Stereographic projection


In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere, onto a plane perpendicular to the diameter through the point. It is a smooth, bijective function from the entire sphere except the center of projection to the entire plane. It maps circles on the sphere to circles or lines on the plane, and is conformal, meaning that it preserves angles at which curves meet and thus locally approximately preserves shapes. It is neither isometric nor equiareal.
The stereographic projection gives a way to represent a sphere by a plane. The metric induced by the inverse stereographic projection from the plane to the sphere defines a geodesic distance between points in the plane equal to the spherical distance between the spherical points they represent. A two-dimensional coordinate system on the stereographic plane is an alternative setting for spherical analytic geometry instead of spherical polar coordinates or three-dimensional cartesian coordinates. This is the spherical analog of the Poincaré disk model of the hyperbolic plane.
Intuitively, 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. Sometimes stereographic computations are done graphically using a special kind of graph paper called a stereographic net, shortened to stereonet, or Wulff net.

History

The origin of the stereographic projection is not known, but it is believed to have been discovered by Ancient Greek astronomers and used for projecting the celestial sphere to the plane so that the motions of stars and planets could be analyzed using plane geometry. Its earliest extant description is found in Ptolemy's Planisphere, but it was ambiguously attributed to Hipparchus by Synesius, and Apollonius's Conics contains a theorem which is crucial in proving the property that the stereographic projection maps circles to circles. Hipparchus, Apollonius, Archimedes, and even Eudoxus have sometimes been speculatively credited with inventing or knowing of the stereographic projection, but some experts consider these attributions unjustified. Ptolemy refers to the use of the stereographic projection in a "horoscopic instrument", perhaps the described by Vitruvius.
By the time of Theon of Alexandria, the planisphere had been combined with a dioptra to form the planispheric astrolabe, a capable portable device which could be used for measuring star positions and performing a wide variety of astronomical calculations. The astrolabe was in continuous use by Byzantine astronomers, and was significantly further developed by medieval Islamic astronomers. It was transmitted to Western Europe during the 11th–12th century, with Arabic texts translated into Latin.
In the 16th and 17th century, the equatorial aspect of the stereographic projection was commonly used for maps of the Eastern and Western Hemispheres. It is believed that already the map created in 1507 by Gualterius Lud was in stereographic projection, as were later the maps of Jean Rotz, Rumold Mercator, and many others. In star charts, even this equatorial aspect had been utilised already by the ancient astronomers like Ptolemy.
François d'Aguilon gave the stereographic projection its current name in his 1613 work Opticorum libri sex philosophis juxta ac mathematicis utiles.
In the late 16th century, Thomas Harriot proved that the stereographic projection is conformal; however, this proof was never published and sat among his papers in a box for more than three centuries. In 1695, Edmond Halley, motivated by his interest in star charts, was the first to publish a proof. He used the recently established tools of calculus, invented by his friend Isaac Newton.

Definition

First formulation

The unit sphere in three-dimensional space is the set of points such that. Let be the "north pole", and let be the rest of the sphere. The plane runs through the center of the sphere; the "equator" is the intersection of the sphere with this plane.
For any point on, there is a unique line through and, and this line intersects the plane in exactly one point, known as the stereographic projection of onto the plane.
In Cartesian coordinates on the sphere and on the plane, the projection and its inverse are given by the formulas
In spherical coordinates on the sphere and polar coordinates on the plane, the projection and its inverse are
Here, is understood to have value when. Also, there are many ways to rewrite these formulas using trigonometric identities. In cylindrical coordinates on the sphere and polar coordinates on the plane, the projection and its inverse are

Other conventions

Some authors define stereographic projection from the north pole onto the plane, which is tangent to the unit sphere at the south pole. This can be described as a composition of a projection onto the equatorial plane described above, and a homothety from it to the polar plane. The homothety scales the image by a factor of 2, hence the values and 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.
Other authors use a sphere of radius and the plane. In this case the formulae become
Image:StereographicGeneric.svg|thumb|right|Stereographic projection of a sphere from a point onto the plane, shown here in cross section
In general, one can define a stereographic projection from any point on the sphere onto any plane such that
  • is perpendicular to the diameter through, and
  • does not contain.
As long as meets these conditions, then for any point other than the line through and meets in exactly one point, which is defined to be the stereographic projection of P onto E.

Generalizations

More generally, stereographic projection may be applied to the unit -sphere in -dimensional Euclidean space. If is a point of and a hyperplane in, then the stereographic projection of a point is the point of intersection of the line with. In Cartesian coordinates on and on, the projection from is given by
Defining
the inverse is given by
Still more generally, suppose that is a quadric hypersurface in the projective space. In other words, is the locus of zeros of a non-singular quadratic form in the homogeneous coordinates. Fix any point on and a hyperplane in not containing. Then the stereographic projection of a point in is the unique point of intersection of with. As before, the stereographic projection is conformal and invertible on a non-empty Zariski open set. The stereographic projection presents the quadric hypersurface as a rational hypersurface. This construction plays a role in algebraic geometry and conformal geometry.

Properties

The first stereographic projection defined in the preceding section sends the "south pole" of the unit sphere to, 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 =. Small neighborhoods of this point are sent to subsets of the plane far away from. The closer is to, the more distant its image is from in the plane. For this reason it is common to speak of 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.
In Cartesian coordinates a point on the sphere and its image on the plane either both are rational points or none of them:
Image:CartesianStereoProj.png|thumb|left|A 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.
Image:PolarStereoProj.png|thumb|left|A 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. 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 coordinates by
Along the unit circle, where, there is no inflation of area in the limit, giving a scale factor of 1. Near areas are inflated by a factor of 4, and near infinity areas are inflated by arbitrarily small factors.
The metric is given in coordinates by
and is the unique formula found in Bernhard Riemann's Habilitationsschrift on the foundations of geometry, delivered at Göttingen in 1854, and entitled Über die Hypothesen welche der Geometrie zu Grunde liegen.
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.
Circles on the sphere that do not pass through the point of projection are projected to circles on the plane. Circles on the sphere that do pass through the point of projection are projected to straight lines on the plane. These lines are sometimes thought of as circles through the point at infinity, or circles of infinite radius. These properties can be verified by using the expressions of in terms of given in : using these expressions for a substitution in the equation of the plane containing a circle on the sphere, and clearing denominators, one gets the equation of a circle, that is, a second-degree equation with as its quadratic part. The equation becomes linear if that is, if the plane passes through the point of projection.
All lines in the plane, when transformed to circles on the sphere by the inverse of stereographic projection, meet at the projection point. Parallel lines, which do not intersect in the plane, are transformed to circles tangent at projection point. Intersecting lines are transformed to circles that intersect transversally at two points in the sphere, one of which is the projection point.
Image:Riemann Sphere.jpg|right|thumb|175px|The sphere, with various loxodromes shown in distinct colors
The loxodromes of the sphere map to curves on the plane of the form
where the parameter measures the "tightness" of the loxodrome. Thus loxodromes correspond to logarithmic spirals. These spirals intersect radial lines in the plane at equal angles, just as the loxodromes intersect meridians on the sphere at equal angles.
The stereographic projection relates to the plane inversion in a simple way. Let and be two points on the sphere with projections and on the plane. Then and are inversive images of each other in the image of the equatorial circle if and only if and are reflections of each other in the equatorial plane.
In other words, if:
  • is a point on the sphere, but not a 'north pole' and not its antipode, the 'south pole',
  • is the image of in a stereographic projection with the projection point and
  • is the image of in a stereographic projection with the projection point,
then and are inversive images of each other in the unit circle.