Find AABB from many 3D planes - that form a convex hull

Question

I have some 3D planes information. When all planes join together, it will form a 3D convex hull.

Here is an example of input.
Each 3D plane is denoted by a point on the plane and its normal.
All normals point to inside :-

- (-1,0,0)              with normal = (1,0,0)
- ( 1,0,0)              with normal = (-1,0,0)
- (0,-1,0)              with normal = (0,1,0)
- (0,1,0)               with normal = (0,-1,0)
- (0,0,-1)              with normal = (0,0,1)
- (0,0,1)               with normal = (0,0,-1)
- (0.142,-7.18,10.12)   with normal = (0.001,0.31,-0.95) 
- note: some planars can be redundant (contribute nothing) e.g. the last one

Question: How to calculate AABB from it?
The solution from the above example is ((-1,-1,-1),(1,1,1)).

(At first, I want center of mass, but I realized that is a Hard problem.
AABB should be good enough for me. )

My poor solution is to find all hull vertices using Constructive solid geometry then do MIN & MAX on them, but the performance is too bad.

In real case, I want to find the center point of 3D hull when two convex hulls are overlapping.
This information can be useful for more accurate Collision Response in Physics Engine.

Answer 1

It"s the classical problem of linear programming. Given a set of linear inequalities (represented by planes in your case, say ax+by+cz+d>=0) and a linear function (say f(x,y,z)=x), find a point that minimises the function while satisfying all the inequalities. The AABB is the solution of 6 such problems, for functions x, -x, y, -y, z and -z.

There are several methods of solving this problem, the most known being the simplex algorithm, and many ready-made libraries (including some that utilise GPUs).