Stereographic Projection : To Map the Stars

The Mathematics of Stereographic Projection

When making a map of the night sky, astronomers don’t have the luxury of mapping only flat land in a relatively limited area; they must create an accurate map that accounts for the spherical geometry of the celestial sphere. This is a problem that astronomers have puzzled over since the mapping of the stars began. One such way, and indeed the most popular for celestial mapping, is stereographic projection, and it is at least 2,000 years old. Hipparchus (~190-120 BCE) is credited with inventing it, though it is reasonable to think that it existed in some form earlier. Stereographic projection survives because it preserves circles and angles.

Maps of a sphere and the stars are always compromises, because no flat sheet can hold a curved surface without stretching something. Cartographers and mathematicians tend to disagree about which compromise is best, since they want different things from a map. The oldest scientific instrument still recognizable today, the astrolabe, is built on it, and the same construction turns up at the center of complex analysis more than two thousand years later.

The Construction

The celestial sphere is conceptualized as a glass orb with the stars painted on the inner surface. Stereographic projection uses it to map those stars onto a flat surface. Picture our celestial sphere suspended in space, with a sheet of paper pinned at the North Pole and a very bright light shining at the South Pole, passing through the sphere and illuminating the paper below; the stars, being opaque, cast a shadow. That shadow becomes the star’s position on the flat paper. The same would hold true for a light situated at the North Pole and the paper pinned at the South Pole, which would map the opposite hemisphere, though stars near the celestial equator would appear in both. The only point that cannot be mapped is the point of light itself, which maps to infinity and falls outside the chart.

Stripped to its geometry, the recipe idea is simple. Start with a sphere. Pick one point on it to be the source of light, the projection point; for a chart of the northern sky you would put it at the celestial south pole. Then take the flat plane on the far side, either tangent to the sphere at the opposite pole or passing through the equator, it changes only the scale. For any star on the sphere, draw the straight line from the projection point through that star and continue it until it strikes the plane. Where it lands is the star’s place on the map. Every point on the sphere has exactly one such landing place, with the single exception already noted: the projection point itself, whose line never reaches the plane and so maps to infinity. That is the projection’s only real flaw, and it is a flaw with an easy fix, since you can project the other hemisphere from the opposite pole.

For those who want the precise math: a point (x, y, z) on a unit sphere with the projection point at the South Pole (0, 0, -1) maps to (x/(1+z), y/(1+z)) on the flat plane. As a star slides toward the projection point, the denominator shrinks toward zero and the mapped point races off to infinity, which is the same fact stated in algebra rather than in light and shadow.

Three Properties of Note

Three properties explain why this projection has outlasted every competitor for charting the sky. The first two are the reasons a star chart looks right. The third is the reason mathematicians care about the projection for its own sake.

Circles project to circles

Great circles are circles whose center is the center of the sphere. The celestial equator and the ecliptic (the apparent yearly path the sun takes against the background stars) are both classed as great circles.

Small circles are circles on the sphere whose center is not the center of the sphere; lines of constant declination are small circles. In stereographic projection, both great circles and small circles project onto the flat plane as circles, which is rare among projections. Mercator, for example, turns small circles into wavy curves. The one exception is a circle that passes through the projection point itself, which projects to a straight line; mathematicians treat this as a circle of infinite radius.

For a star chart this is what keeps the coordinate grid accurate. The celestial equator, the lines of constant declination, the ecliptic, and the arcs that make up constellation boundaries are all circles on the sphere, so they all arrive on the chart as clean arcs rather than the wavy curves a less forgiving projection would produce. Most equal-area projections do worse still, bending small circles into ellipses.

Angles are preserved

Another benefit of stereographic projection is that local angles are preserved, meaning a small constellation retains its recognizable shape on the chart. The grid of right ascension and declination meets at right angles on the sphere, and still meets at right angles on the chart. If two curves cross at some angle up on the sphere, their shadows cross at the same angle down on the plane. Mathematicians call a map with this property conformal.

The reason shapes survive is that, in any small neighborhood, the projection behaves like an accurate zoom. Projection can magnify a region, but it does not stretch one direction more than another and it does not shear. Technically the projection’s Jacobian at every point is a scalar multiple of a rotation, which is the formal way of saying the same thing: locally, just scaling, no distortion of shape. That is why a small constellation looks like itself even when its size on the page is wrong.

It is the conformal map of the sphere

The third property makes stereographic projection the only mathematically accurate solution. There are countless ways to flatten a sphere, but if you insist on conformality, on preserving angles everywhere, there is only one map that works with a sphere minus a single point onto the whole plane. Stereographic projection is that map. Not a good conformal projection among several, but the only conformal one.

This is where the night sky connects to one of the deeper ideas in mathematics. Treat the flat plane as the plane of complex numbers and add a single point at infinity, the very point the projection sent off the edge of the chart, and the sphere becomes what mathematicians call the Riemann sphere: the complex numbers wrapped up into a closed surface with nothing missing. Stereographic projection is the bridge that identifies the two. On that picture, the transformations that move the complex plane around in the gentlest possible way, the Mobius transformations, correspond exactly to rotations and rigid motions of the sphere. Rotating the sky is the same act as a particular transformation of the complex plane. The shadow-casting trick that lets us draw stars on paper turns out to be the natural geometry of the number system itself, which is not something the ancient astronomers could have known they were reaching toward.

The Trade-off

Area distortion is the price to pay: the further from the center of the projection, the more area and scale are exaggerated. This is why star charts using stereographic projection typically show a hemisphere or less, and why stars near the horizon appear more spread out than they do in the sky.

The growth is steep and can be written down exactly. For a unit sphere, a small patch at distance r from the center of the chart is enlarged by a factor of (1 + r-squared/4) squared in area. Near the center that factor is close to one and the map is almost faithful; far out it climbs quickly, and at the projection point itself it becomes infinite, which is the area-language version of the point that maps to infinity.

So the projection trades away any honest sense of size in exchange for the circles and angles it protects. For a sky chart that is usually a fair bargain, because size is rarely what you are reading off a star map, and the stars whose size is most exaggerated are the ones lowest toward the horizon, where they are hardest to see clearly in the first place.

An Ancient History

Stereographic projection has an ancient history. The Greek scientist Hipparchus (~190-120 BCE) is credited with its discovery, though it is possible that earlier Egyptian and Babylonian astronomers used a form as well. Ptolemy (~100-170 CE) wrote about stereographic projection in his work Planisphere, describing how to use it to make star maps and astrolabes.

An astrolabe is a physical implementation of stereographic projection. Its rete, the rotating star map layer, is a stereographic projection of the celestial sphere onto a flat disc, allowing the user to solve astronomical problems such as measuring star altitude and telling time by the stars. During the Islamic golden age (8th-13th centuries CE), scholars, including Al-Biruni and al-Tusi, refined the mathematics and built thousands of astrolabes. Later, European explorers carried the astrolabe across the open ocean, and Mercator (1512-1594) applied stereographic projection to polar regions even on his famous map.

The projection never fell out of use, it only changed fields. Crystallographers plot the symmetry of crystals on stereographic nets, called Wulff nets, for exactly the angle-preserving reason that served the astrolabe. It remains a working tool in cartography, and through the Riemann sphere it sits permanently inside complex analysis. The same construction keeps being rediscovered wherever someone needs to flatten a sphere without losing its angles.

Why Choose This Projection, for the Sky

The projection the ancients relied on is the same math we use today, because the problem hasn’t changed: how do we represent a curved surface on a flat one? For star charts, what matters most is that local shapes are preserved and that the coordinate grid of right ascension and declination remains clean and readable. Stereographic projection handles both well. Because circles project as circles, the grid lines of right ascension and declination stay as clean arcs rather than wavy curves. Because angles are preserved locally, small constellations retain their recognizable shapes. The edge distortion drawback matters less than it might seem. Distortion grows as you move toward the horizon, and the horizon is already the limit of what is visible. More elegantly, the projection point, where distortion becomes infinite, is the nadir, the point directly opposite the observer’s zenith, which is underground and off the chart entirely. The projection’s one flaw lands exactly where we cannot see anyway.