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
