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, the basic ideas are points and lines. In spherical geometry, the basic ideas are points and great circles. Great circles are like the straight lines of a sphere.

We can think of a great circle as the outline you get when a plane cuts straight through the center of a sphere. On the sphere itself, a great circle is the shortest path between two nearby points. In some ways, great circles work like lines in flat geometry, such as forming the sides of triangles. However, angles in spherical geometry work differently. 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 shares some features with what we call non-Euclidean geometry. However, it wasn't enough on its own to solve an old question about Euclid's rules for flat shapes. That answer came from elliptic geometry, which is very close to spherical geometry, and also from hyperbolic geometry. These geometries each change Euclid's parallel rule in different ways.

The ideas from these geometries can be used in many different sizes and shapes. Another geometry related to spheres is the real projective plane, made by connecting opposite points on a sphere. While it looks like spherical geometry up close, it behaves differently overall and can't be shown in regular 3D space without crossing itself. Spherical geometry can also be used for stretched spheres, though some formulas need small changes.

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, who wrote about a spinning ball in ancient Greece.

Later, Greek mathematicians like Theodosius of Bithynia and Menelaus of Alexandria wrote important books about the 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, though some ideas came from earlier work by Jabir ibn Aflah.

The famous mathematician Leonhard Euler also did lots of important work on spherical geometry, writing 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, called antipodal points.
• Two different points that aren't antipodal 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, which help show that the circle is like a regular circle when measuring distance on the sphere's surface.
• 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.
• Flipping or reflecting across a big circle can be thought of as turning around one of the points where the lines cross.
• 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

When we think of lines in spherical geometry, we mean the curves called great circles. This geometry follows only two of Euclid’s five rules. It follows the rule that we can extend a line and the rule 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.