most recent 30 from http://mathoverflow.net 2013-05-21T17:18:51Z http://mathoverflow.net/feeds/question/17048 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/17048/compute-the-centroid-of-a-3d-planar-polygon Graviton 2010-03-04T03:07:39Z 2011-11-07T17:53:48Z

<p>Given a list of 3D coordinates that define the surface( <code>Point3D1</code>, <code>Point3D2</code>, <code>Point3D3</code>, and so on), how to calculate the centroid of the surface?</p> <p>More specifically, I am looking for a natural extension of the following <a href="http://en.wikipedia.org/wiki/Centroid#Centroid_of_polygon" rel="nofollow">2D centroid algorithm</a> in 3 or more dimension:</p> <p><img src="http://upload.wikimedia.org/math/e/e/1/ee14cbb2b170c4bb435f1d84e78f6d66.png" alt="alt text"> <img src="http://upload.wikimedia.org/math/a/4/c/a4cee81a1d18e4d067f66d4d40a8a1fe.png" alt="alt text"></p> <p><img src="http://upload.wikimedia.org/math/0/2/a/02aecb75f67f8c7b2fc11fdcbcb6ea80.png" alt="alt text"></p> <p>Any idea?</p> <p><strong>P/S: All the points are coplanar, this is the assumption.</strong> </p>

http://mathoverflow.net/questions/17048/compute-the-centroid-of-a-3d-planar-polygon/17118#17118 Quimey Vivas 2010-03-04T18:35:39Z 2010-03-04T18:35:39Z

<p>These formulas could be deduced using <a href="http://en.wikipedia.org/wiki/Green_theorem" rel="nofollow"> Green's theorem </a>. For example the formula used to compute the polygon's area is proved using the vector field F=(-y/2,x/2).</p> <p>Maybe you can do the same in the space using analogous vector fields and <a href="http://en.wikipedia.org/wiki/Stokes_theorem" rel="nofollow"> Stokes' theorem </a>.</p>

http://mathoverflow.net/questions/17048/compute-the-centroid-of-a-3d-planar-polygon/20536#20536 Faken 2010-04-06T19:07:26Z 2010-04-06T19:07:26Z

<p>I'm not sure about the centroid of the surface area, but I might be able to help with the centroid of it's enclosed volume under some specific circumstances.</p> <p>If each surface formed by the points creates a triangle (as is the case with the vast majority of computer related applications) then you can use this method that i outlined in StackOverflow (i understand its not in the from a mathematician would like it in, but if you read the explanation the answer is fairly simple).</p> <p><a href="http://stackoverflow.com/questions/2083771/a-method-to-calculate-the-centre-of-mass-from-a-stl-stereo-lithography-file">http://stackoverflow.com/questions/2083771/a-method-to-calculate-the-centre-of-mass-from-a-stl-stereo-lithography-file</a></p> <p>Hopefully that helps</p>

http://mathoverflow.net/questions/17048/compute-the-centroid-of-a-3d-planar-polygon/28381#28381 Joseph O'Rourke 2010-06-16T12:49:50Z 2010-06-29T11:24:32Z

<p>In response to JBL's comment, I offer this answer merely to close out this topic. It has been effectively answered in the comments: Simply project to xy and to xz and compute the centroid there. (One tiny wrinkle not addressed is if the polygon lies in a plane perpendicular to xy or to xz. But then simply chose the coordinate planes in which it does not lie.) </p> <p>On the advice of Andrew Stacey, I am designating this answer "community wiki," and hope that someone will vote it up so it will no longer be bumped to the top of the active list by the MO background process.</p>

http://mathoverflow.net/questions/17048/compute-the-centroid-of-a-3d-planar-polygon/80318#80318 unknown (yahoo) 2011-11-07T17:53:48Z 2011-11-07T17:53:48Z

<p>Related to this topic: what if the points do not lie all in the same plane ? I need to calculate the center of an hexagon. However, the hexagon in not planar (it is a 3D object, a distorted hexagon). Can the above formulas be exended to calculate the centroid of this hexagon ? </p>