48:, consisting of a transparent sheet containing a square grid of dots. To estimate the area of a shape, the sheet is overlaid on the shape and the dots within the shape are counted. The estimate of area is the number of dots counted multiplied by the area of a single grid square. In some variations, dots that land on or near the boundary of the shape are counted as half of a unit. The dots may also be grouped into larger square groups by lines drawn onto the transparency, allowing groups that are entirely within the shape to be added to the count rather than requiring their dots to be counted one by one.
20:
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Greater accuracy can be achieved by using a dot planimeter with a finer grid of dots. Alternatively, repeatedly placing a dot planimeter with different irrational offsets from its previous placement, and averaging the resulting measurements, can lead to a set of sampled measurements whose average
23:
A radius-5 circle overlaid with a grid of dots in the pattern of a dot planimeter. When counting dots near the boundary of the shape as 1/2, there are 69 interior dots and 20 boundary dots for an estimated area of 79, close to the actual area of
105:, a similar technique of counting dots in a grid is applied to cross-sections of rock samples for a different purpose, estimating the relative proportions of different constituent minerals.
145:
in 1914, it is always possible to shift a dot planimeter relative to a given shape without rotating it so that the number of dots within the shape is at least equal to its area.
584:
443:
Dolph, Gary E. (July–September 1977), "The effect of different calculational techniques on the estimation of leaf area and the construction of leaf size distributions",
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417:
Benjamin, D. M.; Freeman, G. H.; Brown, E. S. (February 1968), "The determination of irregularly-shaped areas of leaves destroyed by chewing insects",
129:
in 1899, the version of the dot planimeter with boundary dots counting as 1/2 (and with an added correction term of −1) gives exact results for
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160:. The maximum error is known to be bounded by a fractional power of the radius of the circle, with exponent between 1/2 and 131/208.
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645:, Anneli Lax New Mathematical Library, vol. 41, Mathematical Association of America, Washington, DC, pp. 119–127,
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Frolov, Y. S.; Maling, D. H. (June 1969), "The accuracy of area measurement by point counting techniques",
680:, Problem Books in Mathematics, vol. 1 (3rd ed.), New York: Springer-Verlag, pp. 365–367,
391:
Heinicke, Don R. (October 1963), "Note on estimation of leaf area and leaf distribution in fruit trees",
741:
537:
Sitzungsberichte des deutschen naturwissenschaftlich-medicinischen
Vereines für Böhmen "Lotos" in Prag
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tends towards the true area of the measured shape. The method using a finer grid tends to have better
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is a similar transparency-based device for estimating the length of curves by counting crossings.
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concerns the error that would be obtained by using a dot planimeter to estimate the area of a
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79:, the dot planimeter has been applied to maps to estimate the area of parcels of land. In
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Thomas, Roger K.; Peacock, L. J. (January 1965), "A method of measuring brain lesions",
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Wood, Walter F. (January 1954), "The dot planimeter, a new way to measure map area",
87:, it has been applied directly to sampled leaves to estimate the average leaf area.
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582:(1914), "A new principle in the geometry of numbers, with some applications",
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172:, which measure the area of a shape by passing a device around its boundary.
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or (particularly when the alignment of the grid with the shape is random)
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Bellhouse, D. R. (1981), "Area estimation by point-counting techniques",
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59:. Perhaps because of its simplicity, it has been repeatedly reinvented.
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The estimation of area by means of a dot grid has also been called the
307:"A method of estimating area in irregularly shaped and broken figures"
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Crommer, D. A. N. (January 1949), "Extracting small irregular areas",
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156:. As its name suggests, it was studied in the early 19th century by
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639:(2000), "Chapter 9: A new principle in the geometry of numbers",
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41:
566:
The
Penguin Dictionary of Curious and Interesting Geometry
118:than repeated measurement with random placements.
585:Transactions of the American Mathematical Society
505:Neilson, M. J.; Brockman, G. F. (December 1977),
168:The dot planimeter differs from other types of
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676:(2004), "F1: Gauß's lattice point problem",
399:(4), Canadian Science Publishing: 597–598,
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507:"The error associated with point-counting"
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564:Wells, David (1991), "Pick's theorem",
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744:, Chris Staecker, Fairfield University
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445:Bulletin of the Torrey Botanical Club
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90:In medicine, it has been applied to
678:Unsolved Problems in Number Theory
431:10.1111/j.1744-7348.1968.tb04505.x
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599:10.1090/S0002-9947-1914-1500976-6
568:, Penguin Books, pp. 183–184
393:Canadian Journal of Plant Science
350:10.1111/j.0033-0124.1954.61_12.x
533:"Geometrisches zur Zahlenlehre"
285:Przegląd Matematyczno-Fizyczny
213:10.1080/00049158.1949.10675768
94:as an estimate of the size of
1:
539:, (Neue Folge) (in German),
133:that have the dots as their
338:The Professional Geographer
16:Device used in planimetrics
794:
278:"O mierzeniu pól płaskich"
686:10.1007/978-0-387-26677-0
419:Annals of Applied Biology
143:Hans Frederick Blichfeldt
367:The Cartographic Journal
763:Dimensional instruments
721:Czasopismo Geograficzne
642:The Geometry of Numbers
379:10.1179/caj.1969.6.1.21
116:statistical efficiency
29:
778:Measuring instruments
719:(1931), "Longimetr",
637:Davidoff, Giuliana P.
511:American Mineralogist
305:Abell, C. A. (1939),
22:
177:Steinhaus longimeter
158:Carl Friedrich Gauss
150:Gauss circle problem
139:Blichfeldt's theorem
127:Georg Alexander Pick
36:is a device used in
479:Psychonomic Science
314:Journal of Forestry
201:Australian Forestry
57:systematic sampling
40:for estimating the
773:Mathematical tools
517:(11–12): 1238–1244
492:10.3758/bf03343085
405:10.4141/cjps63-117
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633:Lax, Anneli
629:Olds, C. D.
543:: 311–319,
529:Pick, Georg
73:cartography
63:Application
752:Categories
549:33.0216.01
230:Biometrics
183:References
170:planimeter
103:mineralogy
320:: 344–345
77:geography
531:(1899),
276:(1924),
135:vertices
131:polygons
69:forestry
28:≈ 78.54.
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661:1817689
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608:1988585
465:2484308
258:0673040
250:2530419
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109:Theory
81:botany
75:, and
727:: 1–4
604:JSTOR
461:JSTOR
310:(PDF)
281:(PDF)
246:JSTOR
46:shape
44:of a
758:Area
690:ISBN
647:ISBN
175:The
148:The
83:and
42:area
682:doi
594:doi
545:JFM
487:doi
453:doi
449:104
427:doi
401:doi
375:doi
346:doi
238:doi
209:doi
101:In
67:In
754::
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