Problem 532
Nanobots on Geodesics
Bob is a manufacturer of nanobots and wants to impress his customers by giving them a ball colored by his new nanobots as a present.
His nanobots can be programmed to select and locate exactly one other bot precisely and, after activation, move towards this bot along the shortest possible path and draw a colored line onto the surface while moving. Placed on a plane, the bots will start to move towards their selected bots in a straight line. In contrast, being placed on a ball, they will start to move along a geodesic as the shortest possible path. However, in both cases, whenever their target moves they will adjust their direction instantaneously to the new shortest possible path. All bots will move at the same speed after their simultaneous activation until each bot reaches its goal.
Now Bob places n bots on the ball (with radius 1) equidistantly on a small circle with radius 0.999 and programs each of them to move toward the next nanobot sitting counterclockwise on that small circle. After activation, the bots move in a sort of spiral until they finally meet at one point on the ball.
Using three bots, Bob finds that every bot will draw a line of length 2.84, resulting in a total length of 8.52 for all three bots, each time rounded to two decimal places. The colored ball looks like this:
In order to show off a little with his presents, Bob decides to use just enough bots to make sure that the line each bot draws is longer than 1000. What is the total length of all lines drawn with this number of bots, rounded to two decimal places?
测地线上的纳米机器人
鲍勃是纳米机器人的生产商，他希望送给顾客们一只由他生产的新型纳米机器人上色的球作为礼物，以给他们留下好印象。
纳米机器人内置的程序使得它们能够选定并精确地定位另一只机器人，并在激活时沿着最短路径向它移动，沿途绘制一条彩色的线。如果放在平面上，机器人的移动路径将是直线；如果相反放在球面上，它们将会沿测地线移动，因为此时测地线才是最短路径。无论是哪种情况，当机器人的目标对象也在移动时，机器人将会实时地将自己的方向调整到新的最短路径上。所有纳米机器人将会同时激活，此后保持相同的移动速度，直到每个机器人都与自己的目标相遇。
现在鲍勃在半径为1的球上放置了n个纳米机器人，它们等距地排列在半径为0.999的小圆上，同时设定每只纳米机器人向其逆时针方向的下一个机器人移动。在激活之后，这些机器人的移动轨迹呈螺线形，最后所有机器人都在球上的某一点相遇。
使用三个机器人时，鲍勃发现每个机器人绘制的线长度为2.84，总共的长度为8.52，均保留两位小数。这个上色的球看起来像这样：
为了让礼物显得更加牛逼，鲍勃决定选择合适数量的机器人，使得每个机器人绘制的线长度恰好超过1000。那么，所有机器人绘制的线的总长度保留两位小数是多少？