{"id":10976,"date":"2019-10-25T09:00:11","date_gmt":"2019-10-25T13:00:11","guid":{"rendered":"https:\/\/blogs.mathworks.com\/pick\/?p=10976"},"modified":"2020-03-23T09:21:46","modified_gmt":"2020-03-23T13:21:46","slug":"minimal-bounding-box","status":"publish","type":"post","link":"https:\/\/blogs.mathworks.com\/pick\/2019\/10\/25\/minimal-bounding-box\/","title":{"rendered":"Minimal Bounding Box"},"content":{"rendered":"
\n <\/p>\n

Sean<\/a>‘s pick this week is Minimal Bounding Box<\/a> by Johannes Korsawe<\/a>.\n <\/p>\n

<\/introduction><\/p>\n

I was recently asked by a customer “How can I replicate the flatness measurements from a CMM machine?” A coordinate measuring machine<\/a> measures geometric properties of an object.\n <\/p>\n

A quick internet search, took me to this site<\/a> which introduced two methods. The customer wanted the “Minimum Zone” method, sometimes also referred to as parallel planes. This essentially boils down to “fit a minimum bounding box and calculate the distance between the top and bottom of the box”.\n <\/p>\n

I knew it must be possible to to formulate this as an optimization routine whether something linear that linprog<\/tt> could handle or nonlinear with linear constraints enforcing for fmincon<\/tt>. A quick search of the File Exchange yielded Johannes’ file which does exactly what was needed. It also provides a plotting routine to help validate and understand the output.\n <\/p>\n

Here’s an example of what worked, using random numbers in place of the CMM measurements.<\/p>\n

npts = 10; % Number of points from CMM machine.<\/span>\r\nx = randi(100, npts, 1)+randn(npts, 1);\r\ny = randi(100, npts, 1)+randn(npts, 1);\r\nz = rand(npts, 1);\r\n\r\n[~, cornerpts] = minboundbox(x, y, z);\r\n\r\nscatter3(x, y, z);\r\nhold on<\/span>\r\nplotminbox(cornerpts, 'r'<\/span>);<\/pre>\n
 <\/p>\n
To calculate flatness, I just need to calculate the distance between a top and bottom point.  I wasn’t sure if the point order was deterministic so it’s easier to just calculate the closest point on the opposite plane from a chosen one.\n   <\/p>\n
bottompt = cornerpts(1,:);\r\n[~, topidx] = min(hypot(cornerpts(2:end,1)-bottompt(1), cornerpts(2:end,2)-bottompt(2)));\r\ntoppt = cornerpts(topidx+1,:);\r\n\r\nhold on<\/span>\r\nscatter3(bottompt(1), bottompt(2), bottompt(3), 'b*'<\/span>)\r\nscatter3(toppt(1), toppt(2), toppt(3), 'g*'<\/span>)<\/pre>\n
 <\/p>\n
Computing the distance can be done with a couple calls to hypot<\/tt><\/a>.\n   <\/p>\n
planesep = hypot(hypot(toppt(1)-bottompt(1), toppt(2)-bottompt(2)), toppt(3)-bottompt(3));\r\ndisp(\"Plane separation is: \"<\/span> + planesep)<\/pre>\n
Plane separation is: 0.55053\r\n<\/pre>\n
Comments<\/a><\/h3>\n
Give it a try and let us know what you think here<\/a> or leave a comment<\/a> for Johannes.\n   <\/p>\n