The special case of the nucleator and the circle

Two plots of a circle: with 4-way nucleator and finding the center

The daisy fails, but do things work for a circle

The 4-way nucleator, when clicked inside of a convex profile creates 4 lengths – one for each ray of the nucleator. In this case we examine the circle, the simplest of shapes.

A circle is simple, because it is isotropic. Rotate the circle and nothing changes. The circle looks the same. That makes many questions about the circle easy to answer.

Here an arbitrary point is selected inside of the circle. Four rays start at the arbitrary point and intercept the circle. Because the circle is isotropic it is possible to simplify the math and use rays parallel to the coordinate axes. Also, without loss of generality let’s select the arbitrary point from the first quadrat.

These two simplifications mean that the lengths show in the drawings, a, b, c, and d, can be used to identify the 4 intercepts as (0, a), (b, 0), (0, -c), and (-d, 0). The axes cross at the arbitrarily chosen point. That point has coordinates (0,0). The intercepts on the y-axis are at a and -c. The x-axis intercepts are at -d and b.

So the question is whether or not the information given here is capable of estimating the circumference of the circle? Yes. Let’s see how that is done.

<insert math>

In general, the nucleator cannot be used to estimate perimeter. There is at least 1 case in which the nucleator does work. Are there other cases? The answer to that is unfortunately no. No shape other than a circle has a perimeter that can be estimated by the nucleator  without bias.


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2 Responses to “The special case of the nucleator and the circle”

  1. g Says:

    I am a scientist who uses stereology frequnetly in my projects. I was going to read your blog until I read the first sentence “I a often surprised at how wrong people get the history of stereology. Is it really that hard to get it right?” very very low credibility

    From: Rush Presbyterian-St. Lukes Medical Center

    • stereologist Says:

      Thanks for point out to me that I had a minor mistake in my blog. Seems that making such minor mistakes is common. Here you wrote “frequnetly”. Does that mean that your comment has “very very low credibility”?

      Thanks for the feedback and please take the time to offer feedback that has more substance in the future. Cheers.

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