Homepage of P. Christopher Staecker: Dot planimeter

The dot planimeter is a simple device for measuring the area of a shape drawn on paper.

It's a grid of dots on transparency that is laid over the shape. The number of dots inside the shape tells the area of the shape.

About the dot planimeter

Here's a short video I made about the dot planimeter.

This is part of my series of videos about antique computational devices. Here is the full series playlist

Download

Here are some PDFs of dot planimeters which are meant to be printed on 8½ x 11 transparency paper. If you print it on a different size sheet, be sure not to scale it! The dots are meant to appear at specific distances.

Simple dot planimeter 1cm scale	Fancy dot planimeter 1cm scale
Simple dot planimeter Tenth-cm scale	Fancy dot planimeter Tenth-cm scale

These PDFs were generated by tikz in this LaTeX file:

• dotplan.tex

Academic papers about the dot planimeter

I found these papers very hard to locate, so I'm posting them here. These are obscure old papers, but some of them are still under copyright- I hope the publishers won't mind me reposting them.

• Steinhaus (1924) O mierzeniu pol plaskich (Polish), Przeglad Mat-Fiz. 2 p. 24-29, (1924)
  This is the earliest paper I can find describing the dot planimeter, which proves an interesting theorem about its accuracy: if the dots are placed and counted repeatedly, each time shifting the grid by an irrational vector, then the average of the readings will tend in the limit to the exact area of the shape.
  
• Abell (1939) A method of estimating area in irregularly shaped and broken figures, Journal of Forestry 37 p. 344-345 (1939)
  This is the earliest paper typically cited in the literature. Abell seems to have re-invented the concept with no knowledge of the description by Steinhaus. He gives some empirical measurements of accuracy, but no theoretical treatment.
• Wood (1954) The dot planimeter, a new way to measure map area The Professional Geographer 6 p. 12-14
  doi: 10.1111/j.0033-0124.1954.61_12.x
  Another simple description of the dot planimeter, this time suggesting the filled-in & empty dots for counting the edge cases.
• Heinicke (1963) Note on the distribution of leaf area and leaf distribution in fruit trees Canadian journal of plant science 43 p. 597-598
  Very simple description of the dot planimeter, with a nice photo of one in use.
• Frolov, Maling (1969) The accuracy of area measurement by point counting techniques, The Cartographic Journal 6 p. 21-35
  doi: 10.1179/caj.1969.6.1.21
  A fairly sophisticated treatment that includes both empirical and theoretical measures of accuracy. They seem unaware of Steinhaus's theorem, which would have fit nicely into their discussion.

Other stuff

• Other pages about antique calculating devices