Spherical geometry

Spherical geometry, also called spherics, is the study of the flat parts and curved shapes on the outside of a ball or sphere. People have used this kind of math for a very long time because it helps us understand the stars, find our way on Earth, and measure distances on our planet.

Just like regular geometry that we use on flat paper, spherical geometry has its own special rules and ways to measure angles and distances. But because a sphere is curved, some of the rules are different from what we are used to.

We can look at a sphere from the outside, thinking of it as part of regular 3D space, or we can study just the surface of the sphere by itself, without thinking about what is around it. Both ways help us learn more about the interesting shapes and patterns on round objects.

Principles

In plane (Euclidean) geometry, we learn about points and lines. In spherical geometry, we learn about points and great circles. Great circles are like the straight lines of a sphere.

A great circle is what you see when a plane cuts straight through the center of a sphere. On the sphere, a great circle is the shortest path between two close points. Great circles can act like lines in flat geometry, such as forming the sides of triangles. But angles in spherical geometry are different. For example, the angles inside a spherical triangle always add up to more than 180 degrees.

Relation to similar geometries

Because a sphere and a flat plane are different, spherical geometry is a type of non-Euclidean geometry. It helped solve an old question about rules for flat shapes, but the answer came from elliptic geometry and hyperbolic geometry. These geometries change Euclid's parallel rule in different ways.

These ideas can be used in many sizes and shapes. Another geometry related to spheres is the real projective plane. It looks like spherical geometry up close but behaves differently overall. Spherical geometry can also be used for stretched spheres, with some small changes to the formulas.

Main article: Non-Euclidean geometry

Main articles: Elliptic geometry, Hyperbolic geometry

Main article: Real projective plane

History

The study of shapes on a ball, called spherical geometry, began a long time ago. One of the earliest works was by Autolycus of Pitane. He wrote about a spinning ball in ancient Greece.

Later, Greek mathematicians like Theodosius of Bithynia and Menelaus of Alexandria wrote books about shapes and angles on a ball. In the Islamic world, Al-Jayyani wrote the first book just about spherical trigonometry. In Europe, Regiomontanus wrote the first book only about trigonometry. Some ideas came from earlier work by Jabir ibn Aflah.

The famous mathematician Leonhard Euler did lots of important work on spherical geometry. He wrote many papers about it.

Properties

Spherical geometry has some special features:

• Any two big circles on a sphere meet at two points that are directly opposite each other.
• Two different points decide one special big circle.
• There are natural ways to measure angles, lengths, and areas on a sphere.
• Each big circle has two points called poles.
• Every point has a special big circle called its polar circle.

When looking at triangles made from smaller parts of these big circles, we find more interesting facts:

• The angles in such a triangle always add up to more than 180° but less than 540°.
• The area of a triangle depends on how much more the angle sum is than 180°.
• Triangles with the same angle sum have the same area.
• There is a limit to how big a triangle's area can get.
• Two triangles are exactly the same shape and size if you can match them by flipping or turning them.
• If the angles of two triangles match up, the triangles are the same size and shape.

Relation to Euclid's postulates

In spherical geometry, lines are curves called great circles. This type of geometry uses only two of Euclid’s five rules. It follows the rule that we can extend a line and that all right angles are the same size. But it does not follow the other three rules.

For example, there isn’t just one shortest path between two points on a sphere. Think of the North and South Poles—they have many shortest paths going around the Earth. Also, we can’t draw circles of any size we want on a sphere. Finally, if we try to draw a line that never touches another line, it will always meet the other line at some point.

Because of this, triangles on a sphere work differently. On a flat surface, a triangle’s angles add up to 180 degrees. But on a sphere, the angles add up to more than 180 degrees, depending on how much of the sphere the triangle covers.