Stereographic projection

In mathematics, a stereographic projection is a special way of showing a round ball as a flat piece of paper. Imagine shining a light from one point on the ball and projecting the shadow onto a flat surface. This creates a smooth and exact match between every point on the ball—except the point where the light shines—and points on the paper.

This method keeps special properties, like the angles between lines, which makes it very useful. It turns circles on the ball into circles or straight lines on the paper. Though it doesn’t keep distances or areas the same, it helps us study the ball using flat paper instead of curved space.

Because spheres and flat surfaces appear in many parts of math and science, stereographic projection is used in areas like complex analysis, cartography, geology, and photography. Sometimes people use special graph paper called a stereographic net to help with these calculations.

History

See also: Astrolabe § History

We do not know exactly when the stereographic projection was first used, but it seems that ancient Greek astronomers may have used it. They used it to draw the sky onto flat paper so they could study the stars and planets more easily. The oldest known description of this method appears in a book by Ptolemy from the year 100 AD, called Planisphere[/w/4].

Later, this idea was used in a tool called the astrolabe, which helped astronomers measure the positions of stars. The astrolabe was used for many years and improved by astronomers in many places.

In the 1500s and 1600s, people began using this method to make maps of the Earth. Even today, this way of drawing helps us understand both the sky and the Earth better.

Definition

The unit sphere in three-dimensional space is a set of points where the distances from the center all equal one. Imagine a ball with radius one; every point on its surface is part of this sphere.

Stereographic projection is a way to "flatten" the sphere onto a flat plane. We pick a special point on the sphere, called the "north pole," and imagine lines stretching from this point to every other point on the sphere. Where each line meets a flat plane placed below the sphere becomes the projected point. This creates a special mapping from the round sphere to the flat plane.

This method keeps angles the same, meaning shapes that look uniform on the sphere stay looking smooth and consistent when flattened. It turns circles on the sphere into circles or straight lines on the plane.

Properties

The stereographic projection is a way to draw a sphere on a flat surface. It works by choosing a point on the sphere, called the "south pole," and projecting every other point on the sphere onto a flat plane. This creates a special connection between the sphere and the plane.

This kind of drawing keeps angles the same, which means shapes look similar up close. However, it does not keep areas the same—some parts get stretched more than others. Circles on the sphere become circles or straight lines on the plane, depending on where they are.

Wulff net

Stereographic projection plots can be created using special graph paper called a stereonet or Wulff net, named after the scientist George Wulff. This paper shows a map of a half-sphere, like one side of a planet.

The Wulff net shows lines that help us place points on the sphere. When we look at areas near the center of the net compared to areas far out, we can see the shapes change size. This happens because the way we flatten the sphere onto a flat paper changes how big things look.

To use the Wulff net to draw a point, you can follow a few steps. First, find where the point would be on the edge of the net. Then, move the top layer of the net to line up with the bottom layer. Next, mark where the point should be closer to the center. Finally, move the top layer back to match the bottom layer. The spot you marked is where the point should be drawn.

If the point’s angles are not easy numbers, you will need to guess between the lines on the net. Having a net with smaller lines, like every 2 degrees, can help make this easier.

Applications within mathematics

Complex analysis

A stereographic projection can map almost the entire sphere to a flat plane, missing just one point. By using two projections from different points, the whole sphere can be covered. These projections help describe the sphere as a special kind of surface.

This idea is important in complex analysis. Points on the plane can be linked to complex numbers, and the projection from the north pole to the middle plane creates a special link. Similarly, projecting from the south pole uses another set of complex numbers. These projections help create a useful way to think about infinity in complex numbers and support the study of special functions on what is called the Riemann sphere.

Visualization of lines and planes

Lines through the center of space form a structure that is hard to picture in three dimensions. Stereographic projection helps by turning these lines into points on a disk. Horizontal lines appear at the edge of the disk, and each line through the center matches a point inside or on the edge of the disk.

Planes through the center cut the sphere in circles, which become circles or arcs on the disk. Each plane also has a special line through the center, called its pole, which can be shown as a point on the disk. This makes it easier to picture many planes at once.

Other visualization

Stereographic projection is also used to help visualize shapes with many sides. By projecting a shape from a higher dimension to a sphere and then to a flat space, it becomes easier to see and understand.

Arithmetic geometry

Stereographic projection helps describe special sets of whole-number triples that follow Pythagoras' rule. Projecting from a point on the unit circle to the x-axis links rational points on the circle to points on the axis, leading to a method for finding these triples.

Tangent half-angle substitution

Main article: Tangent half-angle substitution

Stereographic projection offers another way to describe points on the unit circle using trigonometric functions. This change can make solving certain math problems with trigonometric functions simpler.

Applications to other disciplines

Cartography

Main article: Stereographic map projection

Maps try to show the round Earth on a flat piece of paper, but they can’t show everything perfectly. Some maps keep areas the same size, while others keep angles the same. Stereographic projection keeps angles the same, which is useful for navigation. When centered at the North or South Pole, it shows lines of longitude as straight rays and lines of latitude as circles.

Planetary science

Stereographic projection is special because it shows all circles on a sphere as circles on a flat surface. This is helpful when mapping planets, where craters often look like circles. Some circles become very large and look like straight lines.

Crystallography

Main article: Pole figure

In crystallography, scientists study how crystal shapes are arranged in space. They use stereographic projection to draw these arrangements. This helps them understand patterns seen in X-ray and electron studies. A drawing of these points is called a pole figure.

Geology

Geologists study the shapes and directions of rocks and faults underground. They use stereographic projection to draw these directions, usually focusing on the lower half of an imaginary sphere. This helps them understand rock features better.

Rock mechanics

Stereographic projection helps scientists study how rocks might break apart on slopes. It shows the directions of cracks and weaknesses in rocks, helping to predict how a slope might fail. This is useful for building and safety.

Photography

Some special camera lenses use stereographic projection to capture very wide views. This keeps shapes near the edges looking natural and makes straight lines less curved than other wide-angle lenses. Software can change photos taken with other lenses to look like stereographic projections. This way has been used to create special effects in panoramic photos since 1779. People like this method because it keeps shapes looking correct.