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# Stereographic projection

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### Stereographic projection

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 conformal, meaning that it preserves angles. It is neither isometric nor area-preserving: that is, it preserves neither distances nor the areas of figures.

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 stereographic net, shortened to stereonet or Wulff net.

## Contents

• History 1
• Definition 2
• Properties 3
• Wulff net 4
• Other formulations and generalizations 5
• Applications within mathematics 6
• Complex analysis 6.1
• Visualization of lines and planes 6.2
• Other visualization 6.3
• Arithmetic geometry 6.4
• Tangent half-angle substitution 6.5
• Applications to other disciplines 7
• Cartography 7.1
• Crystallography 7.2
• Geology 7.3
• Photography 7.4
• References 9
• Sources 9.1

## 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 known to Hipparchus, Ptolemy and probably earlier to the Egyptians. It was originally known as the planisphere projection.[1] Planisphaerium by Ptolemy is the oldest surviving document that describes it. One of its most important uses was the representation of celestial charts.[1] The term planisphere is still used to refer to such charts.

It is believed that the earliest existing world map, created in 1507 by Gualterius Lud of Saint-Dié, is based upon the stereographic projection, mapping each hemisphere as a circular disk.[2] The equatorial aspect of the stereographic projection, commonly used for maps of the Eastern and Western Hemispheres in the 17th and 18th centuries (and 16th century - Jean Roze 1542; Rumold Mercator 1595),[3] was utilised by the ancient astronomers like Ptolemy.[4]

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).[5]

In 1695, Edmond Halley, motivated by his interest in star charts, published the first mathematical proof that this map is conformal.[6] He used the recently established tools of calculus, invented by his friend Isaac Newton.

## 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 (xyz) on the sphere and (XY) on the plane, the projection and its inverse are given by the formulas

(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 (, θ) on the sphere (with the zenith angle, 0 ≤ ≤ π, and θ the azimuth, 0 ≤ θ ≤ 2 π) 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) = \left(\cot\frac{\varphi}{2}, \theta\right),
(\varphi, \theta) = \left(2 \arctan\left(\frac{1}{R}\right), \Theta\right).

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

## Properties

The stereographic projection defined in the preceding section sends the "south pole" (0, 0, −1) of the unit sphere 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.

In Cartesian coordinates a point P(xyz) on the sphere and its image P′(XY) on the plane either both are rational points or none of them:

P \in \Bbb Q^3 \iff P' \in \Bbb Q^2
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.
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 (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 (XY) 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.

The metric is given in (XY) coordinates by

\frac{4}{(1 + X^2 + Y^2)^2} \; ( dX^2 + dY^2),

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.

The conformality of the stereographic projection implies a number of convenient geometric properties. 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.

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 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 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.
In other words, if:

• P is a point on the sphere, but not a 'north pole' N and not its antipode, the 'south pole' S,
• P' is the image of P in a stereographic projection with the projection point N and
• P" is the image of P in a stereographic projection with the projection point S,

then P' and P" are inversive images of each other in the unit circle.

\triangle NOP^\prime \sim \triangle P^{\prime\prime}OS \implies OP^\prime:ON = OS : OP^{\prime\prime} \implies OP^\prime \cdot OP^{\prime\prime} = r^2

## Wulff net

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. This special graph paper is called a stereonet or Wulff net, after the

• Time Lapse Stereographic Projection
• Weisstein, Eric W., "Stereographic projection", MathWorld.
• Planetmath.org
• Table of examples and properties of all common projections, from radicalcartography.net
• Three dimensional Java Applet
• Stereographic Projection and Inversion from cut-the-knot
• Examples of miniplanet panoramas, majority in UK
• Examples of miniplanet panoramas, majority in Czech Republic
• Examples of miniplanet panoramas, majority in Poland
• DoITPoMS Teaching and Learning Package- "The Stereographic Projection"
• Sphaerica software is capable of displaying spherical constructions in stereographic projection
• Proof about Stereographic Projection taking circles in the sphere to circles in the plane
• Free and open source python program for stereographic projection ---PTCLab