Contouring Structural Data: Kalsbeek Net

Steven Dutch, Professor Emeritus, Natural and Applied Sciences, University of Wisconsin - Green Bay

	If we divide a circle into ten zones of equal width, the innermost circle will contain 1% of the area. The next circle is twice as large and will contain 4%, but 1% is in the inner circle, so the annulus will contain 3% of the area, and so on. If we stack triangles, each row will contain 1, 3, 5... triangles. A stack ten rows high will contain 100 triangles. If we divide a 60 degree sector of the circle into triangles of equal area, each sector will contain 100 triangles, each with 1% of the area of the sector.
	The Kalsbeek counting net is based on this principle. It consists of ten equally spaced circles. Each annulus is divided into triangles. Altogether there are 600 triangles. At each vertex, six triangles meet. The hexagon of triangles around each vertex contains 1% of the area of the net.
	Plot the data on an equal area net then transfer the overlay to the counting net. Of course, the two nets must be the same diameter!
	At each vertex, count the number of points in the surrounding six triangles and plot the number at the vertex. You may want to do this on a second overlay above the data overlay. Each triangle is common to three hexagons so every point is counted three times. (No, this does not mean the densities have to be divided by three.) Be certain to check every vertex close to the data points to be sure of not missing any.
	Remove the numbered overlay and contour the data.

Advantages

Fast and standard
The counting results will be the same for any worker (apart from judgment calls when points are on the edges of triangles)

Disadvantages

Contouring is somewhat approximate.
Results may vary depending on orientation of overlay
Chance location or orientation of the overlay may affect results. A cluster of points that spreads across several hexagons may not give densities as high as the overlapping or moving circle methods.

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Created 16 June 2005, Last Update 12 June 2020