Editing Stereographic projection (section)

===Visualization of lines and planes===
[[Image:Sfsp111.gif|thumb|right|Animation of [[Kikuchi line (solid state physics)|Kikuchi lines]] of four of the eight <111> zones in an fcc crystal. Planes edge-on (banded lines) intersect at fixed angles.]]

The set of all lines through the origin in three-dimensional space forms a space called the [[real projective plane]]. This plane is difficult to visualize, because it cannot be [[Embedding|embedded]] in three-dimensional space.

However, one can visualize it as a disk, as follows. Any line through the origin intersects the southern hemisphere {{math|''z''}}&nbsp;≤&nbsp;0 in a point, which can then be stereographically projected to a point on a disk in the XY plane. Horizontal lines through the origin intersect the southern hemisphere in two [[antipodal point]]s along the equator, which project to the boundary of the disk. Either of the two projected points can be considered part of the disk; it is understood that antipodal points on the equator represent a single line in 3 space and a single point on the boundary of the projected disk (see [[quotient topology]]). So any set of lines through the origin can be pictured as a set of points in the projected disk.  But the boundary points behave differently from the boundary points of an ordinary 2-dimensional disk, in that any one of them is simultaneously close to interior points on opposite sides of the disk (just as two nearly horizontal lines through the origin can project to points on opposite sides of the disk).

Also, every plane through the origin intersects the unit sphere in a great circle, called the ''trace'' of the plane. This circle maps to a circle under stereographic projection. So the projection lets us visualize planes as circular arcs in the disk.  Prior to the availability of computers, stereographic projections with great circles often involved drawing large-radius arcs that required use of a [[beam compass]].  Computers now make this task much easier.

Further associated with each plane is a unique line, called the plane's ''pole'', that passes through the origin and is perpendicular to the plane. This line can be plotted as a point on the disk just as any line through the origin can. So the stereographic projection also lets us visualize planes as points in the disk. For plots involving many planes, plotting their poles produces a less-cluttered picture than plotting their traces.

This construction is used to visualize directional data in crystallography and geology, as described below.