What is the maximum number of planes that can be divided into n lines? __ Computational geometry

N lines, 22 intersect, the intersection points are different, then the number of intersections is a combination of 2 number of N number, at the same time, only in this case (22 intersection, also the intersection point is different), the number of split planes, the number is: 2 + (N-1) (n+2)/2. It doesn't make sense to ask for the least number of planes, because the least number of planes is n+1, that is, when the N line 22 is parallel, the split plane is the least.

Example:

1 line split plane number is up to 2; a1 = 2

2 Line split plane number of up to 4, 1 straight lines when the number of split +2 a2 = A1 + 2

3 Line split plane number of up to 7, 2 straight lines when the number of split +3 a3 = a2 + 3 = a1 + 2 + 3

4 line split plane number of up to 11, 3 straight lines when the number of split +4 A4 = a3 + 4 = a1 + 2 + 3 + 4

5 Line split plane number of up to 16, 4 straight line when the split number +5 A5 = a4 + 5 = a1 + 2 + 3 + 4 + 5

6 line split plane number of up to 22, 5 straight line when the split number +6 A6 = a5 + 6 = a1 + 2 + 3 + 4 + 5 + 6

So an = a1 + 2 + 3 + ... + N.

You can see that the difference in the number of split planes is incremented by arithmetic progression, so the maximum number of planes divided by the nth line is: 2 + (n-2+1) (2+n)/2.

Or

The maximum number of planes is divided into n lines:

(1) using recursion

f (N) = n + f (n-1), n > 1

f (n) = 2, n = 1

(2) using recursion

n = 1, S1 = 2

n > 1, Sn =2+2+3+...... + N = 1 + n * (n+1)/2