Spherical Geometry

Line Segments
In the plane, we use line segments to draw geometric figures such as triangles and squares or, more generally, polygons.
But we cannot draw line segments on the sphere...

So how can we draw polygons on the sphere if we can't draw line segments?

What is a line segment in the plane?
It is the shortest path (in the plane) between two points (the ends of the segment).

What is a line segment on the sphere?
It is the shortest path (on the sphere) between two points (the ends of the segment).

On the sphere, the shortest path between two points is given by an arc of a circle obtained by intersecting the sphere with a plane containing its center. This circle is usually referred to as a great circle.

What is a line in the plane?
A line in the plane is obtained by extending a line segment infinitely in both directions.

What is a line on the sphere?
A line on the sphere is obtained by extending an arc of a great circle infinitely in both directions. 

Two great circles always intersect at two points.

The Circumference on the Sphere
A circumference on the sphere is the set of points on the sphere that are at a constant distance (the radius) from a fixed point on the sphere (the center).

A circumference on the sphere is obtained by intersecting the sphere with a plane not passing through its center.

The Spherical Triangle

A spherical triangle is defined in the same way as a triangle in the plane. It consists of 3 points, not belonging to the same great circle, called vertices, of the great circle arcs that join these vertices called sides, and of the region delimited by them.

There are two distinct spherical triangles with the same vertices.