Ah, now I get it!!
> On Nov 15, 2015, at 12:32 AM, Jim Hurley <jhurley0...@sbcglobal.net> wrote: > > 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 _______________________________________________ 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
