let $x_1$, $y_1$, $x_2$, $y_2$ be points on the sphere $S^n$ such that $x_1$ and $y_1$ are not antipodal and similarly for $x_2$, $y_2$. Why $x_1$ and $y_1$ can be joined by a path that do not cross with a path joining $x_2$ to $y_2$? and why we call these paths "shortest geodesics"?
- Anybody can ask a question
- Anybody can answer
- The best answers are voted up and rise to the top
If two points on a sphere are not identical and not antipodal then they lie on a unique great circle. The shortest distance between the two points will the shorter of the two arcs of the great circle between the two points
Two distinct great circles intersect twice. The two shorter arcs between the pair of points may or may not intersect. This diagram may help.
because i think assume the points on the sphere is antipodal . Line joining these points is the largest chord (diameter) of great circle , it is not a shortestpathand when they points must together then displacement is tends to 0 so conclude that it is the shortest path from their points.it is called geodesic.