In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the pole or center of projection), onto a plane (the projection 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 (distance preserving) nor equiareal (area preserving). 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 (2nd century AD), but it was ambiguously attributed to Hipparchus (2nd century BC) by Synesius (c. 400 AD), and Apollonius's Conics (c. 200 BC) contains a theorem which is crucial in proving the property that the stereographic projection maps circles to circles. Hipparchus, Apollonius, Archimedes, and even Eudoxus (4th century BC) 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 anaphoric clock described by Vitruvius (1st century BC). By the time of Theon of Alexandria (4th century), the planisphere had been combined with a dioptra to form the planispheric astrolabe ("star taker"), 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 (1542), Rumold Mercator (1595), 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 (Six Books of Optics, useful for philosophers and mathematicians alike). 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 S 2 {\displaystyle {\mathcal {S}}^{2}} in three-dimensional space R 3 {\displaystyle \mathbb {R} ^{3}} is the set of points ( x , y , z ) {\displaystyle (x,y,z)} such that x 2 + y 2 + z 2 = 1 {\displaystyle x^{2}+y^{2}+z^{2}=1} . Let N = ( 0 , 0 , 1 ) {\displaystyle N=(0,0,1)} be the "north pole", and let M {\displaystyle {\mathcal {M}}} be the rest of the sphere. The plane z = 0 {\displaystyle z=0} runs through the center of the sphere; the "equator" is the intersection of the sphere with this plane. For any point P {\displaystyle P} on M {\displaystyle {\mathcal {M}}} , there is a unique line through N {\displaystyle N} and P {\displaystyle P} , and this line intersects the plane z = 0 {\displaystyle z=0} in exactly one point P ′ {\displaystyle P'} , known as the stereographic projection of P {\displaystyle P} onto the plane. In Cartesian coordinates ( x , y , z ) {\displaystyle (x,y,z)} on the sphere and ( X , Y ) {\displaystyle (X,Y)} on the plane, the projection and its inverse are given by the formulas
( X , Y ) = ( x 1 − z , y 1 − z ) , ( x , y , z ) = ( 2 X 1 + X 2 + Y 2 , 2 Y 1 + X 2 + Y 2 , − 1 + X 2 + Y 2 1 + X 2 + Y 2 ) . {\displaystyle {\begin{aligned}(X,Y)&=\left({\frac {x}{1-z}},{\frac {y}{1-z}}\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).\end{aligned}}}
In spherical coordinates ( φ , θ ) {\displaystyle (\varphi ,\theta )} on the sphere (with φ {\displaystyle \varphi } the zenith angle, 0 ≤ φ ≤ π {\displaystyle 0\leq \varphi \leq \pi } , and θ {\displaystyle \theta } the azimuth, 0 ≤ θ ≤ 2 π {\displaystyle 0\leq \theta \leq 2\pi } ) and polar coordinates ( R , Θ ) {\displaystyle (R,\Theta )} on the plane, the projection and its inverse are
( R , Θ ) = ( sin φ 1 − cos φ , θ ) = ( cot φ 2 , θ ) , ( φ , θ ) = ( 2 arctan 1 R , Θ ) . {\displaystyle {\begin{aligned}(R,\Theta )&=\left({\frac {\sin \varphi }{1-\cos \varphi }},\theta \right)=\left(\cot {\frac {\varphi }{2}},\theta \right),\\(\varphi ,\theta )&=\left(2\arctan {\frac {1}{R}},\Theta \right).\end{aligned}}}
Here, φ {\displaystyle \varphi } is understood to have value π {\displaystyle \pi } when R = 0 {\displaystyle R=0} . Also, there are many ways to rewrite these formulas using trigonometric identities. In cylindrical coordinates ( r , θ , z ) {\displaystyle (r,\theta ,z)} on the sphere and polar coordinates ( R , Θ ) {\displaystyle (R,\Theta )} on the plane, the projection and its inverse are
( R , Θ ) = ( r 1 − z , θ ) , ( r , θ , z ) = ( 2 R 1 + R 2 , Θ , R 2 − 1 R 2 + 1 ) . {\displaystyle {\begin{aligned}(R,\Theta )&=\left({\frac {r}{1-z}},\theta \right),\\(r,\theta ,z)&=\left({\frac {2R}{1+R^{2}}},\Theta ,{\frac {R^{2}-1}{R^{2}+1}}\right).\end{aligned}}}
Other conventions
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). 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 (a ratio of a diameter to a radius of the sphere); hence, 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. Other authors use a sphere of radius 1/2 and the plane z = −1/2. In this case the formulae become
( x , y , z ) → ( ξ , η ) = ( x 1 2 − z , y 1 2 − z ) , ( ξ , η ) → ( x , y , z ) = ( ξ 1 + ξ 2 + η 2 , η 1 + ξ 2 + η 2 , − 1 + ξ 2 + η 2 2 + 2 ξ 2 + 2 η 2 ) . {\displaystyle {\begin{aligned}(x,y,z)\rightarrow (\xi ,\eta )&=\left({\frac {x}{{\frac {1}{2}}-z}},{\frac {y}{{\frac {1}{2}}-z}}\right),\\(\xi ,\eta )\rightarrow (x,y,z)&=\left({\frac {\xi }{1+\xi ^{2}+\eta ^{2}}},{\frac {\eta }{1+\xi ^{2}+\eta ^{2}}},{\frac {-1+\xi ^{2}+\eta ^{2}}{2+2\xi ^{2}+2\eta ^{2}}}\right).\end{aligned}}}
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.
Generalizations More generally, stereographic projection may be applied to the unit n-sphere Sn in (n + 1)-dimensional Euclidean space En+1. If Q is a point of Sn and E a hyperplane in En+1, then the stereographic projection of a point P ∈ Sn − {Q} is the point P′ of intersection of the line QP with E. In Cartesian coordinates (xi, i from 0 to n) on Sn and (Xi, i from 1 to n) on E, the projection from Q = (1, 0, 0, ..., 0) ∈ Sn is given by
X i = x i 1 − x 0 ( i = 1 , … , n ) . {\displaystyle X_{i}={\frac {x_{i}}{1-x_{0}}}\quad (i=1,\dots ,n).}
Defining
s 2 = ∑ j = 1 n X j 2 = 1 + x 0 1 − x 0 , {\displaystyle s^{2}=\sum _{j=1}^{n}X_{j}^{2}={\frac {1+x_{0}}{1-x_{0}}},}
the inverse is given by
x 0 = s 2 − 1 s 2 + 1 and x i = 2 X i s 2 + 1 ( i = 1 , … , n ) . {\displaystyle x_{0}={\frac {s^{2}-1}{s^{2}+1}}\quad {\text{and}}\quad x_{i}={\frac {2X_{i}}{s^{2}+1}}\quad (i=1,\dots ,n).}
Still more generally, suppose that S is a (nonsingular) quadric hypersurface in the projective space Pn+1. In other words, S is the locus of zeros of a non-singular quadratic form f(x0, ..., xn+1) in the homogeneous coordinates xi. Fix any point Q on S and a hyperplane E in Pn+1 not containing Q. Then the stereographic projection of a point P in S − {Q} is the unique point of intersection of QP with E. 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" (i.e., (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(x, y, z) on the sphere and its image P′(X, Y) on the plane either both are rational points or none of them:
P ∈ Q 3 ⟺ P ′ ∈ Q 2 {\displaystyle P\in \mathbb {Q} ^{3}\iff P'\in \mathbb {Q} ^{2}}
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 (X, Y) coordinates by
d A = 4 ( 1 + X 2 + Y 2 ) 2 d X d Y . {\displaystyle dA={\frac {4}{(1+X^{2}+Y^{2})^{2}}}\;dX\;dY.}
Along the unit circle, where X2 + Y2 = 1, there is no inflation of area in the limit, giving a scale factor of 1. Near (0, 0) areas are inflated by a factor of 4, and near infinity areas are inflated by arbitrarily small factors. The metric is given in (X, Y) coordinates by
4 ( 1 + X 2 + Y 2 ) 2 ( d X 2 + d Y 2 ) , {\displaystyle {\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. 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 x , y , z {\displaystyle x,y,z} in terms of X , Y , Z , {\displaystyle X,Y,Z,} given in § First formulation: using these expressions for a substitution in the equation a x + b y + c z − d = 0 {\displaystyle ax+by+cz-d=0} 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 ( c − d ) ( X 2 + Y 2 ) {\displaystyle (c-d)(X^{2}+Y^{2})} as its quadratic part. The equation becomes linear if c = d , {\displaystyle c=d,} 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. (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
R = e Θ / a , {\displaystyle 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.
△ N O P ′ ∼ △ P ′ ′ O S ⟹ O P ′ : O N = O S : O P ′ ′ ⟹ O P ′ ⋅ O P ′ ′ = r 2 {\displaystyle \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}}
Metric The associated metric tensor of the specific stereographic projection described in the beginning of this section (which sends the "south pole" (0, 0, −1) of the unit sphere to (0, 0), and the equator to the unit circle) is given by
d s 2 = 4 ∑ i d x i 2 ( 1 + ∑ i x i 2 ) 2 = 4 ‖ d x ‖ l 2 ( 1 + ‖ x ‖ l 2 ) 2 {\displaystyle ds^{2}=4{\frac {\sum _{i}dx_{i}^{2}}{\left(1+\sum _{i}x_{i}^{2}\right)^{2}}}={\frac {4\,\lVert d\mathbf {x} \rVert {\vphantom {l}}^{2}}{{\bigl (}1+\lVert \mathbf {x} \rVert {\vphantom {l}}^{2}{\bigr )}^{2}}}}
where the xi are the Cartesian coordinates of the Euclidean plane onto which the sphere is projected. In comparison, the equation for the corresponding metric of the Poincaré disk model of hyperbolic space looks equivalent, except for a sign difference in the denominator.
Wulff net
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 Russian mineralogist George (Yuri Viktorovich) Wulff. The Wulff net shown here is the stereographic projection of the grid of parallels and meridians of a hemisphere centred at a point on the equator (such as the Eastern or Western hemisphere of a planet). 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 or left. The two sectors have equal areas on the sphere. On the disk, the latter has nearly four times the area of the former. If the grid is made finer, this ratio approaches exactly 4. On the Wulff net, the images of the parallels and meridians intersect at right angles. This orthogonality property is a consequence of the angle-preserving property of the stereographic projection. (However, the angle-preserving property is stronger than this property. Not all projections that preserve the orthogonality of parallels and meridians are angle-preserving.)
For an example of the use of the Wulff net, imagine two copies of it on thin paper, one atop the other, aligned and tacked at their mutual center. Let P be the point on the lower unit hemisphere whose spherical coordinates are (140°, 60°) and whose Cartesian coordinates are (0.321, 0.557, −0.766). 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 to plotting P:
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 plot other points, whose angles are not such round numbers as 60° and 5