the equinoxes.”
—Antonin Artaud (1896–1948)
“New York ... is a city of geometric heights, a petrified desert of grids and lattices, an inferno of greenish abstraction under a flat sky, a real Metropolis from which man is absent by his very accumulation.”
—Roland Barthes (1915–1980)
“Assumptions that racism is more oppressive to black men than black women, then and now ... based on acceptance of patriarchal notions of masculinity.”
—bell hooks (b. c. 1955)