limiting triangle

In spherical geometry, a limiting triangle is a great circle of the sphere that is serving as the model for the geometry.

The motivation for this definition is as follows: In Euclidean geometry and hyperbolic geometry, if three collinear points are connected, the result is always a line segment, which does not contain any area. In spherical geometry, if the three points are close to each other, this procedure will produce a great arc (the equivalent to a line segment in this geometry). On the other hand, if the three points are sufficiently spaced from each other, this procedure will yield an entire great circle (the equivalent to a line in this geometry). For example, imagine that the circle shown below is a great circle of a sphere. Then connecting the three plotted points yields the entire great circle.

Thus, limiting triangles are geodesic triangles determined by three collinear points that are sufficiently spaced from each other.

speaking, the resulting figure is not a triangle in spherical geometry; however, it is useful for demonstrating the following facts in spherical geometry:

• ${540}^{\circ }$ is the least upper bound of the angle sum of a triangle;

• $2\pi$ is the least upper bound of the area of a .