Minimal Bounding Box

Minimal bounding box around points in the (x,y,z) space

With this function one can easily compute the minimal box (with right angles) around a set of points in 3d.
The extremal property of the box is determined either in terms of volume, surface or sum of edgelengths.

The calculation is based on heuristics only, but a huge number of tests did not show any counterexamples yet.

The algorithm behind the function is subdivided into three levels of accuracy with differing runtimes.

x = rand(10000,1);
y = rand(10000,1);
z = rand(10000,1);

tic;[rotmat,cornerpoints,volume,surface] = minboundbox(x,y,z,'v',3);toc
Elapsed time is 10.772170 seconds.

volume =
0.9993
surface =
5.9974

My thanks to John d'Errico and Roger Stafford for numerous discussions about proofs and algorithms in this context.

John also wrote minboundrect from the FEX, which heavily influenced this submission.

Also in this archive i included a small plot function (plotminbox) to show the resulting box via connecting the cornerpoints.