Mapping The Sphere Onto The Plane
The unit sphere modulo −1 model of the projective plane is isomorphic (w.r.t. incidence properties) to the planar model: the affine plane extended with a projective line at infinity.
To map a point on the sphere to a point on the plane, let the plane be tangent to the sphere at some point which shall be the origin of the plane's coordinate system (2-D origin). Then construct a line passing through the center of the sphere (3-D origin) and the point on the sphere. This line intersects the plane at a point which is the projection of the point on the sphere onto the plane (or vice versa).
This projection can be used to define a one-to-one onto mapping
If points in are expressed in homogeneous coordinates, then
Also, lines in the planar model are projections of great circles of the sphere. This is so because through any line in the plane pass an infinitude of different planes: one of these planes passes through the 3-D origin, but a plane passing through the 3-D origin intersects the sphere along a great circle.
As we have seen, any great circle in the unit sphere has a projective point perpendicular to it, which can be defined as its dual. But this point is a pair of antipodal points on the unit sphere, through both of which passes a unique 3-D line, and this line extended past the unit sphere intersects the tangent plane at a point, which means that there is a geometric way to associate a unique point on the plane to every line on the plane, such that the point is the dual of the line.
Read more about this topic: Duality (projective Geometry)
Famous quotes containing the words sphere and/or plane:
“Don’t feel guilty if you don’t immediately love your stepchildren as you do your own, or as much as you think you should. Everyone needs time to adjust to the new family, adults included. There is no such thing as an “instant parent.”
Actually, no concrete object lies outside of the poetic sphere as long as the poet knows how to use the object properly.”
—Johann Wolfgang Von Goethe (1749–1832)
“Even though I had let them choose their own socks since babyhood, I was only beginning to learn to trust their adult judgment.. . . I had a sensation very much like the moment in an airplane when you realize that even if you stop holding the plane up by gripping the arms of your seat until your knuckles show white, the plane will stay up by itself. . . . To detach myself from my children . . . I had to achieve a condition which might be called loving objectivity.”
—Anonymous Parent of Adult Children. Ourselves and Our Children, by Boston Women’s Health Book Collective, ch. 5 (1978)