# Working with Surface of Revolution of Inverse Square

1. Nov 15, 2007

### natai

The problem below is actually in reference to determining the location of a unknown gamma radiation source. However, I believe the solution lies with relatively simple calculus.

First, the equation that defines the relationship between the radiation exposure rate and the distance from the source is defined by:
y = k / x^2
Where y is the dose rate, x is the distance from the source, and k is a constant which may vary depending on the specific situation and source.

Next, this needs to be applied in three dimensions (x,y,z). So I believe it needs to be rotated around the y-axis to form a surface of revolution. If I remember correctly, the equation for such a surface should look something like:
x^2 + z^2 = 1 / y^2
On this curve your distance from the y-axis on the x-z plane would be equivalent to your distance frlom the radiation source, and the y-coord corresponding to each (x,z) would be the dose rate at that distance.

Think of it this way. At point (x,y,z) x and z are like latitude and longitude while y is the radiation dose rate.

This curve could be rotated around any line parallel to the y-axis, so the highest y reading would not always be at x-z point (0,0).

Here is what I am trying to accomplish:
If I have multiple (x,y,z) points where x and z are latitude and longitude (or just points on the x-z plane measured in feet) and y is the radiation rate, I want to be able to locate the highest y-reading on the x-z plane. In other words, if I know a radiation reading at two or three points and the latitude and longitude at those points, I want to be able to locate latitude and longitude of the radiation source.

Any ideas?

2. Nov 21, 2007

### EnumaElish

I am not sure whether you have one question or two questions:

1. "If I have multiple (x,y,z) points where x and z are latitude and longitude (or just points on the x-z plane measured in feet) and y is the radiation rate, I want to be able to locate the highest y-reading on the x-z plane."
2. "if I know a radiation reading at two or three points and the latitude and longitude at those points, I want to be able to locate latitude and longitude of the radiation source."

3. Nov 21, 2007

### natai

Actually, those questions are one and the same as the radiation source will be the highest reading.

4. Nov 21, 2007

### EnumaElish

I am interpreting your questions. Let's say you estimated a least squares regression of the form y = a + b x + c z + u where a, b, c are parameters and u is the random error. Now for any (x, z) pair, you can estimate the corresponding y. However, one of your expressions reads "I want to be able to locate latitude and longitude of the radiation source." Which makes me think that maybe you wish to go the other way around: from y to (x, z), which is not insolvable, but it is somewhat more complicated.

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