Planimeter types and digital planimeter
Planimeter types and digital planimeter:
A planimeter, also known as a platometer, is a measuring instrument used to determine the area of an arbitrary two-dimensional shape.
There are several kinds of planimeters, but all operate in a similar way. There are Three types of planimeters available however all operate in a similar way. They are:
1.Prytz hatchet planimeter,
2.linear rolling planimeter
3.Amsler (polar) planimeter and compensating planimeter.
The Swiss mathematician Jakob Amsler-Laffon built the first modern planimeter in 1854, the concept having been pioneered by Johann Martin Hermann in 1814. Many developments followed Amsler's famous planimeter, including electronic versions.
The first planimeter (Prytz) is incredibly simple, but it provides only approximation of the total area. The other two of the planimeters are based on Green's theorem which relates line integral of second kind with double integral (the vector field and the domains of integration are supposed to be sufficiently regular).
Prytz hatchet planimeter
Incredibly simple device. Very rare! Hard to find or buy this planimeter. Perhaps many people do not recognize it as a scientific instrument (looks like strange ruler). Invented about 1894 by Captain Holger Prytz for Danish army.
Prytz planimeter:
Prytz planimeter gives only approximation of the total area, since it includes some principal error. To measure the area with minimal principal error first estimate the position of the center of mass. Then start with the pointer at the center of mass, follow the line to boundary, go around the boundary and using the same line return from the boundary to the center of mass. The second end is displaced by this procedure and the area is approximately equal to the product of the displacement and the length of the planimeter. For the planimeter on the picture, the distance is 20cm, therefore the planimeter should not be used to measure areas with diameter bigger than 10cm and the area is a product of the displacement (in cm) multiplied by 20. The result is in square cm.
Linear (rolling) planimeter:
The arm with a perpendicular wheel and tracing point is attached to a cylinder, this cylinder moves along a line.
There is an excellent mathematical description by Tanya Leise which explains in details how does the linear planimeter work. Note that in that description there is not written the explicit formula for the direction field which is being integrated by the linear planimeter. This is caused by the fact that as one intermediate step there is shown that some components of the motion have in total zero influence to the final value and it is possible to neglect them
Amsler (polar) planimeter:
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 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 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.
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 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.
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.
The working of the linear planimeter may be explained by measuring the area of a rectangle ABCD. 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 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 Between the yellow and green areas, which is the area of ABCD.
Digital Planimeter:
Placom Digital Planimeter KP-90N-Series with the most advanced functions and perfect reliability.
The most standard model of “Koizuimi Placom” N-Series has made the excitingly highest attainment in the history of Placom Digital Planimeter KP-90N ever made. Measuring capacity is remarkably enlarged. Measuring by 6-digit pulse count enables to measure 100 times larger accumulative area than ordinary planimeters. Easy calculations of Cumulative and Average Value Measurement.
Applications
· Construction and civil engineering
· Land developing planning, Machine designing
· Data analysis in statistics
· Architecture and interior designing
· Forestry area, Ship designing
· Medical research, Agriculture
Product Description
Key Features of Placom KP-90N Digital Planimeter
Measurement by 6 digit pulse count
This N-Series can measure the area by pulse count of 6-digits. This method enables a larger cumulative measurement of area (Up to 10m²)
Maximum cumulative measurement value of 10m²
With the measurement by 6-digit pulse count, the N-Series Planimeter can measure a larger area by 100 times than ordinary Planimeter.
The conventional planimeter has a maximum cumulative area to measure only 0.1m². If measurement value shows over 0.1m2 when measuring, the display is to be reset to “O”.
This requires to register the numbers of how many times these re-settings have occurred.
With the New PLACOM N-Series, there is no need of these troublesome procedures.
Easy conversion function of unit and scale
Placom KP-90N can convert easily the measured area value into a new area value in a newly desired unit and with a newly desired scale value.
Placom KP-92N and KP-82N can convert the unit between in² and cm²
Hold memory function
In cumulative measurement, this Hold Memory function is effectively used. If an error occurs in measuring, press C/AC key. Then last measured value is appeared by pulse count. Since previous measured values before error are stored, there is no need to another measurement from the beginning.
Automatic shifting of unit to upper unit
When the measured area value overflows the displayed figures of 8 digits, the measured area value is calculated and displayed by an upper unit. When the overflows is occurred in the uppermost unit, km², acre, the measurement can be continued by pulse count display.
Average value measurement
In order to get better result of measurement, Mean value measurement is recommended. This measurement can be easily done on this N-Series.
Measurement of an area with different lateral and longitudinal Scale.
Auto-power-off function for energy saving.
A planimeter, also known as a platometer, is a measuring instrument used to determine the area of an arbitrary two-dimensional shape.
There are several kinds of planimeters, but all operate in a similar way. There are Three types of planimeters available however all operate in a similar way. They are:
1.Prytz hatchet planimeter,
2.linear rolling planimeter
3.Amsler (polar) planimeter and compensating planimeter.
The Swiss mathematician Jakob Amsler-Laffon built the first modern planimeter in 1854, the concept having been pioneered by Johann Martin Hermann in 1814. Many developments followed Amsler's famous planimeter, including electronic versions.
The first planimeter (Prytz) is incredibly simple, but it provides only approximation of the total area. The other two of the planimeters are based on Green's theorem which relates line integral of second kind with double integral (the vector field and the domains of integration are supposed to be sufficiently regular).
Prytz hatchet planimeter
Incredibly simple device. Very rare! Hard to find or buy this planimeter. Perhaps many people do not recognize it as a scientific instrument (looks like strange ruler). Invented about 1894 by Captain Holger Prytz for Danish army.
Prytz planimeter:
Prytz planimeter gives only approximation of the total area, since it includes some principal error. To measure the area with minimal principal error first estimate the position of the center of mass. Then start with the pointer at the center of mass, follow the line to boundary, go around the boundary and using the same line return from the boundary to the center of mass. The second end is displaced by this procedure and the area is approximately equal to the product of the displacement and the length of the planimeter. For the planimeter on the picture, the distance is 20cm, therefore the planimeter should not be used to measure areas with diameter bigger than 10cm and the area is a product of the displacement (in cm) multiplied by 20. The result is in square cm.
Linear (rolling) planimeter:
The arm with a perpendicular wheel and tracing point is attached to a cylinder, this cylinder moves along a line.
There is an excellent mathematical description by Tanya Leise which explains in details how does the linear planimeter work. Note that in that description there is not written the explicit formula for the direction field which is being integrated by the linear planimeter. This is caused by the fact that as one intermediate step there is shown that some components of the motion have in total zero influence to the final value and it is possible to neglect them
Amsler (polar) planimeter:
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 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 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.
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 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.
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.
The working of the linear planimeter may be explained by measuring the area of a rectangle ABCD. 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 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 Between the yellow and green areas, which is the area of ABCD.
Digital Planimeter:
Placom Digital Planimeter KP-90N-Series with the most advanced functions and perfect reliability.
The most standard model of “Koizuimi Placom” N-Series has made the excitingly highest attainment in the history of Placom Digital Planimeter KP-90N ever made. Measuring capacity is remarkably enlarged. Measuring by 6-digit pulse count enables to measure 100 times larger accumulative area than ordinary planimeters. Easy calculations of Cumulative and Average Value Measurement.
Applications
· Construction and civil engineering
· Land developing planning, Machine designing
· Data analysis in statistics
· Architecture and interior designing
· Forestry area, Ship designing
· Medical research, Agriculture
Product Description
Key Features of Placom KP-90N Digital Planimeter
Measurement by 6 digit pulse count
This N-Series can measure the area by pulse count of 6-digits. This method enables a larger cumulative measurement of area (Up to 10m²)
Maximum cumulative measurement value of 10m²
With the measurement by 6-digit pulse count, the N-Series Planimeter can measure a larger area by 100 times than ordinary Planimeter.
The conventional planimeter has a maximum cumulative area to measure only 0.1m². If measurement value shows over 0.1m2 when measuring, the display is to be reset to “O”.
This requires to register the numbers of how many times these re-settings have occurred.
With the New PLACOM N-Series, there is no need of these troublesome procedures.
Easy conversion function of unit and scale
Placom KP-90N can convert easily the measured area value into a new area value in a newly desired unit and with a newly desired scale value.
Placom KP-92N and KP-82N can convert the unit between in² and cm²
Hold memory function
In cumulative measurement, this Hold Memory function is effectively used. If an error occurs in measuring, press C/AC key. Then last measured value is appeared by pulse count. Since previous measured values before error are stored, there is no need to another measurement from the beginning.
Automatic shifting of unit to upper unit
When the measured area value overflows the displayed figures of 8 digits, the measured area value is calculated and displayed by an upper unit. When the overflows is occurred in the uppermost unit, km², acre, the measurement can be continued by pulse count display.
Average value measurement
In order to get better result of measurement, Mean value measurement is recommended. This measurement can be easily done on this N-Series.
Measurement of an area with different lateral and longitudinal Scale.
Auto-power-off function for energy saving.
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