The stereographic projection is one way of projecting the points that lie on a spherical surface onto a plane. Such projections are commonly used in Earth and space mapping where the geometry is often inherently spherical and needs to be displayed on a flat surface such as paper or a computer display. Any attempt to map a sphere onto a plane requires distortion, stereographic projections are no exception and indeed it is not an ideal approach if minimal distortion is desired.
A physical model of stereographic projections is to imagine a transparent sphere sitting on a plane. If we call the point at which the sphere touches the plane the south pole then we place a light source at the north pole. Each ray from the light passes through a point on the sphere and then strikes the plane, this is the stereographic projection of the point on the sphere.
In order to derive the formulae for the projection of a point (x,y,z) lying on the sphere assume the sphere is centered at the origin and is of radius r. The plane is all the points z = -r, and the light source is at point (0,0,r). The cross section of this arrangement is shown below in what is commonly called a Schlegal diagram.
Consider the equation of the line from P1 = (0,0,r) through a point
P2 = (x,y,z) on the sphere,
The following example is taken from the mapping of EEG data recorded on an approximate hemisphere (human head). The data can be rendered on a virtual hemisphere but as such the whole field is not readily visible from any particular viewpoint. The best option is to view the data from the top of the head but the effects around the rim are hard to interpret due to the compression of information as a result of the curvature of the surface.
The following shows a planar projection from the hemisphere on the left and the same data with a stereographic projection. The compression near the rim is clearly reduced greatly improving the visibility of the results in that region.
Note that in the above, after the projection has been performed, the resulting disk is scaled by a factor of 0.5 in order to retain the same dimensions as the hemisphere.
The general cylindrical projection is one where lines of latitude are projected to equally spaced parallel lines and lines of longitude are projected onto not necessarily equally spaced parallel lines. The diagram below illustrates the basic projection, a line is projected from the centre of the sphere through each point on the sphere until it intersects the cylinder.
The equations are quite straightforward, if the cylinder is unwrapped and the horizontal axis is x and the vertical axis is y (origin in the vertical center and on the left side horizontally) then:
A Mercator projection is related but has a different distortion in the spacing of the lines of longitude. This is one of the more common projections used in mapping the Earth onto a flat surface. The equations are:
Cylindrical projections in general have an increased vertical stretching as one moves towards either of the poles. Indeed, the poles themselves can't be represented (except at infinity). This stretching is reduced in the Mercator projection by the natural logarithm scaling.
While not strictly a projection, a common way of representing spherical surfaces in a rectangular form is to simply use the polar angles directly as the horizontal and vertical coordinates. Since longitude varies over 2 pi and latitude only over pi, such polar maps are normally presented in a 2:1 ratio of width to height. The most noticeable distortion in these maps is the horizontal stretching that occurs as one approaches the poles from the equator, this culminates in the poles (a single point) being stretched to the whole width of the map.
An example of such a map is given below for the Earth.
While such maps are rarely used in cartography, they are very popular in computer graphics since it is the standard way of texture mapping a sphere.....hence the popularity of maps of the Earth as shown above.
An Aitoff map projection (attributed to David Aitoff circa 1889) is a class of azimuthal projection, basically an azimuthal equidistant projection where the longitude values are doubled (squeezing 2pi into pi) and the resulting 2D map is stretched in the horizontal axis to form a 2:1 ellipse. In a normal azimuthal projection all distances are preserved from the tangent plane point, this is not the case for a Aitoff projection, except along the vertical and horizontal axis. A modification to the Aitoff projection is the Hammer-Aitoff projection which has the property of preserving equal area over the whole map.
Conversion from longitude/latitude to Hammer-Aitoff coordinates (x,y)Consider longitude to range between -pi and pi, latitude between -pi/2 and pi/2.
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z2 = 1 + cos(latitude) cos(longitude/2)
x = cos(latitude) sin(longitude/2) / z y = sin(latitude) / z |
(x,y) are each normalised coordinates, -1 to 1.
Conversion of Hammer-Aitoff coordinates to longitude/latitude
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z2 = 1 - x2/2 - y2/2
longitude = 2 atan(sqrt(2) x z / (2 z2 - 1)) latitude = asin(sqrt(2) y z) |
The Hammer-Aitoff map is limited to where (x longitude) >= 0.
Example: Conversion of longitude/latitude to Hammer-Aitoff coordinates
 Grid test pattern, eg: spherical panoramic map |
 Resulting Hammer-Aitoff projection |
Example: Conversion of Hammer-Aitoff coordinates to longitude/latitude
 Cosmic microwave background |
 Spherical projection |