Stereographic projection Totally Explained

Everything about Wulff Net totally explained

In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point — the projection point. Where it's defined, the mapping is smooth and bijective. It is also conformal, meaning that it preserves angles. On the other hand, it doesn't preserve area, especially near the projection point.
Intuitively, then, the stereographic projection is a way of picturing the sphere as the plane, with some inevitable compromises. Because the sphere and the plane appear in many areas of mathematics and its applications, so does the stereographic projection; it finds use in diverse fields including complex analysis, cartography, geology, and photography. In practice, the projection is carried out by computer or by hand using a special kind of graph paper called a Wulff net or stereonet.

History

The stereographic projection was known to Hipparchus, Ptolemy and probably earlier to the Egyptians. It was originally known as the planisphere projection. Planisphaerium by Ptolemy is the oldest surviving document that describes it. One of its most important uses was the representation of celestial charts. The equatorial aspect of the stereographic projection was commonly used for maps of the Eastern and Western Hemispheres in the 17th and 18th centuries. François d'Aiguillon gave the stereographic projection its current name in his 1613 work Opticorum libri sex philosophis juxta ac mathematicis utiles (Six Books of Optics, useful for philosophers and mathematicians alike).

Definition

This section focuses on the projection of the unit sphere from the north pole onto the plane through the equator. Other formulations are treated in later sections.
   The unit sphere in three-dimensional space R³ is the set of points (x, y, z) such that x² + y² + z² = 1. Let N = (0, 0, 1) be the "north pole", and let M be the rest of the sphere. The plane z = 0 runs through the center of the sphere; the "equator" is the intersection of the sphere with this plane.
   For any point P on M, there's a unique line through N and P, and this line intersects the plane z = 0 in exactly one point P'. Define the stereographic projection of P to be this point P' in the plane.
   For the stereographic projection to be performed on a computer, it must be expressed by explicit formulas. In Cartesian coordinates (x, y, z) on the sphere and (X, Y) on the plane, the projection and its inverse are given by the formulas ť

(X, Y) = left(frac
ight).

define a stereographic projection from the south pole onto the equatorial plane. The transition maps between the ζ- and ξ-coordinates are then ζ = 1 / ξ and ξ = 1 / ζ, with ζ approaching 0 as ξ goes to infinity, and vice versa. This facilitates an elegant and useful notion of infinity for the complex numbers and indeed an entire theory of meromorphic functions mapping to the Riemann sphere. The standard metric on the unit sphere agrees with the Fubini-Study metric on the Riemann sphere.

Visualization of lines and planes

The set of all lines through the origin in three-dimensional space forms a space called the real projective plane. This space is difficult to visualize, because it can't be embedded in three-dimensional space.
However, one can "almost" visualize it as a disk, as follows. Any line through the origin intersects the southern hemisphere

z leq 0

in a point, which can then be stereographically projected to a point on a disk. Horizontal lines intersect the southern hemisphere in two antipodal points along the equator, either of which can be projected to the disk; it's understood that antipodal points on the boundary of the disk represent a single line. (See quotient topology.) So any set of lines through the origin can be pictured, almost perfectly, as a set of points in a 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.

Other visualization

Stereographic projection is also applied to the visualization of polytopes. In a Schlegel diagram, an n-dimensional polytope in R^n + 1 is projected onto an n-dimensional sphere, which is then stereographically projected onto Rⁿ. The reduction from R^n + 1 to Rⁿ can make the polytope easier to visualize and understand.

Applications to other disciplines

Cartography

The fact that no map from the sphere to the plane can accurately represent both angles (and thus shapes) and areas is the fundamental problem of cartography. In general, area-preserving map projections are preferred for statistical applications, because they behave well with respect to integration, while angle-preserving (conformal) map projections are preferred for navigation.
Stereographic projection falls into the second category. When the projection is centered at the Earth's north or south pole, it has additional desirable properties: It sends meridians to rays emanating from the origin and parallels to circles centered at the origin.

Crystallography

In crystallography, the orientations of crystal axes and faces in three-dimensional space are a central geometric concern, for example in the interpretation of X-ray and electron diffraction patterns. These orientations can be visualized as in the section Visualization of lines and planes above. That is, crystal axes and poles to crystal planes are intersected with the northern hemisphere and then plotted using stereographic projection. A plot of poles is called a pole figure.
In electron diffraction, Kikuchi line pairs appear as bands decorating the intersection between lattice plane traces and the Ewald sphere thus providing experimental access to a crystal's stereographic projection. Model Kikuchi maps in reciprocal space, and fringe visibility maps for use with bend contours in direct space, thus act as road maps for exploring orientation space with crystals in the transmission electron microscope.

Geology

Researchers in structural geology are concerned with the orientations of planes and lines for a number of reasons. The foliation of a rock is a planar feature that often contains a linear feature called lineation. Similarly, a fault plane is a planar feature that may contain linear features such as slickensides which indicate the direction of the fault's movement.
These orientations of lines and planes at various scales can be plotted using the methods of the Visualization of lines and planes section above. As in crystallography, planes are plotted by their poles. Unlike crystallography, the southern hemisphere is used instead of the northern one (because the geological features in question lie below the Earth's surface). In this context the stereographic projection is often referred to as the equal-angle lower-hemisphere projection. The equal-area lower-hemisphere projection defined by the Lambert azimuthal equal-area projection is also used, especially when the plot is to be subjected to subsequent statistical analysis such as density contouring.

Photography

Some fisheye lenses use a stereographic projection to capture a wide angle view. These lenses are usually preferred to more traditional fisheye lenses, which use an equal-area projection. This is probably a result of the conformal property of the stereographic: even areas close to the edge retain their shape, and straight lines are less curved. Unfortunately stereographic fisheye lenses are expensive to manufacture (none are currently being produced). Image remapping software, such as Panotools, allows the automatic remapping of photos from an equal-area fisheye to a stereographic projection.
The stereographic projection has been used to map spherical panoramas. This results in interesting effects: the area close to the point opposite to the center of projection becomes significantly enlarged, resulting in an effect known as little planet (when the center of projection is the nadir) and tube (when the center of projection is the zenith).
Compared to other azimuthal projections, the stereographic projection tends to produce especially visually pleasing panoramas; this is due to the excellent shape preservation that's a result of the conformality of the projection.

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