Comments on: The land surveyor’s algorithm http://bit-player.org/2007/the-land-surveyors-algorithm An amateur's outlook on computation and mathematics. Thu, 17 May 2012 09:07:50 +0000 http://wordpress.org/?v=2.6.3 By: mandy http://bit-player.org/2007/the-land-surveyors-algorithm#comment-1495 mandy Sun, 09 Sep 2007 19:47:15 +0000 http://bit-player.org/?p=85#comment-1495 this was so awesome. i deff. used it to do a project. wicked cool. thanx this was so awesome. i deff. used it to do a project. wicked cool. thanx

]]> By: Brie Finegold http://bit-player.org/2007/the-land-surveyors-algorithm#comment-1418 Brie Finegold Tue, 06 Feb 2007 01:06:48 +0000 http://bit-player.org/?p=85#comment-1418 The simplest version of this is to consider a parallelogram and notice that if one corner is the origin then only two other points are needed to define the parallelogram. Furthermore, these points can be regarded as vectors and be entered as rows or columns in a matrix whose determinant is then the area of the parallelogram. The area of a convex polygon can then be found by placing the origin in the center, chopping the polygon into triangles (which are halves of parallelograms), and then doing the above parallelogram trick over and over. I wonder if the same thing can be done with three dimensional polytopes using three by three matricies. The simplest version of this is to consider a parallelogram and notice that if one corner is the origin then only two other points are needed to define the parallelogram. Furthermore, these points can be regarded as vectors and be entered as rows or columns in a matrix whose determinant is then the area of the parallelogram.
The area of a convex polygon can then be found by placing the origin in the center, chopping the polygon into triangles (which are halves of parallelograms), and then doing the above parallelogram trick over and over. I wonder if the same thing can be done with three dimensional polytopes using three by three matricies.

]]> By: brian http://bit-player.org/2007/the-land-surveyors-algorithm#comment-1417 brian Fri, 02 Feb 2007 22:48:48 +0000 http://bit-player.org/?p=85#comment-1417 Thanks for both the illumination and the pointers. "A version of Green's theorem" -- Conceptually, sure, but I trust you don't mean to suggest that the line-integral formulation came first historically, and the discrete algorithm was derived from it? For one thing, that would put the origin sometime after 1828. Thanks for both the illumination and the pointers.

“A version of Green’s theorem” — Conceptually, sure, but I trust you don’t mean to suggest that the line-integral formulation came first historically, and the discrete algorithm was derived from it? For one thing, that would put the origin sometime after 1828.

]]> By: D. Eppstein http://bit-player.org/2007/the-land-surveyors-algorithm#comment-1416 D. Eppstein Fri, 02 Feb 2007 20:51:12 +0000 http://bit-player.org/?p=85#comment-1416 It's a version of Green's theorem (turning the integral of the constant function 1 over the area of the polygon into something over the perimeter). It can be interpreted as summing the areas of triangles determined by the origin and the two endpoints of each edge. I usually teach something like this in my algorithms and computational geometry classes, but I use the similar formula 1/2 sum((x_{i+1}-x_i)(y_i+y_{i+1})) which uses half as many multiplication operations. It can be interpreted as summing the areas of with vertical sides between each edge and the x axis. See also the comp.graphics.algorithms faq, question 2: http://www.faqs.org/faqs/graphics/algorithms-faq/ It’s a version of Green’s theorem (turning the integral of the constant function 1 over the area of the polygon into something over the perimeter). It can be interpreted as summing the areas of triangles determined by the origin and the two endpoints of each edge.

I usually teach something like this in my algorithms and computational geometry classes, but I use the similar formula 1/2 sum((x_{i+1}-x_i)(y_i+y_{i+1})) which uses half as many multiplication operations. It can be interpreted as summing the areas of with vertical sides between each edge and the x axis.

See also the comp.graphics.algorithms faq, question 2: http://www.faqs.org/faqs/graphics/algorithms-faq/

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