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measuring wheel measures the distance PQ (perpendicular to EM). Moving from C to D the arm EM moves through the green parallelogram, with area equal to the area of the rectangle D"DCC". The measuring wheel now moves in the opposite direction, subtracting this reading from the former. The movements along BC and DA are the same but opposite, so they cancel each other with no net effect on the reading of the wheel. The net result is the measuring of the difference of the yellow and green areas, which is the area of ABCD.
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278:-axis. For the polar planimeter the "elbow" is connected to an arm with its other endpoint O at a fixed position. Connected to the arm ME is the measuring wheel with its axis of rotation parallel to ME. A movement of the arm ME can be decomposed into a movement perpendicular to ME, causing the wheel to rotate, and a movement parallel to ME, causing the wheel to skid, with no contribution to its reading.
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calibrated diameter, with a scale to show fine rotation, and worm gearing for an auxiliary turns counter scale. As the area outline is traced, this wheel rolls on the surface of the drawing. The operator sets the wheel, turns the counter to zero, and then traces the pointer around the perimeter of the shape. When the tracing is complete, the scales at the measuring wheel show the shape's area.
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883:{\displaystyle {\begin{aligned}&\oint _{C}(N_{x}\,dx+N_{y}\,dy)=\iint _{S}\left({\frac {\partial N_{y}}{\partial x}}-{\frac {\partial N_{x}}{\partial y}}\right)\,dx\,dy\\={}&\iint _{S}\left({\frac {\partial x}{\partial x}}-{\frac {\partial (b-y)}{\partial y}}\right)\,dx\,dy=\iint _{S}\,dx\,dy=A,\end{aligned}}}
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When the planimeter's measuring wheel moves perpendicular to its axis, it rolls, and this movement is recorded. When the measuring wheel moves parallel to its axis, the wheel skids without rolling, so this movement is ignored. That means the planimeter measures the distance that its measuring wheel
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The working of the linear planimeter may be explained by measuring the area of a rectangle ABCD (see image). Moving with the pointer from A to B the arm EM moves through the yellow parallelogram, with area equal to PQ×EM. This area is also equal to the area of the parallelogram A"ABB". The
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The Amsler (polar) type consists of a two-bar linkage. At the end of one link is a pointer, used to trace around the boundary of the shape to be measured. The other end of the linkage pivots freely on a weight that keeps it from moving. Near the junction of the two links is a measuring wheel of
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The polar planimeter is restricted by design to measuring areas within limits determined by its size and geometry. However, the linear type has no restriction in one dimension, because it can roll. Its wheels must not slip, because the movement must be constrained to a straight line.
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The images show the principles of a linear and a polar planimeter. The pointer M at one end of the planimeter follows the contour C of the surface S to be measured. For the linear planimeter the movement of the "elbow" E is restricted to the
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There are several kinds of planimeters, but all operate in a similar way. The precise way in which they are constructed varies, with the main types of mechanical planimeter being polar, linear, and Prytz or "hatchet" planimeters. The Swiss
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built the first modern planimeter in 1854, the concept having been pioneered by Johann Martin
Hermann in 1818. Many developments followed Amsler's famous planimeter, including electronic versions.
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travels, projected perpendicularly to the measuring wheel's axis of rotation. The area of the shape is proportional to the number of turns through which the measuring wheel rotates.
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as the rotation is proportional to the distance traveled, which at any point in time is proportional to radius and to change in angle, as in the circumference of a circle (
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The justification for the above derivation lies in noting that the linear planimeter only records movement perpendicular to its measuring arm, or when
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Napier
Tercentenary Celebration: Handbook of the Exhibition of Napier Relics and of Books, Instruments, and Devices for facilitating Calculation
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1371:{\displaystyle \int _{t}{\tfrac {1}{2}}(r(t))^{2}\,d(\theta (t))=\int _{t}{\tfrac {1}{2}}(r(t))^{2}\,{\dot {\theta }}(t)\,dt.}
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enclosed by the contour, is proportional to the distance measured by the measuring wheel, with proportionality factor
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meaning that the rate at which area changes with respect to change in angle varies quadratically with the radius.
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A linear planimeter. Wheels permit measurement of long areas without restriction.
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Gatterdam, R. W. (1981), "The planimeter as an example of Green's theorem",
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For a polar planimeter the total rotation of the wheel is proportional to
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A planimeter (1908) measuring the indicated area by tracing its perimeter
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Hodgson, John L. (1 April 1929), "Integration of flow meter diagrams",
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is non-zero. When this quantity is integrated over the closed curve C,
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The left hand side of the above equation, which is equal to the area
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The connection with Green's theorem can be understood in terms of
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The operation of a linear planimeter can be justified by applying
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Three planimeters: digital, Prytz's (hatchet) and Amsler's (polar)
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How Round is your Circle?: Where
Engineering and Mathematics Meet
478:{\displaystyle {\overrightarrow {EM}}\cdot N=xN_{x}+(y-b)N_{y}=0}
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Developments of the planimeter can establish the position of the
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can be recognized as the derivative of the earlier integrand
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This vector field is perpendicular to the measuring arm EM:
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Lowell, L. I. (1954), "Comments on the polar planimeter",
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1113:: in polar coordinates, area is computed by the integral
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1907:Photo: Geographers using planimeters (1940–1941)
1708:, Princeton University Press, pp. 138–171,
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77:Learn how and when to remove this message
1700:"Chapter 8: In pursuit of coat-hangers"
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201:Prytz planimeter with wheel at the left
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105:of an arbitrary two-dimensional shape.
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1698:Bryant, John; Sangwin, Chris (2007),
7:
1486:{\textstyle \int r\,d\theta =2\pi r}
1187:where the form being integrated is
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373:{\displaystyle \!\,N(x,y)=(b-y,x),}
1914:Green's Theorem and the Planimeter
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1832:, New York: Keuffel & Esser,
1800:The American Mathematical Monthly
1791:Modern Geometry with Applications
1752:Journal of Scientific Instruments
1724:The American Mathematical Monthly
1202:in polar coordinates, where both
1784:, The Royal Society of Edinburgh
1111:integration in polar coordinates
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1896:As the Planimeter’s Wheel Turns
1859:P. Kunkel: Whistleralley site,
1882:Computer model of a planimeter
1877:Robert Foote's planimeter page
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391:-coordinate of the elbow E.
238:Various types of planimeters
126:Various types of planimeters
309:onto the components of the
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1867:Larry's Planimeter Platter
1872:Wuerzburg Planimeter Page
1778:Horsburgh, E. M. (1914),
1772:10.1088/0950-7671/6/4/302
1912:O. Knill and D. Winter:
1902:Make a simple planimeter
1826:Wheatley, J. Y. (1908),
32:This article includes a
1935:Technical drawing tools
1930:Dimensional instruments
1891:planimeter explanations
1650:Mathematical instrument
301:Mathematical derivation
165:Amsler polar planimeter
61:more precise citations.
16:Tool for measuring area
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1764:1929JScI....6..116H
1200:parametric equation
119:Jakob Amsler-Laffon
1940:Mathematical tools
1854:Hatchet Planimeter
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1294:
1290:
1286:
1277:
1271:
1265:
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1248:
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1233:
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1209:
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1196:
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1190:
1174:
1171:
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1136:
1133:
1125:
1121:
1112:
1104:
1099:
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1080:
1075:
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848:
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837:
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830:
827:
822:
815:
804:
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798:
786:
780:
772:
762:
756:
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749:
742:
735:
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728:
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713:
703:
699:
689:
683:
673:
669:
658:
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629:
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313:N, given by:
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116:mathematician
108:
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81:
78:
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56:
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43:
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21:
20:
1913:
1895:
1860:
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1799:
1790:
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1188:
1108:
1010:
1005:
1001:
999:
892:
573:
489:
487:
393:
388:
384:
382:
311:vector field
304:
295:
275:
272:
220:
216:
212:
208:
112:
109:Construction
94:
90:
88:
73:
64:
53:Please help
45:
1887:Tanya Leise
59:introducing
1924:Categories
1793:, Springer
1666:References
1640:Curvimeter
1628:derivative
95:platometer
91:planimeter
1655:Integraph
1598:˙
1595:θ
1523:˙
1520:θ
1478:π
1469:θ
1459:∫
1417:˙
1414:θ
1389:∫
1344:˙
1341:θ
1291:∫
1272:θ
1219:∫
1189:quadratic
1172:θ
1152:θ
1126:θ
1122:∫
1025:⋅
964:−
943:∂
939:∂
925:−
910:∂
906:∂
893:because:
845:∬
813:∂
802:−
793:∂
787:−
778:∂
770:∂
753:∬
711:∂
696:∂
690:−
681:∂
666:∂
649:∬
593:∮
525:−
511:‖
505:‖
451:−
420:⋅
415:→
353:−
282:Principle
67:June 2022
1634:See also
1820:2308082
1760:Bibcode
1744:2320679
1691:Sources
387:is the
97:, is a
55:improve
1836:
1818:
1742:
1712:
1198:For a
574:Then:
383:where
1816:JSTOR
1740:JSTOR
40:, or
1950:Area
1893:and
1834:ISBN
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1206:and
103:area
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1028:(
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1006:m
1002:A
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934:=
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816:y
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781:x
773:x
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733:d
729:x
726:d
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714:y
704:x
700:N
684:x
674:y
670:N
659:(
653:S
645:=
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639:y
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630:y
626:N
622:+
619:x
616:d
610:x
606:N
602:(
597:C
559:m
556:=
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519:(
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508:N
490:m
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411:M
408:E
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356:y
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347:(
344:=
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329:(
326:N
276:y
225:(
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74:(
69:)
65:(
51:.
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