Convex hull#1044
A 3-dimensional object is called convex if, for any two points in the object, the line segment connecting them is entirely inside the object. For a given set of points in 3-dimensional space, the convex hull of is the smallest convex shape that contains all the points in .
For example, consider X = \{(0, 0, 0), (10, 0, 0), (0, 10, 0), (0, 0, 10)\}. The convex hull of is the tetrahedron with vertices at the points in . Note that this tetrahedron also contains the point (1, 1, 1), so adding this point to would not change the convex hull.
Given a set of points , your task is to find the surface area of the convex hull of , rounded to the nearest integer.
Note: Every convex hull consists of polygonal faces. In this problem, you may assume that there will be at most points from on any face of the convex hull.
The input contains multiple test cases. Each test case begins with an integer (4 \le n \le 25) indicating the number of points in . Then lines follow, each containing integers giving the , and coordinates of a point. All coordinates are between and , inclusive. The end of input is indicated by a line where n = 0, which should not be processed.
For each test case, print a single line with the surface area of the convex hull, rounded to the nearest integer (for example, rounds to , and rounds to ).
Hint
All test cases are chosen so that, when rounding to a precision of , there is no ambiguity (i.e., you may assume that the area will never be ).
Example 1