I am going through this particular topic "intrinsic metric of sphere" . What i have learned so far is that the intrinsic metric is the infimum of all the arclengths of the curves(paths) joining two points . Now for a sphere this metric is defined to be a segment of some great circle . Here i am stucked to understand this . Can we discuss this or if you have some books or notes about this ?
intrinsic metric of sphere
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$\begingroup$
geometry
1 Answers
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A great circle is intersection of the sphere with a plane that goes through its origin. You can also think of it as the "cutting line" for any division of the sphere into two hemispheres. The equator of the Earth (if the Earth were a sphere) is a great circle.
The shortest path between two points on the sphere lies along a great circle connecting the two points. If the two points are not antipodal, there is a unique great circle connecting them.