Lake-swimming immigrants

As shown in the above picture: Let A and B be located at a latitude in the northern hemisphere (as shown in the left picture).

First of all, the great circle of the sphere refers to the tangent circle passing through the center of the sphere. That is to say, if the center of any tangent circle on the sphere is the center of the sphere, then this circle is a great circle. As the picture shows, the red circle (equator), the green circle and the outline of the earth (black) all belong to the great circle. On the contrary, the latitude (blue circle) where AB is located is not a big circle.

Secondly, the great circle of the sphere connects the shortest distance between two points on the sphere. We know that in plane geometry, between two points, the line segment is the shortest, which is called geodesic connecting two points, and the line segment is unique. In fact, our ground is spherical. In spherical geometry, the geodesic line connecting two points on the sphere is the great circle between them, and the shortest path is unique. As shown in the middle figure, the shortest distance connecting AB is the thick green line in the figure, not the weft (thick blue line) between them. For example, in real life, flights from Beijing (like point A) to San Francisco (like point B) often follow the principle of great circle route, that is, bypassing the Pacific Ocean and flying along the east coast of Eurasia and the west coast of North America.

Finally, because the principle of the great circle route is locked, it is more accurate to say that AB divides the great circle into two parts. Obviously, the shorter part is the shortest spherical distance (the red part on the right), and the central angle of this arc is called "bad arc" because it is less than 180. The other part of the great circle is called the "optimal arc". Obviously, its path is too long, not the shortest route.