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I am neither a physicist nor a mathematician, I am an archaeologist trying to develop a mathematical model for archaeological site detection. The problem is set up like this (and hopefully this will make some sense):

The detector moves in parallel transects that can be of any interval but is consistent within a given test.

The detector has a swath width of a fixed distance.

The target is an archaeological site typically elliptical in shape which has an average density of artifacts which are visible to the detector per certain unit of area.

The orientation of transects and the site are independent of each other.

Assume an even (non clustered) distribution of artifacts within the site.

To find the chance of detection I have been using the formula for a Poisson distribution:

p(x) = exp(-1*lambda) lambda^x / x! for x=0,1,2,...

lamda is the expected density of artifacts per unit of space

Which I believe I can then multiply by the average area of intersection between the transects (rectangles) and the site (ellipse). I have found formulas for determining the probability of intersection of ellipses with transects but they do not calculate the area of intersection:

P=(1-b/l)(2l/πs(1-sqr root((1-s/l)^{2}+ s/l(Acs(s/l)))) + (b/l)(2l/πs)

where P is probability of intersection

b=width of ellipse

l is length of the ellipse

s is the transect spacing or interval

This formula should account for multiple intersections of transects if the ellipse is much larger than the transect spacing.

I assume that the formula might be adapted to provide spatial data but I do not know quite how to make that work to hook it into the Poisson distribution formula.

I suspect that I might be forgetting something fundamental so if so let me know.

Thanks very much in advance for your help!

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# Probability and average area of intersection of rectangles and ellipse

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