Q:

Smallest enclosing cylinder for an irregular body

I have an $3$-dimensional irregular body composed of 162 points $(x,y,z)$.
I need to find the smallest enclosing cylinder for this body. Is there a standard algorithm for achieving this?

A:

I'll assume it's to be a right circular cylinder, and "smallest" is in terms of volume.  Consider a cylinder of radius $r$ and height $h$, with one of the ends centred at point $P$, and axis in the direction of unit vector $U$.  Then a point $X = (x,y,z)$ is in the cylinder if $0 \le (X - P)\cdot U \le h$ and $\|X - P - ((X -P)\cdot U) U \|^2 \le r^2$.  Thus you want to minimize $r^2 h$ subject to constraints
$r \ge 0$, $h \ge 0$, $\|U\|^2 = 1$, and for all $X$, $(X - P) \cdot U \ge 0$, $(X - P)\cdot U \le h$, $\|X - P - ((X -P)\cdot U) U \|^2 \le r^2$.  You only need to consider those $X$ that are extreme points of their convex hull.
  You can also require, say, $U_1 \ge 0$ because you have your choice of which end of the cylinder $P$ is on.  This is a (non-convex) nonlinear constrained optimization problem.
I might try Maple's Global Optimization Toolbox.

