Stereographic Projection
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{{SwitchPreviewHideMessage=Click here to hide.ShowMessage=Click here to show.  {{SwitchPreviewHideMessage=Click here to hide.ShowMessage=Click here to show.  
  PreviewText=Derivation of Eq. 2.FullText=:This inverse  +  PreviewText=Derivation of Eq. 2.FullText=:This inverse function is derived by substituting the coordinates from {{EquationNoteEq. 1}} back into the equation for the unit sphere and solving for ''z''. 
::<math>x^2 + y^2 + z^2 = 1</math>  ::<math>x^2 + y^2 + z^2 = 1</math>  
  ::<math>(X(1z))^2 + (Y(1z))^2 + z^2 = 1</math>  +  ::<math>(X(1z))^2 + (Y(1z))^2 + z^2 = 1</math> (substitute) 
  ::<math>(X^2 + Y^2)(z^2  2z + 1) + z^2  1 = 0</math>  +  ::<math>(X^2 + Y^2)(z^2  2z + 1) + z^2  1 = 0</math> (factor and expand) 
  ::<math>(X^2 + Y^2 + 1)z^2  2(X^2 + Y^2)z + (X^2 + Y^2  1) = 0</math>  +  ::<math>(X^2 + Y^2 + 1)z^2  2(X^2 + Y^2)z + (X^2 + Y^2  1) = 0</math> (distribute and factor) 
+  
+  :This latter equation is is a quadratic in ''z'', so we can solve for ''z'' using the quadratic formula, obtaining:  
+  
::<math>z = 1, \frac{X^2 + Y^2  1}{X^2 + Y^2 + 1}</math>  ::<math>z = 1, \frac{X^2 + Y^2  1}{X^2 + Y^2 + 1}</math>  
:The first solution may be discarded because ''T''(0, 0, 1) is the pole of the circle; it is the one point on the sphere for which stereographic projection is not defined. (Think about why it would not make sense to map ''T'' onto the plane. We would have to draw a line from ''T'' to ''T'', but The plane tangent to the sphere at ''T'' is parallel to the ''x''''y'' plane onto which we are projecting, so any line tangent to the sphere at ''T'' will never pass through the plane.) The second solution is what we expected.  :The first solution may be discarded because ''T''(0, 0, 1) is the pole of the circle; it is the one point on the sphere for which stereographic projection is not defined. (Think about why it would not make sense to map ''T'' onto the plane. We would have to draw a line from ''T'' to ''T'', but The plane tangent to the sphere at ''T'' is parallel to the ''x''''y'' plane onto which we are projecting, so any line tangent to the sphere at ''T'' will never pass through the plane.) The second solution is what we expected.  
+  
+  :An alternate route to getting this result is to note that the line drawn from ''T'' to ''Q'' intersects the sphere at two points: at ''T'' and ''P''! We know that one result will be ''z''<sub>1</sub> = 1, and we must find the other. If we divide the previous quadratic by the coefficient of the secondorder term, we get:  
+  
+  ::<math>z^2  \frac{2(X^2 + Y^2)}{X^2 + Y^2 + 1}z + \frac{X^2 + Y^2  1}{X^2 + Y^2 + 1} = 0</math>  
+  
+  :The constant term ''c'' in a quadratic of the form ''x''<sup>2</sup> + ''bx'' + ''c'' = 0 must be equal to the product of the roots. As noted, ''z''<sub>1</sub> = 1 is one solution, so  
+  
+  ::<math>z_1 \cdot z_2 = c</math>  
+  ::<math>1 \cdot z_2 = \frac{X^2 + Y^2  1}{X^2 + Y^2 + 1}</math>  
+  ::<math>z_2 = \frac{X^2 + Y^2  1}{X^2 + Y^2 + 1}</math>  
+  
+  The other solution is, again, what we expected for the coordinate of ''z''.  
+  
:In order to find inverse expressions for ''x'' and ''y'', we return once again to the equation for our sphere.  :In order to find inverse expressions for ''x'' and ''y'', we return once again to the equation for our sphere. 
Revision as of 16:58, 26 June 2013
 Stereographic projection maps each point on a sphere onto a plane.
Stereographic Projection of a Sphere 

Contents 
Basic Description
Stereographic projection is a method of mapping the surface of a sphere onto a plane.
A map corresponds each point on the sphere with a point on the plane. The process for mapping to the plane is to draw a line from the north pole, passing through both a point on the sphere and a point on the plane. The point on the sphere is mapped to the point on the plane.
The image to the left shows this process for a twodimensional crosssection of the sphere. In a way, the figure is an example of the unit circle being stereographically projected onto the x axis. The rest of this page will examine the threedimensional stereographic projection of the unit sphere onto the xy plane.
The main image illustrates stereographic projection. Here, the plane is drawn under the sphere instead of cutting through its equator. The coloring demonstrates how regions of the sphere are mapped to corresponding the plane. The projection is still from the north pole of the sphere, and the bands of color are not centered around the vertical axis, the projection forms some interesting ellipses on the plane.
The following applet demonstrates how a sphere is projected onto a plane. A sphere with coaxial bands of color is stereographically projected onto a plane in the background. Rotating the sphere with the mouse will change the orientation of the colors on the sphere relative to the north pole changes the projection on the plane. The sphere and north pole remain fixed; only the colors are shifted.
A More Mathematical Explanation
Coordinates
A stereographic projection maps the points of a sphere onto a plane.
Specifically, let's consider the unit sphere centered at the origin. We define S as the set of all points (x, y, z) that satisfy the equation x^{2} + y^{2} + z^{2} = 1. We define the north pole of the sphere, the point (0, 0, 1), as T. We define H as the xy plane, the horizontal plane that passes through the origin, or the set of all points (x, y, z) that satisfy the equation z = 0.
A line drawn from T through some point P on the sphere S (for P ≠ T) will also intersect the plane H at a point which we define as Q. We say that Q is the stereographic projection of P, and P is mapped to Q by stereographic projection. Figure 2 to the right shows how these points are projected. Our goal in this section is to derive the formulas that map P to Q.
Using rectilinear coordinates, we will refer generally to point P (x, y, z) on the sphere S and point Q (X, Y, 0) on the xy plane H.
Mapping of S to H in rectangular coordinates:
 .
Derivation of Eq. 1.
 Let's restate what is happening by returning to our definition of the sphere. The sphere's pole is at the point T (0, 0, 1). A line can be drawn through some point P (x, y, z) on the sphere and some point Q (X, Y, 0) on the plane. We consider the vectors drawn from T to P and from T to Q. By construction, these two vectors are colinear and parallel:
 .
 Since the vectors are parallel, their cross product is the 0 vector.
 We get three equations:
 These are the coordinates in Eq. 1.
Inverse of rectangular coordinates: Since coordinates on the sphere are mapped uniquely, or onetoone, to coordinates on the plane, this function is invertible. The explicit inverse is:
 .
Derivation of Eq. 2.
 This inverse function is derived by substituting the coordinates from Eq. 1 back into the equation for the unit sphere and solving for z.
 (substitute)
 (factor and expand)
 (distribute and factor)
 This latter equation is is a quadratic in z, so we can solve for z using the quadratic formula, obtaining:
 The first solution may be discarded because T(0, 0, 1) is the pole of the circle; it is the one point on the sphere for which stereographic projection is not defined. (Think about why it would not make sense to map T onto the plane. We would have to draw a line from T to T, but The plane tangent to the sphere at T is parallel to the xy plane onto which we are projecting, so any line tangent to the sphere at T will never pass through the plane.) The second solution is what we expected.
 An alternate route to getting this result is to note that the line drawn from T to Q intersects the sphere at two points: at T and P! We know that one result will be z_{1} = 1, and we must find the other. If we divide the previous quadratic by the coefficient of the secondorder term, we get:
 The constant term c in a quadratic of the form x^{2} + bx + c = 0 must be equal to the product of the roots. As noted, z_{1} = 1 is one solution, so
The other solution is, again, what we expected for the coordinate of z.
 In order to find inverse expressions for x and y, we return once again to the equation for our sphere.
 This is certainly a mess to simplify. But in the end we obtain the expected result:
 Since x and y are interchangeable for the purpose of these formulas, the same may be repeated for x to obtain:
 These are the coordinates in Eq. 2.
Projection in terms of spherical coordinates: A point on the sphere has rectangular coordinates , where is the polar angle formed from the positive z axis and is the azimuthal angle formed from the positive x axis. In spherical coordinates, we say equivalently (radius 1 for unit sphere). We can write the coordinates of the projection in terms of these spherical coordinates:
 where are polar coordinates of the stereographic projection.
This formulation of the coordinates is revealing. R, the distance of a projected point from the origin, depends entirely on , the polar angle, while the azimuthal angle is preserved by the projection.
Derivation of Eq. 3.
 By substituting spherical coordinates into the Cartesian coordinates of Eq. 1, we can use the Pythagorean Theorem to find R:
 That the spherical coordinate is the same as the polar coordinate should be unsurprising if we visualize the projection. Both are the azimuthal angle.
Inverse of polar coordinates: The inverse can be formulated in terms of spherical coordinates as well.
Derivation of Eq. 4.
 Using trigonometric identities, we can rearrange in terms of .
 We will make use of the trigonometric identity:
 The righthand side of Eq. 3 is the reciprocal of this identity, so
 .
 This can easily be rewritten:
 .
 Again, we complete our derivation by noting that the radius of the spherical point must be 1, and that .
Properties
Points on the upper hemisphere (z > 0) of the unit sphere are mapped outside of the unit circle on the xy plane (X ^{2} + Y ^{2} > 1). Points on the lower hemisphere (z < 0) of the unit sphere are mapped inside of the unit circle on the xy plane (X ^{2} + Y ^{2} < 1).
 We prove this by rewriting the righthand side of the equation R ^{2} = X ^{2} + Y ^{2} in terms of z using the coordinates in Eq. 1:
 For positive z, the numerator is greater than the denominator, so the projection falls outside of the unit circle. For negative z, the denominator is greater than the numerator, so the projection falls inside of the unit circle.
 This completes the proof.
The stereographic projection preserves circles. We distinguish between two possible cases:
 The circle on the sphere does not contain T. Then the projection of the circle onto the xy plane is a circle.
 This may be understood intuitively, if one visualizes a cone with its apex at T. The cone's intersection with the sphere would form a circle, as would the cone's intersection with the plane (although the two circles are generally not of the same radius).
 The circle on the sphere contains the north pole T. Then the stereographic projection of the circle onto the xy plane is a line.
 This case is probably less obvious. Recall that we've said that the stereographic projection of T is undefined. This fact can be seen in a variety of ways; for instance, the coordinates for X and Y in Eq. 1 would have denominators of 0 with z = 1. Geometrically, it is apparent that points very close to T will be projected very far from the origin on the xy plane. So we would expect projections of circles on the sphere containing T to extend very far away from the origin. Rather than circles, they are lines which extend infinitely.
Here we will also prove this theorem analytically.
 We recall that our sphere is defined:
 .
 Planes in are defined:
 .
 Any circle on the surface of a sphere is the intersection of a plane with the sphere:
 .
 Note that we are only interested in cases in which the plane intersects the sphere at more than one point. Some planes of this form either do not intersect the unit sphere or are tangent to the unit sphere, in which cases does not contain a circle.
 We want to show that the stereographic projection of W is a line on the plane when it includes T and a circle on the plane when it excludes T.
 Recall that T (0, 0, 1), P (x, y, z), Q (X, Y, 0) lie on a line, so
 for some real, nonzero t.
 In vector notation,
 so
 , , .
 In the previous proof, we showed that
 where R is the distance of the projected point Q from the origin, and Q (X, Y) are rectangular coordinates.
 By manipulating this relationship, we get
 and
 Now we consider the equation of the plane which intersects the sphere:
 By substituting, this becomes:
 Multiply by :
 Since ,
 To conclude, note that when , because . Furthermore, .
 If , then coefficients of and in Eq. 5 are 0, so Eq. 5 is the formula for a line in . Therefore, when , the stereographic projection of is a line.
 If , then the coefficients of and in Eq. 5 are the same, so Eq. 5 is the formula for a circle in . Therefore, when , the stereographic projection of is a circle.
 This completes the proof.
 Figure 6 is an illustration of this result. The circles which pass through the north pole are projected to lines on the plane. The circles which do not pass through the north pole are projected to circles on the plane.
 However, it is possible to rearrange Eq. 5 further. In the latter case where , we have
 By completing the square, this becomes
 Therefore, the stereographic projection of onto is a circle centered at
 with radius
 .
The stereographic projection is conformal, meaning that any angles formed on the surface of the sphere are the same as those projected onto the plane. This means that the projection preserves shape locally; the angles on the sphere and on the plane are the same at the point of intersection.
However, the stereographic projection does not preserve the area of regions on the sphere. This makes sense because the sphere has finite surface area, but is projected onto the whole of the xy plane (the projection can be inverted and any point on the xy plane can be mapped to a point on the sphere).
The image to the left shows this property. It is clear that the area of each enclosed region on the rectangular grid is not equal to that of its projection on the sphere. It also appears that the angles are preserved locally. These relationships can be proven.
An important mathematical application of stereographic projections in complex analysis is the Riemann Sphere. In the Riemann sphere, the north pole T of the sphere is the point at infinity (recall that T cannot be projected onto the plane, while the coordinates of the projection become much larger as z approaches 1). The Riemann sphere therefore constitutes the extended complex plane, the union of the complex numbers with infinity.
Why It's Interesting
Stereographic projections are used in cartography and photography.Cartography
Cartographers have always struggled to create maps that are as accurate and usable as possible. Map projections are projections of earth's surface as an idealized threedimensional sphere onto a twodimensional plot. (The earth is actually closer to an oblate spheroid, and is of course contoured.) Each type of map projection has its merit, but unfortunately, no map projection can be ideal; they cannot be both conformal and equiareal, but cartographers and geometers have left us many to choose from.
The stereographic projection is one such method of mapping. Stereographic projections are chosen because they are conformal and are not distorted much locally. By locally, we mean near the pole opposite of the projection.
The two stereographic maps in Figure X are polar stereographic projections. The center of the map is the north pole; the points on the sphere of the earth are projected from the south pole. (In other words, the point T from the More Mathematical Explanation is the south pole.)
Since these two stereographic maps are are both projected from the same point, they are technically the same projection. They appear different because the second is more "zoomed in." It is limited to the upper hemisphere. The red circle in the first map denotes the equator, outside of which (in the southern hemisphere) the map becomes rather distorted. The second, more local map is contained within the red circle on the first map.
This shows how the map can be distorted at points further from the pole. While the United States, Greenland, and Russia may appear fairly "normal" on these maps, one would probably not use it, for instance, around Chile or Antarctica. However, if one wanted to navigate the lower hemisphere, then one could create another polar stereographic map projection from the north pole. Such a map would be accurate around the south pole and inaccurate in the upper hemisphere.
Another type of stereographic projection is transverse, as in the two maps of Figure Z.
The process for creating a transverse stereographic projection is no different. One projects from some point on the equator rather than from one of the poles. The advantages of such a map are obvious: it is accurate in those places where most people live and travel. One might observe that the second map in Figure Z perhaps most closely resembles other common map projections.
This map is centered at 0°N 0°E.
Mathematically, the general case discussed in the More Mathematical Explanation is a projection onto the plane z = 0. Often, for the purpose of map projections, it makes sense to project the sphere representing the earth onto a plane tangent to a pole of the sphere (for example, z = 1 or z = 1, assuming the unit sphere).
How do stereographic maps compare to other map projections? Shown below are two other map projections, the Mercator projection and the GallPeters projection, both of which have been popular in the past. Readers are perhaps familiar with the the Mercator projection, which, like a stereographic projection, is conformal. The GallPeters projection is not conformal. It is equiareal, so although the shapes of the landmasses appear distorted, their sizes are all properly relative.


We can see that different map projections produce very different maps! Depending on what the map is being used for, these differences can be very relevant. Conformal maps tend to be useful for navigation but do not give the best perspective of the relative size of landmasses. Compare the sizes of Greenland and Antarctica between these two maps.
Stereographic map projections are azimuthal. Azimuthal projections are so called because they preserve the azimuth of each point on the sphere. In other words, direction is preserved at the point around which the map is centered; great circles passing through the central point on the sphere are projected to lines passing through the central point on the plane.
Azimuthal maps may be thought of as projections from some "light source" on the axis of the sphere onto a flat piece of paper tangent to the sphere. The light source, in stereographic projection, is at one of the poles of the sphere.
Figure C compares stereographic projections to two other types of azimuthal projections. In gnomonic projection, the light source is at the center of the sphere. In orthographic projection, the light source is at an "infinite" distance, so the light beams are parallel. As such, gnomonic projection and orthographic projection are both limited to mapping just one hemisphere to the plane, while stereographic projection can map the entire sphere. Furthermore, gnomonic projection, like stereographic projection, can map to the whole of the xy plane, but orthographic projection only maps within the unit circle on the xy plane.
Photography
Stereographic projections are used in photography to produce impressive, beautiful images.^{[1]}
Such an image as the one to the left is created by taking a panoramic photo. This involves taking a photo, rotating by some fixed angle, and taking another photo. When the photos cover 360°, they may be strung together to form a panoramic photo. It is possible then to map the photo as a texture on a sphere, which can then be projected onto a plane to produce the image on the right of the figure.
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About the Creator of this Image
Thomas F. Banchoff is a geometer, and a professor at Brown University since 1967.
References
 ↑ Bourke, Paul (2011). "Little Planet" Photographs. University of Western Australia.
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