SPHERICAL GEOMETRY

The shortest path between Sao Paulo and Tokyo on a flat map. Source: Google Maps.

The shortest path between two points on a plane is a line segment. What is the shortest path between São Paulo and Tokyo?

Thinking of the Earth as a sphere, the answer to that question is a straight line segment of the sphere. But what we call "straight" in a sphere surface is quite different from the straight line in a plane: it is a maximum circle. For example, the equator line is a straight line on the sphere as well as any of the meridians.

Note that, given any two points on the sphere surface, there can be more than one straight line segment joining them. Think, for example, of the poles: all meridians are "straight lines" through these two points. It is also clear that two "straight lines" always intersect at two points and it becomes difficult to define the concept of “parallel lines” in this geometry.

An interesting curiosity: In the spherical geometry, the sum of the internal angles of a triangle is always greater than 180 degrees and less than 540 degrees.

The shortest path between Sao Paulo and Tokyo on a globe. Source: Google Earth.

USEFUL LINKS

This Wikipedia page explains a little about spherical geometry, bringing a little of its history, some properties and applications of this geometry, so important in areas such as mathematics, physics, astronomy, among others. Spherical trigonometry, shown in this other Wikipedia page, studies the geometric properties of spherical triangles, essential for the study of this topic.

This video from the Institute of Mathematics, Statistics and Scientific Computing at Unicamp presents non-Euclidean geometry, including explaining the difference in relation to Euclidean geometry, and presents spherical geometry in a very engaging video, showing the usefulness of this geometry in determination of the shortest distance between two points on a spherical surface.

Spherical geometry is widely used when talking about positional astronomy. This Wikipedia page talks about the celestial coordinate system, used to specify the position of satellites and planets, for example. This page, in addition to talking about coordinate systems, also shows how to perform some conversions between them, in a slightly more formal way in some cases.

Another topic where this geometry is widely used is cartography. The flat representation of the sphere that is the planet Earth can bring some problems, so some cartographic projections were created, which use the spherical geometry. This Mundo Educação article provides an overview of these possible problems and the most commonly used projections.

This article by Marlon Mülhbauer and Mateus Bernardes reports the experience resulting from the introduction of concepts of non-Euclidean Geometries to high school students. They were introduced to basic principles of Cartography, History of Mathematics, spherical coordinates and metric spaces. The article discusses the validity of teaching this type of content in basic education.