I have a polygon shapefile with about a dozen polygons (Area.shp). I have an equation that calculates a value for points (or small areas) as a function of distance to the edge of these polygons. I would ultimately like to apply this function to all points within each of these polygons and sum them (within each polygon) to get a single value for the polygon.
It seems like one of those things that shouldn't be too difficult, but I can't seem to get a working solution. I had an idea of converting the shapefile to raster then raster to points then taking the distance of each point to the nearest polygon edge, then apply the equation. It seemed like a reasonable, if clunk solution but unfortunately, I get all sorts of errors during various conversions using ArcMap 10.
Does anyone have any suggestions on how to do this? I am happy to use ArcGIS, QGIS, GRASS, Python, or export to R or any other solution. I am a novice in all of these systems (except R) but am willing to use whatever works.
[UPDATE] Additional Information: When I was thinking of points, it was really as a way to approximate numerical integration. I have an equation where the number of animals within a particular patch of habitat (polygon) is a function of distance to the edge of the habitat (nearest edge of the polygon). The equation is a 4pt logistic function
N(i) = C + A/(1 + e^-(x-D)/B)
where all values are known (previously estimated) constants except x which is distance to the edge. N(i) is the number of animals in a small area, roughly 2 m^2. I'm interested in how the size and shape of habitat patches (polygons) affects the total N (sum of N(i) within a polygon) in a patch. I have a variety of patches identified from GIS layers as a single shape file with about a dozen polygons. For the purpose of this study, I can assume that no polygons overlap.