A dot planimeter is a device used in

planimetrics Planimetrics is the study of plane measurements, including angles, distances, and areas. History To measure planimetrics a planimeter or dot planimeter is used. This rather advanced analog technology is being taken over by simple image measu ...

for estimating the

area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...

of a

shape A shape is a graphics, graphical representation of an object's form or its external boundary, outline, or external Surface (mathematics), surface. It is distinct from other object properties, such as color, Surface texture, texture, or material ...

, 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. The estimation of area by means of a dot grid has also been called the dot grid method or (particularly when the alignment of the grid with the shape is random) systematic sampling. Perhaps because of its simplicity, it has been repeatedly reinvented.

Application

forestry Forestry is the science and craft of creating, managing, planting, using, conserving and repairing forests and woodlands for associated resources for human and Natural environment, environmental benefits. Forestry is practiced in plantations and ...

cartography Cartography (; from , 'papyrus, sheet of paper, map'; and , 'write') is the study and practice of making and using maps. Combining science, aesthetics and technique, cartography builds on the premise that reality (or an imagined reality) can ...

, and

geography Geography (from Ancient Greek ; combining 'Earth' and 'write', literally 'Earth writing') is the study of the lands, features, inhabitants, and phenomena of Earth. Geography is an all-encompassing discipline that seeks an understanding o ...

, the dot planimeter has been applied to maps to estimate the area of parcels of land. In

botany Botany, also called plant science, is the branch of natural science and biology studying plants, especially Plant anatomy, their anatomy, Plant taxonomy, taxonomy, and Plant ecology, ecology. A botanist or plant scientist is a scientist who s ...

and

horticulture Horticulture (from ) is the art and science of growing fruits, vegetables, flowers, trees, shrubs and ornamental plants. Horticulture is commonly associated with the more professional and technical aspects of plant cultivation on a smaller and mo ...

, it has been applied directly to sampled leaves to estimate the average leaf area. In medicine, it has been applied to Lashley diagrams as an estimate of the size of

brain lesion Brain injury (BI) is the destruction or degeneration of brain cells. Brain injuries occur due to a wide range of internal and external factors. In general, brain damage refers to significant, undiscriminating trauma-induced damage. A common ...

s. In

mineralogy Mineralogy is a subject of geology specializing in the scientific study of the chemistry, crystal structure, and physical (including optical mineralogy, optical) properties of minerals and mineralized artifact (archaeology), artifacts. Specific s ...

, 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.

Theory

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 tends towards the true area of the measured shape. The method using a finer grid tends to have better

statistical efficiency In statistics, efficiency is a measure of quality of an estimator, of an experimental design, or of a hypothesis testing procedure. Essentially, a more efficient estimator needs fewer input data or observations than a less efficient one to achiev ...

than repeated measurement with random placements. According to

Pick's theorem In geometry, Pick's theorem provides a formula for the area of a simple polygon with integer vertex coordinates, in terms of the number of integer points within it and on its boundary. The result was first described by Georg Alexander Pick in 1 ...

, published by

Georg Alexander Pick Georg Alexander Pick (10 August 1859 – 26 July 1942) was an Austrian Jewish mathematician who was murdered during The Holocaust. He was born in Vienna to Josefa Schleisinger and Adolf Josef Pick and died at Theresienstadt concentration camp. T ...

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

polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...

s that have the dots as their vertices. According to Blichfeldt's theorem, published by Hans Frederick Blichfeldt 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. The

Gauss circle problem In mathematics, the Gauss circle problem is the problem of determining how many integer lattice points there are in a circle centered at the origin and with radius r. This number is approximated by the area of the circle, so the real problem is t ...

concerns the error that would be obtained by using a dot planimeter to estimate the area of a

circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...

. As its name suggests, it was studied in the early 19th century by

Carl Friedrich Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and ...

. 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.

Related devices

The dot planimeter differs from other types of

planimeter A planimeter, also known as a platometer, is a measuring instrument used to determine the area of an arbitrary two-dimensional shape. Construction There are several kinds of planimeters, but all operate in a similar way. The precise way in whic ...

, which measure the area of a shape by passing a device around its boundary. The

Steinhaus longimeter The Steinhaus longimeter, patented by the professor Hugo Steinhaus, is an instrument used to measure the arc length, lengths of curves on maps. Description It is a transparent sheet of three grids, turned against each other by 30 degrees, each ...

is a similar transparency-based device for estimating the length of curves by counting crossings.

References

{{reflist, refs= {{citation , last = Abell , first = C. A. , journal = Journal of Forestry , pages = 344–345 , title = A method of estimating area in irregularly shaped and broken figures , url = https://cstaecker.fairfield.edu/~cstaecker/files/machines/filer.php?name=dotplanpaperabell.pdf , volume = 37 , year = 1939 {{citation , last = Bellhouse , first = D. R. , doi = 10.2307/2530419 , issue = 2 , journal = Biometrics , jstor = 2530419 , mr = 673040 , pages = 303–312 , title = Area estimation by point-counting techniques , volume = 37 , year = 1981 {{citation , last = Blichfeldt , first = H. F. , author-link = Hans Frederick Blichfeldt , doi = 10.1090/S0002-9947-1914-1500976-6 , doi-access = free , jstor = 1988585 , issue = 3 , journal =

Transactions of the American Mathematical Society The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of pure and applied mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must ...

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CiteBank:47270
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External links

Chris Staecker, Fairfield University Area Dimensional instruments Lattice points Mathematical tools Measuring instruments