Stereographic projection

File:Stereographic Projection Northern Hemisphere.png

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

In geometry, the stereographic projection is a certain mapping (function) that projects a sphere onto a plane. Intuitively, it gives a planar picture of the sphere.

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 accurately represents angular relationships. On the other hand, it does not accurately represent area, especially near the projection point.

Stereographic projection finds use in several areas of mathematics, such as differential geometry and complex analysis, and in many other fields including cartography, geology, and crystallography.

History

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 . 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. 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).

Definition

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

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

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

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

$(R,\Theta )=\left({\frac {\sin \varphi }{1-\cos \varphi }},\theta \right),$

$(\varphi ,\theta )=\left(2\arctan \left({\frac {1}{R}}\right),\Theta \right).$

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

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

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

Effect on subsets of the sphere and plane

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 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 sphere, with various loxodromes shown in distinct colors.

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

$R=e^{\Theta /a},\,$

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.

Conformality and area-preservation

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

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

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 an isometry and would preserve Gaussian curvature. The sphere and the plane have different Gaussian curvatures, so this is impossible.

Other formulations

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

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

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

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 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.

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.

Generalizations

Stereographic projection may also be applied to the n-sphere Sⁿ in (n + 1)-dimensional Euclidean space E^n + 1 in much the same way. 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 $\scriptstyle {\overline {QP}}$ 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 $\scriptstyle {\overline {QP}}$ with E. As before, the stereographic projection is conformal and invertible outside of a "small" set.

In this general setting, the stereographic projection presents the quadric hypersurface as a rational hypersurface. This generalization plays a role in algebraic geometry and conformal geometry.

Applications within mathematics

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

$\zeta ={\frac {x+iy}{1-z}},$

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

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

$\xi ={\frac {x-iy}{1+z}},$

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

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.

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 fields

Cartography

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 a fundamental problem in 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.

Geology

The set of all lines through the origin in three-dimensional space forms a space called the real projective plane. In some scientific disciplines, such as structural geology, the orientations of lines are a central geometric concern, and concrete graphical representations of the real projective plane are desirable.

Every line through the origin intersects the unit sphere in exactly two points, one of which is 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 gives a way to plot lines as points in the unit disk. This kind of plot is called an equal-angle stereonet, because it preserves angular relationships between lines (and not areas).

There is a related notion of equal-area stereonet that preserves areas but not angles; it does not arise from a stereographic projection in the sense of this article.

Crystallography

Stereographic projection of a crystallographic pole figure demonstrated on the diamond lattice in [111] direction.

The orientation of a plane (through the origin) in three-dimensional space may be recorded as the orientation of the line (through the origin) that is perpendicular to that plane. Hence planes can also be plotted by stereographic projection. In crystallography, a stereographic plot of crystal lattice planes is called a pole figure.

Photography

File:StereographicProjectionTube.jpg

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 is 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 being 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).

References

• Apostol, Tom (1974). Mathematical Analysis (2 ed.). Addison-Wesley.
• Brown, James and Churchill, Ruel (1989). Complex variables and applications. New York: McGraw-Hill. ISBN 0070109052.{{cite book}}: CS1 maint: multiple names: authors list (link)
• German, Daniel (June 2007). "Flattening the Viewable Sphere". "Proceedings of Computational Aesthetics 2007". Banff: Eurographics. pp. 23--28. {{cite conference}}: Unknown parameter |booktitle= ignored (|book-title= suggested) (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)
• 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.

See also

• Astrolabe
• Astronomical clock

External links

• http://mathworld.wolfram.com/StereographicProjection.html
• 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