In geometry the

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

The stereographic projection was known to Hipparchus Ptolemy and probably earlier to the Egyptians It was originally known as the planisphere projectionSnyder (1993)

It is believed that the earliest existing world map created by Gualterious Lud in 1507 is based upon the stereographic projection mapping each hemisphere as a circular diskAccording to (Snyder 1993) although he acknowledges he did not personally see it The equatorial aspect of the stereographic projection was commonly used for maps of the Eastern and Western Hemispheres in the 17th and 18th centuriesSnyder (1989)

François d'Aiguillon gave the stereographic projection its current name in his 1613 work

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

For any point

For the stereographic projection to be performed on a computer it must be expressed by explicit formulas In Cartesian coordinates (

- $(X\; Y)\; =\; left(frac\{x\}\{1\; -\; z\},\; frac\{y\}\{1\; -\; z\}right)$

- $(x\; y,\; z)\; =\; left(frac\{2\; X\}\{1\; +\; X^2\; +\; Y^2\}\; frac\{2\; Y\}\{1\; +\; X^2\; +\; Y^2\}\; frac\{-1\; +\; X^2\; +\; Y^2\}\{1\; +\; X^2\; +\; Y^2\}right)$

In spherical coordinates (φ θ) on the sphere (with φ the zenith and θ the azimuth) and polar coordinates (

- $(R\; Theta)\; =\; left(frac\{sin\; varphi\}\{1\; -\; cos\; varphi\}\; thetaright)$

- $(varphi\; theta)\; =\; left(2\; arctanleft(frac\{1\}\{R\}right)\; Thetaright)$

Here φ is understood to have value π when

- $(R\; Theta)\; =\; left(frac\{r\}\{1\; -\; z\},\; thetaright)$

- $(r\; theta\; z)\; =\; left(frac\{2\; R\}\{1\; +\; R^2\}\; Theta\; frac\{R^2\; -\; 1\}\{R^2\; +\; 1\}right)$

The projection is not defined at the projection point

perpendicular but the areas of the grid squares shrink as they approach the north pole

perpendicular but the areas of the grid sectors shrink as they approach the north pole

Stereographic projection is conformal meaning that it preserves the angles at which curves cross each other (see figures) On the other hand stereographic projection does not preserve area; in general the area of a region of the sphere does not equal the area of its projection onto the plane The area element is given in (

- $dA\; =\; frac\{4\}\{(1\; +\; X^2\; +\; Y^2)^2\}\; ;\; dX\; ;\; dY$

No map from the sphere to the plane can be both conformal and area-preserving If it were then it would be a local isometry and would preserve Gaussian curvature The sphere and the plane have different Gaussian curvatures so this is impossible

The conformality of the stereographic projection implies a number of convenient geometric properties Circles on the sphere that do

All lines in the plane when transformed to circles on the sphere by the inverse of stereographic projection intersect each other at infinity Parallel lines which do not intersect in the plane are tangent at infinity Thus all lines in the plane intersect somewhere in the sphere — either transversally at two points or tangently at infinity (Similar remarks hold about the real projective plane but the intersection relationships are different there)

shown in distinct colors

The loxodromes of the sphere map to curves on the plane of the form

- $R\; =\; e^\{Theta\; /\; a\}$

The stereographic projection relates to the plane inversion in a simple way Let P and Q be two points on the sphere with projections P' and Q' on the plane Then P' and Q' are inversive images of each other in the image of the equatorial circle if and only if P and Q are reflections of each other in the equatorial plane

Stereographic projection plots can be carried out by a computer using the explicit formulas given above However for graphing by hand these formulas are unwieldy; instead it is common to use graph paper designed specifically for the task To make this graph paper one places a grid of parallels and meridians on the hemisphere and then stereographically projects these curves to the disk The result is called a

In the figure the area-distorting property of the stereographic projection can be seen by comparing a grid sector near the center of the net with one at the far right of the net The two sectors have equal areas on the sphere On the disk the latter has nearly four times the area as the former; if one uses finer and finer grids on the sphere then the ratio of the areas approaches exactly 4

The angle-preserving property of the projection can be seen by examining the grid lines Parallels and meridians intersect at right angles on the sphere and so do their images on the Wulff net

For an example of the use of the Wulff net imagine that we have two copies of it on thin paper one atop the other aligned and tacked at their mutual center Suppose that we want to plot the point (0321 0557 -0766) on the lower unit hemisphere This point lies on a line oriented 60° counterclockwise from the positive

- Using the grid lines which are spaced 10° apart in the figures here mark the point on the edge of the net that is 60° counterclockwise from the point (1, 0) (or 30° clockwise from the point (0, 1))
- Rotate the top net until this point is aligned with (1, 0) on the bottom net
- Using the grid lines on the bottom net mark the point that is 50° toward the center from that point
- Rotate the top net oppositely to how it was oriented before to bring it back into alignment with the bottom net The point marked in step 3 is then the projection that we wanted

To find the central angle between two points on the sphere based on their stereographic plot overlay the plot on a Wulff net and rotate the plot about the center until the two points lie on or near a meridian Then measure the angle between them by counting grid lines along that meridian

In general one can define a stereographic projection from any point

*E*is perpendicular to the diameter through*Q*and*E*does not contain*Q*

All of the formulations of stereographic projection described thus far have the same essential properties They are smooth bijections (diffeomorphisms) defined everywhere except at the projection point They are conformal and not area-preserving

More generally stereographic projection may be applied to the

Still more generally suppose that

This construction has special significance in complex analysis The point (

- $zeta\; =\; frac\{x\; +\; i\; y\}\{1\; -\; z\}$
- $(x\; y,\; z)\; =\; left(frac\{2\; mathrm\{Re\}(zeta)\}\{1\; +\; bar\; zeta\; zeta\}\; frac\{2\; mathrm\{Im\}(zeta)\}\{1\; +\; bar\; zeta\; zeta\}\; frac\{-1\; +\; bar\; zeta\; zeta\}\{1\; +\; bar\; zeta\; zeta\}right)$

Similarly letting ξ =

- $xi\; =\; frac\{x\; -\; i\; y\}\{1\; +\; z\}$
- $(x\; y,\; z)\; =\; left(frac\{2\; mathrm\{Re\}(xi)\}\{1\; +\; bar\; xi\; xi\}\; frac\{2\; mathrm\{Im\}(xi)\}\{1\; +\; bar\; xi\; xi\}\; frac\{1\; -\; bar\; xi\; xi\}\{1\; +\; bar\; xi\; xi\}right)$

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

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 cannot 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 is 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

Further associated with each plane is a unique line called the plane's

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

- $left(frac\{2mn\}\{n^2+m^2\}\; frac\{n^2-m^2\}\{n^2+m^2\}right)$

which gives Euclid's formula for a Pythagorean triple

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 meridian to rays emanating from the origin and parallel to circles centered at the origin

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

In electron diffraction Kikuchi line pairs appear as bands decorating the intersection between lattice plane traces and the Ewald sphere thus providing

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

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 typically 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

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 projectionSee http://wwwbrunopostlenet/neatstuff/fisheye-to-stereographic/ for examples and further discussion

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)German

Compared to other azimuthal projections the stereographic projection tends to produce especially visually pleasing panoramas; this is due to the excellent shape preservation that is a result of the conformality of the projection

- Astrolabe
- Astronomical clock

- http://planetmathorg/encyclopedia/StereographicProjectionhtml
- Table of examples and properties of all common projections from radicalcartographynet
- three dimensional Java Applet
- Stereographic Projection and Inversion from cut-the-knot