Stereographic projection

Imagine a round ball.

We can turn its shape into a flat map. We do this by using a special point on the ball.

Lines go from that point to a flat sheet. This helps us see circles on the flat map. It also keeps the angles of shapes the same.

Long ago, people used this to study the stars. It is a great way to draw the sky.

This makes big, round things easy to study on paper.

Imagine a round ball. You want to turn its shape into a flat map.

You can do this with a special way called stereographic projection.

Pick one point on the ball. We often call this the North Pole.

From that point, draw lines to every other spot on the ball. These lines pass through a flat sheet below the ball. Where the lines hit the sheet, you mark a dot. These dots make your flat map.

This way has a few special rules. It keeps angles the same. This means shapes look right where they meet.

However, it does not keep the size the same. Some areas will look much bigger or smaller than they are.

People have used this for a long time. Ancient Greeks used it to study the stars.

They used it to map the sky on flat paper. Later, map makers used it to draw the Earth. Even today, it helps in science and photography. It is a clever way to see a round world on a flat page.

Imagine you have a round ball and a flat sheet of paper. You want to turn the round shape into a flat map.

This is a hard job because a sphere is curved and a plane is flat. Stereographic projection is a special way to do this. It uses a single point on the sphere to guide the process. We often call this point the North Pole.

By using this point, we can find a way to represent the whole sphere on a flat surface.

To make the map, you start at the North Pole. You draw a straight line from that pole through any other point on the sphere. This line will eventually hit the flat plane below.

The spot where the line hits the plane is the new location on your map. This process creates a smooth connection between the sphere and the plane. It is a very special kind of math called a bijective function. This means every point on the sphere has exactly one partner on the plane.

Every circle on the sphere becomes either a circle or a line on the flat map.

People have used this idea for a very long time. We do not know exactly who first discovered it. Many believe Ancient Greek astronomers found it to study the stars. They used it to turn the sky into a flat map.

This helped them study how stars and planets move. A writer named Synesius said the mathematician Hipparchus hinted at this idea long ago. The famous Ptolemy also wrote about using it in a special instrument. Later, the planispheric astrolabe became a useful tool for measuring stars.

This method has many important rules and properties. One key rule is that it is conformal. This means it preserves the angles where curves meet.

Because of this, small shapes look mostly correct on the map. However, the projection is not isometric or equiareal. This means it does not keep distances or areas the same.

Some parts of the map will look much larger than they really are. The North Pole itself is sent to a place called infinity. As you get closer to the North Pole, the points move much farther away on the plane.

Today, this math is used in many different jobs. Map makers have used it to draw the Earth for hundreds of years.

In the 16th and 17th centuries, it was used for maps of the two hemispheres. Scientists use it in fields like geology and photography. Some people even use special graph paper called a stereonet to do math.

It remains a clever way to see a round world on a flat page.

Stereographic projection is a specific type of perspective projection. It allows us to represent a sphere on a flat plane.

This mathematical method is a smooth, bijective function. This means every point on the sphere, except the center of projection, maps to exactly one point on the plane. It is a vital tool for visualizing curved surfaces in a flat format. While it provides a unique way to view the sphere, it involves certain geometric compromises.

To understand the mechanism, imagine a unit sphere in three-dimensional space. We select a specific point on the sphere to act as the center of projection, often called the North Pole.

Below the sphere, we place a projection plane. This plane is perpendicular to the diameter that passes through the North Pole. To project a point, you draw a straight line from the North Pole through that point on the sphere. The exact spot where this line intersects the plane is the projection of that point.

This process can be described using Cartesian, spherical, or cylindrical coordinates. For example, the South Pole maps to the origin of the plane. The equator maps to a unit circle on that plane.

There are different ways to define this projection depending on the chosen plane. Some mathematicians define the projection onto the equatorial plane. Others use a plane that is tangent to the sphere at the South Pole.

If the plane is tangent to the South Pole, the projection is scaled by a factor of two. This specific version sends the equator to a circle with a radius of two. This version produces no area distortion at the South Pole. However, the equatorial projection produces no infinitesimal area distortion along the equator. The projection is not isometric, so it does not preserve distances. It is also not equiareal, meaning it does not preserve the area of shapes.

The projection has several unique geometric properties. Most importantly, it is conformal. This means it preserves the angles at which curves meet.

Because angles are preserved, the projection locally approximates the original shapes. It also maps circles on the sphere to either circles or lines on the plane. A significant feature is how it handles the North Pole. The projection is not defined at the North Pole itself. As points on the sphere move closer to the North Pole, their images move farther away on the plane. In this sense, the North Pole is often said to map to infinity.

The history of this concept is ancient but somewhat mysterious. The exact origin is unknown, though it likely began with Ancient Greek astronomers. They needed to project the celestial sphere onto a plane to study star motions. The earliest written description appears in Ptolemy's Planisphere from the 2nd century AD. The writer Synesius later claimed that Hipparchus had hinted at this method long ago. He suggested Hipparchus was the first to apply it to a spherical surface. However, some experts doubt these attributions to Hipparchus or Archimedes. By the 4th century, Theon of Alexandria helped combine the planisphere with a dioptra. This created the planispheric astrolabe, a portable tool for astronomical calculations.

Over the centuries, the use of stereographic projection expanded into many fields. In the 16th and 17th centuries, it was used to create maps of the Earth's hemispheres.

Notable mapmakers like Rumold Mercator used this projection. The name "stereographic projection" was actually given by François d'Aguilon in 1613. Regarding its mathematical proof, Thomas Harriot discovered the projection was conformal in the late 16th century. However, his work remained unpublished for three hundred years. Edmond Halley finally published a proof in 1695. He used the new tools of calculus developed by Isaac Newton.

Today, the application of stereographic projection reaches far beyond simple mapmaking. It is used in complex analysis, geology, and photography. In geology, scientists use a special kind of graph paper called a stereonet or Wulff net.

This allows them to plot planar and linear data from the Earth's surface. The projection also connects to higher mathematics through the study of the Riemann sphere. This relates the sphere to the one-point compactification of the plane. It also serves as a spherical analog to the Poincaré disk model used in hyperbolic geometry. These connections make it a fundamental concept in algebraic and conformal geometry.