## A more general way to examine the nucleator and the circle

The nucleator is a local stereological probe.  Being a local probe means that the nucleator is based on the selection of a reference point. The probe, i.e. the geomtrical shape that is used to sample, is constructed relative to the reference point. A nucleator probe is a ray that starts at the reference points and projects outward. The nucleator is an unbiased probe because there is some essence of randomness to the sampling. That is accomplished by making the ray IUR. The abbreviation IUR stands for isotropic uniform random. That’s a fancy way of saying that all directions in space have the same chance of being chosen. On a flat surface a ray should have the same chance of choosing any of the directions from 0 to 360 degrees.

The measure of the nucleator is the positions along the ray where the ray crosses the boundaries of whatever is of interest. If the nucleator’s reference point is inside of the object of interest, and the ray goes out and never crosses the boundary again, then the measure is sort of like the length of the ray inside of the object.

In the case of the circle it is possible to come up with a formula that relates the circle and nucleator and the intercepts.

1. The circle is defined by its size which is its radius. That will be called r.
2. The nucleator is defined by its reference point and the orientation of the ray. These will be shown to be c and theta.

The first step is to work out the circle. The circle is defined to be centered at (0,0) and to have a radius r. A rather simple equation defines the point (x,y) that lie on the circle. Remember that a circle is the curve. A disc is the area inside of a circle.

The nucleator is a bit trickier. A circle is radially symmetric. No matter how it is rotated it is still the same. That allows us to simplify the choice of point inside of a circle. All points a distance c from the center of the circle are really the same. Therefore, all reference points inside of the circle a distance c from the center are essentially the same. Think about drawing a line through the reference point and the center of the circle. The line is a diameter of the circle. Use this line as the x-axis of a coordinate system.  For another point also a distance c from the center use a different coordinate system. The angle theta is the angle between the ray and the x-axis.

Knowing r, c, and theta it is possible to derive a formula that gives the intercept distance I. The mean of this formula is found as follows:

It should be fairly obvious that this is not related to the circumference of the circle. Actually this result gives the correct value if c=0. But what if I squared was used instead of I? The mean of I squared is:

This result, r squared, is in units-squared. Length requires units. The square root of this result is proportional to the circumference of the circle.

The question is: Does this work for shapes other than the circle?

The answer is seen to be a resounding NO. All stereologists already know that perimeter is related to another measure, not the mean intercept length or the mean intercept squared, or on any variation of that except for a few trivial cases.