3.6.7. Geodesics on a right circular cone. Let S' be the right circular cone r(r,6) = 5" cos 61+ 5" sin 6j + m5" k, (3.126) with D S 6' :1 2w, 1' Z 0, and m a constant (the slope of the cone). A curve 7, on the surface of the cone, is given by a function, r = 1(9), that relates 'r and 9 along 7. We wish to nd the geodesic that minimizes the arc-length integral between two points on the surface of the cone. (a) Write down the arc-length integral for the curve 7. (b) Show that if 7 is a geodesic, 'r : TU?) must satisfy the differential equation 2 (1+ m2) 0:2 (g) = r2 (r2 (12) (3.127) for some suitable constant oz (0 3 oz 5 1'). 68 3. Cases and Examples (c) Integrate this differential equation to get the equation rm) 2 '1' @ (3.128) W for a geodesic curve that winds around and up the cone (with nonconstant radius). Here a and are appropriate constants of integration