Cumulative Arc Length on a Polyline

A polyline route on the sphere is a sequence of waypoints $P_0, P_1, P_2, \ldots, P_n$ joined by great-circle segments. The $i$-th leg runs from $P_{i-1}$ to $P_i$ along the shorter great-circle arc, and has arc length $d_i$.

The total arc length of the route is the sum of all leg lengths:

$$L = d_1 + d_2 + \cdots + d_n = \sum_{i=1}^{n} d_i.$$

For example, a 3-leg route with leg lengths $d_1 = 250$ km, $d_2 = 380$ km, and $d_3 = 195$ km has total length

$$L = 250 + 380 + 195 = 825 \text{ km}.$$