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Stereographic projection of the northern hemisphere of the Earth from the south pole onto the plane tangent at the north pole.

3D illustration of a stereographic projection from the north pole onto a plane below the sphere.

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 is defined, the mapping is smooth and bijective. It is also conformal, meaning that it preserves angles. On the other hand, it does not preserve area, especially near the projection point. The projection can be carried out by computer or by hand using a special kind of graph paper called a Wulff net or stereonet.

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.

Illustration by Rubens for "Opticorum libri sex philosophis juxta ac mathematicis utiles", by François d'Aiguillon. It demonstrates how the projection is computed.

The stereographic projection was originally known as the planisphere projection (Snyder, 1993); in 1613 François d'Aiguillon gave it its current name in his "Opticorum libri sex philosophis juxta ac mathematicis utiles" (Six Books of Optics, useful for philosophers and mathematicians alike), Anvers, 1613 ^[1]. The term planisphere is still used to refer to celestial charts.

The stereographic projection is one of the oldest. It was known to Hipparchus, Ptolemy and probably earlier to the Egyptians. Planisphaerium by Ptolemy is the oldest surviving document that describes it. One of its most important uses was the representation of celestial charts (Snyder, 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 circle^[2]. The equatorial aspect of the stereographic was commonly used for maps of the Eastern and Western hemispheres in the 17th and 18th centuries (Snyder, 1989).

Stereographic projection of the unit sphere from the north pole onto the plane z = 0, shown here in cross section.

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 is 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.

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

The projection has simpler formulas in other coordinate systems. In spherical coordinates (φ, θ) on the sphere (with φ the zenith and θ the azimuth) and polar coordinates(R, Θ) on the plane, the projection and its inverse are

Here, φ is understood to have value π when R = 0. Also, there are many ways to rewrite these formulas using trigonometric identities. In cylindrical coordinates (r, θ, z) on the sphere and polar coordinates (R, Θ) on the plane, the projection and its inverse are

The stereographic projection defined in the preceding section sends the "south pole" (0, 0, −1) to (0, 0), the equator to the unit circle, the southern hemisphere to the region inside the circle, and the northern hemisphere to the region outside the circle.

The sphere, with various loxodromes shown in distinct colors.

The projection is not defined at the projection point N = (0, 0, 1). Small neighborhoods of this point are sent to subsets of the plane far away from (0, 0). The closer P is to (0, 0, 1), the more distant its image is from (0, 0) in the plane. For this reason it is common to speak of (0, 0, 1) as mapping to "infinity" in the plane, and of the sphere as completing the plane by adding a "point at infinity". This notion finds utility in projective geometry and complex analysis. On a merely topological level, it illustrates how the sphere is homeomorphic to the one point compactification of the plane.

Stereographic projection transforms those circles on the sphere that do not pass through the point of projection to circles on the plane. It transforms circles on the sphere that do pass through the point of projection to straight lines on the plane. These are sometimes thought of as circles through the point at infinity, or circles of infinite radius.

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.)

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

where the parameter a measures the "tightness" of the loxodrome. Thus loxodromes correspond to equiangular spirals. These spirals intersect radial lines in the plane at equal angles, just as the loxodromes intersect meridians on the sphere at equal angles.

A 10x10 Cartesian grid on the plane appears distorted on the sphere. The grid lines are still perpendicular, but the areas of the grid squares shrink as they approach the north pole.

A radius-5 polar grid on the plane appears distorted on the sphere. The grid curves are still 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. This is the underlying reason why loxodromes and circles on the sphere map to equiangular spirals and circles, respectively, on the plane.

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 (X, Y) coordinates by

Along the unit circle, where X² + Y² = 1, there is no infinitesimal distortion of area. Near (0, 0) areas are distorted by a factor of 4, and near infinity areas are distorted by arbitrarily small factors.

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.

Wulff net or stereonet, used for making plots of the stereographic projection by hand.

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 Wulff net or stereonet. Its use was first proposed by the Russian mineralogist George (Yuri Viktorovich) Wulff ^[3].

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.

Illustration of steps 1-4 for plotting a point on a 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 (0.321, 0.557, -0.766) on the lower unit hemisphere. This point lies on a line oriented 60° counterclockwise from the positive x-axis (or 30° clockwise from the positive y-axis) and 50° below the horizontal plane z = 0. Once these angles are known, there are four steps:

1. 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)).
2. Rotate the top net until this point is aligned with (1, 0) on the bottom net.
3. Using the grid lines on the bottom net, mark the point that is 50° toward the center from that point.
4. Rotate the top net oppositely to how it was rotated 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 plot other points, whose angles are not such round numbers as 60° and 50°, one must visually interpolate between the nearest grid lines. It is helpful to have a net with finer spacing than 10°; spacings of 2° are common.

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.

Instead of projecting onto the equatorial plane from the north pole, one may project from the south pole S = (0, 0, −1). A Cartesian formula is

Like the projection of the preceding section, this projection takes the equator to the unit circle. The inverse, viewed as a parametrization (x, y, z) of the sphere, induces the same orientation on the sphere as the one above. Thus the sphere can be viewed as an oriented surface (or two-dimensional manifold) covered by two stereographic charts.

Stereographic projection of the unit sphere from the north pole onto the plane z = −1, shown here in cross section.

Some authors^[4] define stereographic projection from the north pole (0, 0, 1) onto the plane z = −1, which is tangent to the unit sphere at the south pole (0, 0, −1). The values X and Y produced by this projection are exactly twice those produced by the equatorial projection described in the preceding section. For example, this projection sends the equator to the circle of radius 2 centered at the origin. While the equatorial projection produces no infinitesimal area distortion along the equator, this pole-tangent projection instead produces no infinitesimal area distortion at the south pole.

Stereographic projection of a sphere from a point Q onto the plane E, shown here in cross section.

In general, one can define a stereographic projection from any point Q on the sphere onto any plane E such that

• E is perpendicular to the diameter through Q, and
• E does not contain Q.

As long as E meets these conditions, then for any point P other than Q the line through P and Q meets E in exactly one point P^′, which is defined to be the stereographic projection of P onto E.^[5]

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 n-sphere Sⁿ in (n + 1)-dimensional Euclidean space E^n + 1. If Q is a point of Sⁿ and E a hyperplane in E^n + 1, then the stereographic projection of a point P ∈ Sⁿ − {Q} is the point P^′ of intersection of the line with E.

Still more generally, suppose that S is a (nonsingular) quadric hypersurface in the projective space P^n + 1. By definition, S is the locus of zeros of a non-singular quadratic form f(x₀, ..., x_n + 1) in the homogeneous coordinates x_i. Fix any point Q on S and a hyperplane E in P^n + 1 not containing Q. Then the stereographic projection of a point P in S − {Q} is the unique point of intersection of with E. As before, the stereographic projection is conformal and invertible outside of a "small" set. The stereographic projection presents the quadric hypersurface as a rational hypersurface.^[6] This construction plays a role in algebraic geometry and conformal geometry.

The fact that the sphere is covered by two stereographic parametrizations from a plane through the equator has special significance in complex analysis. The point (X, Y) in the real plane can be identified with the complex number ζ = X + iY. The stereographic projection from the north pole can then be written

Similarly, letting ξ = X − iY be the complex coordinate corresponding to the coordinates (X, Y) of the other parametrization, stereographic projection becomes

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 the construction of an elegant 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.

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; in fact, it cannot be embedded in three-dimensional space at all. Every line through the origin intersects the unit sphere in exactly two points, one of which is on the southern hemisphere. So lines are in one-to-one correspondence with points on the southern hemisphere. (Horizontal lines intersect the equator in two points. It is understood that antipodal points on the equator represent a single line. See quotient topology.) Stereographic projection from the north pole onto the plane through the equator sends the southern hemisphere to the unit disk. Thus it lets us visualize the real projective plane as the unit disk. This construction finds use in geology and crystallography, as described below.

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.

Stereographic projection is used to map the Earth, especially near the poles, but also near other points of interest.

The fact that no map from the sphere to the plane can accurately represent both angles 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.

A crystallographic pole figure for the diamond lattice in [111] direction.

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.

An axis corresponds to a line through the origin, which intersects the unit lower (or upper) hemisphere in a point, which can then be stereographically projected to a disk. Hence stereographic projection gives us a means for visualizing crystal axes as points in a disk.

Similarly, a crystal face corresponds to a plane through the origin, which intersects the hemisphere in a circular arc, which stereographically projects to a circular arc (because the projection sends circles to circles). So planes can be visualized as arcs in a disk.

However, instead of plotting the plane itself it is common to plot the line perpendicular to the plane, called the pole of the plane. In this way the stereographic projection lets us visualize planes as points in a disk as well. This kind of plot is called a pole figure.

Use of lower hemisphere stereographic projection to plot planar and linear data in structural geology, using the example of a fault plane with a slickenside lineation.

Researchers in structural geology use stereographic projection to make plots of crystallographic axes and faces, and also of lines and planes at other scales. The foliation of a rock is a planar feature that often contains a linear feature called lineation. Similarly, the a fault plane is a planar feature that may contain linear features such as slickensides. As in crystallography, planes are often plotted by their poles.

In geology, such a stereographic plot is called an equal-angle lower-hemisphere projection. There is also an equal-area lower-hemisphere projection, defined by the Lambert azimuthal equal-area projection. That projection is usually more appropriate for statistical analyses.

Spherical panorama projected using the stereographic projection where the center of projection is the zenith. This type of panorama is colloquially known as tube.

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).^{[citation needed]} Image remapping software, such as Panotools, allows the automatic remapping of photos from an equal-area fisheye to a stereographic projection ^[7]

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 et al, 2007).^[8]

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.

1. ^ According to (Elkins, 1988) who references Eckert, "Die Kartenwissenschaft", Berlin 1921, pp 121--123
2. ^ According to (Snyder 1993), although he acknowledges he did not personally see it
3. ^ Wulff, George, Untersuchungen im Gebiete der optischen Eigenschaften isomorpher Kristalle: Zeits. Krist.,36, l-28 (1902)
4. ^ Cf. Apostol (1974) p. 17.
5. ^ Cf. Pedoe (1988).
6. ^ Cf. Shafarevich (1995).
7. ^ See http://www.bruno.postle.net/neatstuff/fisheye-to-stereographic/ for examples and further discussion
8. ^ See http://www.flickr.mud.yahoo.com/photos/gadl/sets/72157594279945875/?page=2 for examples of little planets.

• Apostol, Tom (1974). Mathematical Analysis, 2, Addison-Wesley. 
• Brown, James and Churchill, Ruel (1989). Complex variables and applications. New York: McGraw-Hill. ISBN 0070109052. 
• German, Daniel; Burchill, L.; Duret-Lutz, A.; Pérez-Duarte, S. ; Pérez-Duarte, E. and Sommers, J. (June 2007). "Flattening the Viewable Sphere". "Proceedings of Computational Aesthetics 2007": 23--28, Banff: Eurographics. 
• Do Carmo, Manfredo P. (1976). Differential geometry of curves and surfaces. Englewood Cliffs, New Jersey: Prentice Hall. ISBN 0-13-212589-7. 
• Elkins, James (1988). "Did Leonardo Develop a Theory of Curvilinear Perspective?: Together with Some Remarks on the 'Angle' and 'Distance' Axioms". Journal of the Warburg and Courtauld Institutes 51: 190--196. 
• Oprea, John (2003). Differential geometry and applications. Englewood Cliffs, New Jersey: Prentice Hall. ISBN 0130652466. 
• Pedoe, Dan (1988). Geometry. Dover. ISBN 0-486-65812-0. 
• Shafarevich, Igor (1995). Basic Algebraic Geometry I. Springer. ISBN 0387548122. 
• Snyder, John P. (1989). An Album of Map Projections, Professional Paper 1453. US Geological Survey. 
• Snyder, John P. (1993). Flattening the Earth. University of Chicago. ISBN 0-226-76746-9. 
• Spivak, Michael (1999). A comprehensive introduction to differential geometry, Volume IV. Houston, Texas: Publish or Perish. ISBN 091409873X. 

• Astrolabe
• Astronomical clock

• Eric W. Weisstein, Stereographic projection at MathWorld.
• http://planetmath.org/encyclopedia/StereographicProjection.html
• Panoramic Image Projections - Interactive visual comparison between the stereographic image projection and other types, for primary use in panoramic photography.
• Table of examples and properties of all common projections, from radicalcartography.net
• three dimensional Java Aplett