The shortest path between two points on a curved surface, such as the surface of a sphere, is called a geodesic. To find a geodesic, one has first to set up an integral that gives the length of a path on the surface in question. This will always be similar to the integral $(6\mathrm{.}2)$ but may be more complicated $($depending on the nature of the surface$)$ and may involve different coordinates than x and y$.$ To illustrate this, use spherical polar coordinates $($r$,0,0)$ to show that the length of a path joining two points on a sphere of radius R is $02$ L $=$ R f $\backslash 11+$ sin$20\text{'}(9)2$ d$0$ e$,(6\mathrm{.}41)$ if $(01,01)$ and $(82,02)$ specify the two points and we assume that the path is expressed as $0=(0).($You will find how to minimize this length in Problem $6\mathrm{.}16.)$