Hi Roger, Actually, the answer to that question I raised follows in the next paragraph. All those terms cancel in pairs.
As someone pointed out, that paragraph is a proof of the validity of the method for calculating the area of a polygon. You’re probably right about the minus sign. Jim > > Message: 23 > Date: Sat, 14 Nov 2015 18:37:13 -0800 > From: Roger Guay <i...@mac.com> > To: How to use LiveCode <use-livecode@lists.runrev.com> > Subject: Re: Area of Irregular Polygon > Message-ID: <868cedf8-5e56-46dc-b88c-bb6a68cd4...@mac.com> > Content-Type: text/plain; charset=utf-8 > > Jim, > > I'm just now trying to catch up on this discussion and I see that no one has > answered your question. I can?t answer either and wonder what?s going on??? > > BTW, I believe you should have a negative sign in front of the square bracket > . . . not that that helps at all! > > Cheers, > > Roger > > > > >> On Nov 11, 2015, at 2:35 PM, Jim Hurley <jhurley0...@sbcglobal.net> wrote: >> >> Very interesting discussion. >> >> However, I was puzzled by the following term in the sum used to calculate >> the area of a polygon--labeled the centroid method. >> >> x(i)*y(i+1) - x(i+1)y(i) >> >> Where does this come from? If one were using the traditional method of >> calculating the area under a curve (perhaps a polygon) the i'th term in the >> sum would be: >> >> [x(i+1) - x(i)] * [y(i+1) + y(i)] / 2 >> >> That is, the base times the average height. This was the original method >> employed by many--the non-centroid method >> >> Multiplying this out you get: >> >> x(i)*y(i+1) - x(i+1)y(i) + [x(i+1)* y(i+1) - x(i)* y(i)] >> >> So THE SAME EXPRESSION as in the centroid method EXCEPT for the added term >> in square brackets. So, WHY THE DIFFERENCE? >> >> In calculating this sum all of the intermediate terms in the sum cancel out, >> leaving just the end terms: x(n)y(n) - x(1)* y(1) >> But if the figure is closed, they too cancel each other. >> For example: (x3 * y3 - x2*y2) + (x4*y4 - x3*y3)... Notice that the x3 * y3 >> terms cancel. >> >> Curiously, if one were attempting to calculate the area under a curve >> (non-polygon) using this method, the principle contribution could come from >> just these end point terms. As an extreme example, if the curve began at the >> origin (x(0) = y(0) = 0),. there would be a substantial contribution from >> the end point x(n) * y(n), where n is the last point. If it were a straight >> line, ALL the contribution would come from the end point: x(n) y(n) . >> Divided by 2 of course. >> >> Jim >> >> P.S. the centroid of a closed curve might be liken eo the center of gravity >> in physics. >> Two closed curves could have the same centroid but very different areas >> (masses) >> The center of mass bears no relation to the mass, and the centroid of a >> closed curved bears no relation to the area. >> _______________________________________________ >> use-livecode mailing list >> use-livecode@lists.runrev.com >> Please visit this url to subscribe, unsubscribe and manage your subscription >> preferences: >> http://lists.runrev.com/mailman/listinfo/use-livecode > _______________________________________________ use-livecode mailing list use-livecode@lists.runrev.com Please visit this url to subscribe, unsubscribe and manage your subscription preferences: http://lists.runrev.com/mailman/listinfo/use-livecode
